What is prepositional logic
https://discrete.openmathbooks.org/dmoi2/sec_propositional.html
Propositional Logic
1995-01-01
Oscar Levin
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<main><section><h2>
</h2>
<a href="https://discrete.openmathbooks.org/dmoi2/sec_propositional.html">¶</a><section><article><h6>
</h6>
<p>You stumble upon two trolls playing Stratego®. They tell you:</p>
<blockquote>
<p>Troll 1: If we are cousins, then we are both knaves.</p>
<p>Troll 2: We are cousins or we are both knaves.</p>
</blockquote>
<p>Could both trolls be knights? Recall that all trolls are either always-truth-telling knights or always-lying knaves.</p></article><p>A <dfn>proposition</dfn> is simply a statement. <dfn>Propositional logic</dfn> studies the ways statements can interact with each other. It is important to remember that propositional logic does not really care about the content of the statements. For example, in terms of propositional logic, the claims, “if the moon is made of cheese then basketballs are round,” and “if spiders have eight legs then Sam walks with a limp” are exactly the same. They are both implications: statements of the form, \(P \imp Q\text{.}\)</p></section><section><h3>
</h3>
<a href="https://discrete.openmathbooks.org/dmoi2/sec_propositional.html#subsection-27">¶</a><p>Here's a question about playing Monopoly:</p>
<blockquote><p>If you get more doubles than any other player then you will lose, or if you lose then you must have bought the most properties.</p></blockquote>
<p>True or false? We will answer this question, and won't need to know anything about Monopoly. Instead we will look at the logical <em>form</em> of the statement.</p>
<p>We need to decide when the statement \((P \imp Q) \vee (Q \imp R)\) is true. Using the definitions of the connectives in <a href="https://discrete.openmathbooks.org/dmoi2/sec_intro-statements.html">Section 0.2</a>, we see that for this to be true, either \(P \imp Q\) must be true or \(Q \imp R\) must be true (or both). Those are true if either \(P\) is false or \(Q\) is true (in the first case) and \(Q\) is false or \(R\) is true (in the second case). So—yeah, it gets kind of messy. Luckily, we can make a chart to keep track of all the possibilities. Enter <dfn>truth tables</dfn>. The idea is this: on each row, we list a possible combination of T's and F's (for true and false) for each of the sentential variables, and then mark down whether the statement in question is true or false in that case. We do this for every possible combination of T's and F's. Then we can clearly see in which cases the statement is true or false. For complicated statements, we will first fill in values for each part of the statement, as a way of breaking up our task into smaller, more manageable pieces.</p>
<p>Since the truth value of a statement is completely determined by the truth values of its parts and how they are connected, all you really need to know is the truth tables for each of the logical connectives. Here they are:</p>
<div>
<div><table>
<tbody><tr>
<td>\(P\)</td>
<td>\(Q\)</td>
<td>\(P\wedge Q\)</td>
</tr>
<tr>
<td>T</td>
<td>T</td>
<td>T</td>
</tr>
<tr>
<td>T</td>
<td>F</td>
<td>F</td>
</tr>
<tr>
<td>F</td>
<td>T</td>
<td>F</td>
</tr>
<tr>
<td>F</td>
<td>F</td>
<td>F</td>
</tr>
</tbody></table></div>
<div><table>
<tbody><tr>
<td>\(P\)</td>
<td>\(Q\)</td>
<td>\(P\vee Q\)</td>
</tr>
<tr>
<td>T</td>
<td>T</td>
<td>T</td>
</tr>
<tr>
<td>T</td>
<td>F</td>
<td>T</td>
</tr>
<tr>
<td>F</td>
<td>T</td>
<td>T</td>
</tr>
<tr>
<td>F</td>
<td>F</td>
<td>F</td>
</tr>
</tbody></table></div>
<div><table>
<tbody><tr>
<td>\(P\)</td>
<td>\(Q\)</td>
<td>\(P\imp Q\)</td>
</tr>
<tr>
<td>T</td>
<td>T</td>
<td>T</td>
</tr>
<tr>
<td>T</td>
<td>F</td>
<td>F</td>
</tr>
<tr>
<td>F</td>
<td>T</td>
<td>T</td>
</tr>
<tr>
<td>F</td>
<td>F</td>
<td>T</td>
</tr>
</tbody></table></div>
<div><table>
<tbody><tr>
<td>\(P\)</td>
<td>\(Q\)</td>
<td>\(P\iff Q\)</td>
</tr>
<tr>
<td>T</td>
<td>T</td>
<td>T</td>
</tr>
<tr>
<td>T</td>
<td>F</td>
<td>F</td>
</tr>
<tr>
<td>F</td>
<td>T</td>
<td>F</td>
</tr>
<tr>
<td>F</td>
<td>F</td>
<td>T</td>
</tr>
</tbody></table></div>
</div>
<p>The truth table for negation looks like this:</p>
<p>None of these truth tables should come as a surprise; they are all just restating the definitions of the connectives. Let's try another one.</p>
<article><h6>
</h6>
<p>Make a truth table for the statement \(\neg P \vee Q\text{.}\)</p>
<div>
<p></p><p></p>
</div></article><p>Now let's answer our question about monopoly:</p>
<article><h6>
</h6>
<p>Analyze the statement, “if you get more doubles than any other player you will lose, or that if you lose you must have bought the most properties,” using truth tables.</p>
<div>
<p></p><p></p>
</div></article><p>The statement about monopoly is an example of a <dfn>tautology</dfn>, a statement which is true on the basis of its logical form alone. Tautologies are always true but they don't tell us much about the world. No knowledge about monopoly was required to determine that the statement was true. In fact, it is equally true that “If the moon is made of cheese, then Elvis is still alive, or if Elvis is still alive, then unicorns have 5 legs.”</p></section><section><h3>
</h3>
<a href="https://discrete.openmathbooks.org/dmoi2/sec_propositional.html#subsection-28">¶</a><p>You might have noticed that the final column in the truth table from \(\neg P \vee Q\) is identical to the final column in the truth table for \(P \imp Q\text{:}\)</p>
<div><table>
<tbody><tr>
<td>\(P\)</td>
<td>\(Q\)</td>
<td>\(P \imp Q\)</td>
<td>\(\neg P \vee Q\)</td>
</tr>
<tr>
<td>T</td>
<td>T</td>
<td>T</td>
<td>T</td>
</tr>
<tr>
<td>T</td>
<td>F</td>
<td>F</td>
<td>F</td>
</tr>
<tr>
<td>F</td>
<td>T</td>
<td>T</td>
<td>T</td>
</tr>
<tr>
<td>F</td>
<td>F</td>
<td>T</td>
<td>T</td>
</tr>
</tbody></table></div>
<p>This says that no matter what \(P\) and \(Q\) are, the statements \(\neg P \vee Q\) and \(P \imp Q\) either both true or both false. We therefore say these statements are <dfn>logically equivalent</dfn>.</p>
<article><h6></h6>
<p>Two (molecular) statements \(P\) and \(Q\) are <dfn>logically equivalent</dfn> provided \(P\) is true precisely when \(Q\) is true. That is, \(P\) and \(Q\) have the same truth value under any assignment of truth values to their atomic parts.</p>
<p>To verify that two statements are logically equivalent, you can make a truth table for each and check whether the columns for the two statements are identical.</p></article><p>Recognizing two statements as logically equivalent can be very helpful. Rephrasing a mathematical statement can often lends insight into what it is saying, or how to prove or refute it. Using truth tables we can systematically verify that two statements are indeed logically equivalent.</p>
<article><h6>
</h6>
<p>Are the statements, “it will not rain or snow” and “it will not rain and it will not snow” logically equivalent?</p>
<div>
<p></p><p></p>
</div></article><p>Notice that this example gives us a way to “distribute” a negation over a disjunction (an “or”). We have a similar rule for distributing over conjunctions (“and”s):</p>
<article><h6></h6>
\begin{equation*}
\neg(P \wedge Q) \text{ is logically equivalent to } \neg P \vee \neg Q.
\end{equation*}
\begin{equation*}
\neg(P \vee Q) \text{ is logically equivalent to } \neg P \wedge \neg Q.
\end{equation*}
</article><p>This suggests there might be a sort of “algebra” you could apply to statements (okay, there is: it is called <em>Boolean algebra</em>) to transform one statement into another. We can start collecting useful examples of logical equivalence, and apply them in succession to a statement, instead of writing out a complicated truth table. We will probably also want a way to deal with double negation:</p>
<article><h6></h6>
\begin{equation*}
\neg \neg P \mbox{ is logically equivalent to } P.
\end{equation*}
<p>Example: “It is not the case that \(c\) is not odd” means “\(c\) is odd.”</p></article><p>Let's see how we can apply the equivalences we have encountered so far.</p>
<article><h6>
</h6>
<p>Prove that the statements \(\neg(P \imp Q)\) and \(P\wedge \neg Q\) are logically equivalent without using truth tables.</p>
<div>
<p></p><p></p>
</div></article><p>Notice that the above example illustrates that the negation of an implication is NOT an implication: it is a conjunction!</p>
<p>To verify that two statements are logically equivalent, you can use truth tables or a sequence of logically equivalent replacements. The truth table method, although cumbersome, has the advantage that it can verify that two statements are NOT logically equivalent.</p>
<article><h6>
</h6>
<p>Are the statements \((P \vee Q) \imp R\) and \((P \imp R) \vee (Q \imp R)\) logically equivalent?</p>
<div>
<p></p><p></p>
</div></article></section><section><h3>
</h3>
<a href="https://discrete.openmathbooks.org/dmoi2/sec_propositional.html#subsection-29">¶</a><article><h6>
</h6>
<p>Holmes owns two suits: one black and one tweed. He always wears either a tweed suit or sandals. Whenever he wears his tweed suit and a purple shirt, he chooses to not wear a tie. He never wears the tweed suit unless he is also wearing either a purple shirt or sandals. Whenever he wears sandals, he also wears a purple shirt. Yesterday, Holmes wore a bow tie. What else did he wear?</p></article><p>Earlier we claimed that the following was a valid argument:</p>
<blockquote><p>If Edith eats her vegetables, then she can have a cookie. Edith ate her vegetables. Therefore Edith gets a cookie.</p></blockquote>
<p>How do we know this is valid? Let's look at the form of the statements. Let \(P\) denote “Edith eats her vegetables” and \(Q\) denote “Edith can have a cookie.” The logical form of the argument is then:</p>
<div><table>
<tbody><tr>
<td></td>
<td>\(P \imp Q\)</td>
</tr>
<tr>
<td></td>
<td>\(P\)</td>
</tr>
<tr>
<td>\(\therefore\)</td>
<td>\(Q\)</td>
</tr>
</tbody></table></div>
<p>This is an example of a <dfn>deduction rule</dfn>, an argument form which is always valid. This one is a particularly famous rule called <i>modus ponens</i>. Are you convinced that it is a valid deduction rule? If not, consider the following truth table:</p>
<div><table>
<tbody><tr>
<td>\(P\)</td>
<td>\(Q\)</td>
<td>\(P\imp Q\)</td>
</tr>
<tr>
<td>T</td>
<td>T</td>
<td>T</td>
</tr>
<tr>
<td>T</td>
<td>F</td>
<td>F</td>
</tr>
<tr>
<td>F</td>
<td>T</td>
<td>T</td>
</tr>
<tr>
<td>F</td>
<td>F</td>
<td>T</td>
</tr>
</tbody></table></div>
<p>This is just the truth table for \(P \imp Q\text{,}\) but what matters here is that all the lines in the deduction rule have their own column in the truth table. Remember that an argument is valid provided the conclusion must be true given that the premises are true. The premises in this case are \(P \imp Q\) and \(P\text{.}\) Which <em>rows</em> of the truth table correspond to both of these being true? \(P\) is true in the first two rows, and of those, only the first row has \(P \imp Q\) true as well. And lo-and-behold, in this one case, \(Q\) is also true. So if \(P\imp Q\) and \(P\) are both true, we see that \(Q\) must be true as well.</p>
<p>Here are a few more examples.</p>
<article><h6>
</h6>
<p>Show that</p>
<div><table>
<tbody><tr>
<td></td>
<td>\(P \imp Q\)</td>
</tr>
<tr>
<td></td>
<td>\(\neg P \imp Q\)</td>
</tr>
<tr>
<td>\(\therefore\)</td>
<td>\(Q\)</td>
</tr>
</tbody></table></div>
<p>is a valid deduction rule.</p>
<div>
<p></p><p></p>
</div></article><article><h6>
</h6>
<p>Decide whether</p>
<div><table>
<tbody><tr>
<td></td>
<td>\(P \imp R\)</td>
</tr>
<tr>
<td></td>
<td>\(Q \imp R\)</td>
</tr>
<tr>
<td></td>
<td>\(R\)</td>
</tr>
<tr>
<td>\(\therefore\)</td>
<td>\(P \vee Q\)</td>
</tr>
</tbody></table></div>
<p>is a valid deduction rule.</p>
<div>
<p></p><p></p>
</div></article></section><section><h3>
</h3>
<a href="https://discrete.openmathbooks.org/dmoi2/sec_propositional.html#subsection-30">¶</a><p>As we saw in <a href="https://discrete.openmathbooks.org/dmoi2/sec_intro-statements.html">Section 0.2</a>, not every statement can be analyzed using logical connectives alone. For example, we might want to work with the statement:</p>
<blockquote><p>All primes greater than 2 are odd.</p></blockquote>
<p>To write this statement symbolically, we must use quantifiers. We can translate as follows:</p>
\begin{equation*}
\forall x ((P(x) \wedge x \gt 2) \imp O(x)).
\end{equation*}
<p>In this case, we are using \(P(x)\) to denote “\(x\) is prime” and \(O(x)\) to denote “\(x\) is odd.” These are not propositions, since their truth value depends on the input \(x\text{.}\) Better to think of \(P\) and \(O\) as denoting <em>properties</em> of their input. The technical term for these is <dfn>predicates</dfn> and when we study them in logic, we need to use <dfn>predicate logic</dfn>.</p>
<p>It is important to stress that predicate logic <em>extends</em> propositional logic (much in the way quantum mechanics extends classical mechanics). You will notice that our statement above still used the (propositional) logical connectives. Everything that we learned about logical equivalence and deductions still applies. However, predicate logic allows us to analyze statements at a higher resolution, digging down into the individual propositions \(P\text{,}\) \(Q\text{,}\) etc.</p>
<p>A full treatment of predicate logic is beyond the scope of this text. One reason is that there is no systematic procedure for deciding whether two statements in predicate logic are logically equivalent (i.e., there is no analogue to truth tables here). Rather, we end with a couple of examples of logical equivalence and deduction, to pique your interest.</p>
<article><h6>
</h6>
<p>Suppose we claim that there is no smallest number. We can translate this into symbols as</p>
\begin{equation*}
\neg \exists x \forall y (x \le y)
\end{equation*}
<p>(literally, “it is not true that there is a number \(x\) such that for all numbers \(y\text{,}\) \(x\) is less than or equal to \(y\)”).</p>
<p>However, we know how negation interacts with quantifiers: we can pass a negation over a quantifier by switching the quantifier type (between universal and existential). So the statement above should be <em>logically equivalent</em> to</p>
\begin{equation*}
\forall x \exists y (y \lt x).
\end{equation*}
<p>Notice that \(y \lt x\) is the negation of \(x \le y\text{.}\) This literally says, “for every number \(x\) there is a number \(y\) which is smaller than \(x\text{.}\)” We see that this is another way to make our original claim.</p></article><article><h6>
</h6>
<p>Can you switch the order of quantifiers? For example, consider the two statements:</p>
\begin{equation*}
\forall x \exists y P(x,y) \qquad \mathrm{ and } \qquad \exists y \forall x P(x,y).
\end{equation*}
<p>Are these logically equivalent?</p>
<div>
<p></p><p></p>
</div></article></section><section><h3>
</h3>
<a href="https://discrete.openmathbooks.org/dmoi2/sec_propositional.html#exercises_sec-logic">¶</a><article><h6></h6>
<p>Consider the statement about a party, “If it's your birthday or there will be cake, then there will be cake.”</p>
<ol>
<li><p>Translate the above statement into symbols. Clearly state which statement is \(P\) and which is \(Q\text{.}\)</p></li>
<li><p>Make a truth table for the statement.</p></li>
<li><p>Assuming the statement is true, what (if anything) can you conclude if there will be cake?</p></li>
<li><p>Assuming the statement is true, what (if anything) can you conclude if there will not be cake?</p></li>
<li><p>Suppose you found out that the statement was a lie. What can you conclude?</p></li>
</ol>
<div>
<p></p><p></p>
</div></article><article><h6></h6>
<p>Make a truth table for the statement \((P \vee Q) \imp (P \wedge Q)\text{.}\)</p>
<div>
<p></p><p></p>
</div></article><article><h6></h6>
<p>Make a truth table for the statement \(\neg P \wedge (Q \imp P)\text{.}\) What can you conclude about \(P\) and \(Q\) if you know the statement is true?</p>
<div>
<p></p><p></p>
</div></article><article><h6></h6>
<p>Make a truth table for the statement \(\neg P \imp (Q \wedge R)\text{.}\)</p>
<div>
<p></p><p></p>
</div></article><article><h6></h6>
<p>Determine whether the following two statements are logically equivalent: \(\neg(P \imp Q)\) and \(P \wedge \neg Q\text{.}\) Explain how you know you are correct.</p>
<div>
<p></p><p></p>
</div></article><article><h6></h6>
<p>Are the statements \(P \imp (Q\vee R)\) and \((P \imp Q) \vee (P \imp R)\) logically equivalent?</p></article><article><h6></h6>
<p>Simplify the following statements (so that negation only appears right before variables).</p>
<ol>
<li>\(\neg(P \imp \neg Q)\text{.}\)</li>
<li>\((\neg P \vee \neg Q) \imp \neg (\neg Q \wedge R)\text{.}\)</li>
<li>\(\neg((P \imp \neg Q) \vee \neg (R \wedge \neg R))\text{.}\)</li>
<li><p>It is false that if Sam is not a man then Chris is a woman, and that Chris is not a woman.</p></li>
</ol>
<div>
<p></p><p></p>
</div></article><article><h6></h6>
<p>Use De Morgan's Laws, and any other logical equivalence facts you know to simplify the following statements. Show all your steps. Your final statements should have negations only appear directly next to the sentence variables or predicates (\(P\text{,}\) \(Q\text{,}\) \(E(x)\text{,}\) etc.), and no double negations. It would be a good idea to use only conjunctions, disjunctions, and negations.</p>
<ol>
<li>\(\neg((\neg P \wedge Q) \vee \neg(R \vee \neg S))\text{.}\)</li>
<li>\(\neg((\neg P \imp \neg Q) \wedge (\neg Q \imp R))\) (careful with the implications).</li>
</ol></article><article><h6></h6>
<p>Tommy Flanagan was telling you what he ate yesterday afternoon. He tells you, “I had either popcorn or raisins. Also, if I had cucumber sandwiches, then I had soda. But I didn't drink soda or tea.” Of course you know that Tommy is the worlds worst liar, and everything he says is false. What did Tommy eat?</p>
<p>Justify your answer by writing all of Tommy's statements using sentence variables (\(P, Q, R, S, T\)), taking their negations, and using these to deduce what Tommy actually ate.</p></article><article><h6></h6>
<p>Determine if the following deduction rule is valid:</p>
<div><table>
<tbody><tr>
<td></td>
<td>\(P \vee Q\)</td>
</tr>
<tr>
<td></td>
<td>\(\neg P\)</td>
</tr>
<tr>
<td>\(\therefore\)</td>
<td>\(Q\)</td>
</tr>
</tbody></table></div>
<div>
<p></p><p></p>
</div></article><article><h6></h6>
<p>Determine if the following is a valid deduction rule:</p>
<div><table>
<tbody><tr>
<td></td>
<td>\(P \imp (Q \vee R)\)</td>
</tr>
<tr>
<td></td>
<td>\(\neg(P \imp Q)\)</td>
</tr>
<tr>
<td>\(\therefore\)</td>
<td>\(R\)</td>
</tr>
</tbody></table></div></article><article><h6></h6>
<p>Determine if the following is a valid deduction rule:</p>
<div><table>
<tbody><tr>
<td></td>
<td>\((P \wedge Q) \imp R\)</td>
</tr>
<tr>
<td></td>
<td>\(\neg P \vee \neg Q\)</td>
</tr>
<tr>
<td>\(\therefore\)</td>
<td>\(\neg R\)</td>
</tr>
</tbody></table></div></article><article><h6></h6>
<p>Can you chain implications together? That is, if \(P \imp Q\) and \(Q \imp R\text{,}\) does that means the \(P \imp R\text{?}\) Can you chain more implications together? Let's find out:</p>
<ol>
<li>
<p>Prove that the following is a valid deduction rule:</p>
<div><table>
<tbody><tr>
<td></td>
<td>\(P \imp Q\)</td>
</tr>
<tr>
<td></td>
<td>\(Q \imp R\)</td>
</tr>
<tr>
<td>\(\therefore\)</td>
<td>\(P \imp R\)</td>
</tr>
</tbody></table></div>
</li>
<li>
<p>Prove that the following is a valid deduction rule for any \(n \ge 2\text{:}\)</p>
<div><table>
<tbody><tr>
<td></td>
<td>\(P_1 \imp P_2\)</td>
</tr>
<tr>
<td></td>
<td>\(P_2 \imp P_3\)</td>
</tr>
<tr>
<td></td>
<td>\(\vdots\)</td>
</tr>
<tr>
<td></td>
<td>\(P_{n-1} \imp P_n\)</td>
</tr>
<tr>
<td>\(\therefore\)</td>
<td>\(P_1 \imp P_n\text{.}\)</td>
</tr>
</tbody></table></div>
<p>I suggest you don't go through the trouble of writing out a \(2^n\) row truth table. Instead, you should use part (a) and mathematical induction.</p>
</li>
</ol></article><article><h6></h6>
<p>We can also simplify statements in predicate logic using our rules for passing negations over quantifiers, and then applying propositional logical equivalence to the “inside” propositional part. Simplify the statements below (so negation appears only directly next to predicates).</p>
<ol>
<li>\(\neg \exists x \forall y (\neg O(x) \vee E(y))\text{.}\)</li>
<li>\(\neg \forall x \neg \forall y \neg(x \lt y \wedge \exists z (x \lt z \vee y \lt z))\text{.}\)</li>
<li><p>There is a number \(n\) for which no other number is either less \(n\) than or equal to \(n\text{.}\)</p></li>
<li><p>It is false that for every number \(n\) there are two other numbers which \(n\) is between.</p></li>
</ol>
<div>
<p></p><p></p>
</div></article><article><h6></h6>
<p>Suppose \(P\) and \(Q\) are (possibly molecular) propositional statements. Prove that \(P\) and \(Q\) are logically equivalent if any only if \(P \iff Q\) is a tautology.</p>
<div>
<p></p><p></p>
</div></article><article><h6></h6>
<p>Suppose \(P_1, P_2, \ldots, P_n\) and \(Q\) are (possibly molecular) propositional statements. Suppose further that</p>
<div><table>
<tbody><tr>
<td></td>
<td>\(P_1\)</td>
</tr>
<tr>
<td></td>
<td>\(P_2\)</td>
</tr>
<tr>
<td></td>
<td>\(\vdots\)</td>
</tr>
<tr>
<td></td>
<td>\(P_n\)</td>
</tr>
<tr>
<td>\(\therefore\)</td>
<td>\(Q\)</td>
</tr>
</tbody></table></div>
<p>is a valid deduction rule. Prove that the statement</p>
\begin{equation*}
(P_1 \wedge P_2 \wedge \cdots \wedge P_n) \imp Q
\end{equation*}
<p>is a tautology.</p></article></section></section></main>
</div></div>||||I||||Skip to main content
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 Discrete Mathematics An Open Introduction

 Oscar Levin

 Contents
 IndexPrevUpNext
 Annotations
 ContentsPrevUpNext

 Front Matter

 * Colophon
 * Dedication
 * Acknowledgements
 * Preface
 * How to use this book

 0 Introduction and Preliminaries

 * What is Discrete Mathematics?
 * Mathematical Statements
 * Sets
 * Functions

 1 Counting

 * Additive and Multiplicative Principles
 * Binomial Coefficients
 * Combinations and Permutations
 * Combinatorial Proofs
 * Stars and Bars
 * Advanced Counting Using PIE
 * Chapter Summary

 2 Sequences

 * Definitions
 * Arithmetic and Geometric Sequences
 * Polynomial Fitting
 * Solving Recurrence Relations
 * Induction
 * Chapter Summary

 3 Symbolic Logic and Proofs

 * Propositional Logic
 * Proofs
 * Chapter Summary

 4 Graph Theory

 * Definitions
 * Planar Graphs
 * Coloring
 * Euler Paths and Circuits
 * Matching in Bipartite Graphs
 * Chapter Summary

 5 Additional Topics

 * Generating Functions
 * Introduction to Number Theory

 Backmatter

 * Selected Solutions
 * List of Symbols
 * Index
 * Colophon
 Authored in PreTeXt

 Section 3.1 Propositional Logic

 ¶

 Investigate! 25

 You stumble upon two trolls playing Stratego®. They tell you:

 Troll 1: If we are cousins, then we are both knaves.

 Troll 2: We are cousins or we are both knaves.

 Could both trolls be knights? Recall that all trolls are either always-truth-telling knights or always-lying knaves.

 A proposition is simply a statement. Propositional logic studies the ways statements can interact with each other. It is important to remember that propositional logic does not really care about the content of the statements. For example, in terms of propositional logic, the claims, “if the moon is made of cheese then basketballs are round,” and “if spiders have eight legs then Sam walks with a limp” are exactly the same. They are both implications: statements of the form, \(P \imp Q\text{.}\)

 Subsection Truth Tables

 ¶

 Here's a question about playing Monopoly:

 If you get more doubles than any other player then you will lose, or if you lose then you must have bought the most properties.

 True or false? We will answer this question, and won't need to know anything about Monopoly. Instead we will look at the logical form of the statement.

 We need to decide when the statement \((P \imp Q) \vee (Q \imp R)\) is true. Using the definitions of the connectives in Section 0.2, we see that for this to be true, either \(P \imp Q\) must be true or \(Q \imp R\) must be true (or both). Those are true if either \(P\) is false or \(Q\) is true (in the first case) and \(Q\) is false or \(R\) is true (in the second case). So—yeah, it gets kind of messy. Luckily, we can make a chart to keep track of all the possibilities. Enter truth tables. The idea is this: on each row, we list a possible combination of T's and F's (for true and false) for each of the sentential variables, and then mark down whether the statement in question is true or false in that case. We do this for every possible combination of T's and F's. Then we can clearly see in which cases the statement is true or false. For complicated statements, we will first fill in values for each part of the statement, as a way of breaking up our task into smaller, more manageable pieces.

 Since the truth value of a statement is completely determined by the truth values of its parts and how they are connected, all you really need to know is the truth tables for each of the logical connectives. Here they are:

 \(P\) \(Q\) \(P\wedge Q\)
 T T T 
 T F F 
 F T F 
 F F F 
 
 \(P\) \(Q\) \(P\vee Q\)
 T T T 
 T F T 
 F T T 
 F F F 
 
 \(P\) \(Q\) \(P\imp Q\)
 T T T 
 T F F 
 F T T 
 F F T 
 
 \(P\) \(Q\) \(P\iff Q\)
 T T T 
 T F F 
 F T F 
 F F T 
 

 The truth table for negation looks like this:

 \(P\) \(\neg P\)
 T F 
 F T 
 
 

 None of these truth tables should come as a surprise; they are all just restating the definitions of the connectives. Let's try another one.

 Example 3.1.1

 Make a truth table for the statement \(\neg P \vee Q\text{.}\)

 Solution

 Note that this statement is not \(\neg(P \vee Q)\text{,}\) the negation belongs to \(P\) alone. Here is the truth table:

 \(P\) \(Q\) \(\neg P\) \(\neg P \vee Q\)
 T T F T 
 T F F F 
 F T T T 
 F F T T 
 

 We added a column for \(\neg P\) to make filling out the last column easier. The entries in the \(\neg P\) column were determined by the entries in the \(P\) column. Then to fill in the final column, look only at the column for \(Q\) and the column for \(\neg P\) and use the rule for \(\vee\text{.}\)

 Now let's answer our question about monopoly:

 Example 3.1.2

 Analyze the statement, “if you get more doubles than any other player you will lose, or that if you lose you must have bought the most properties,” using truth tables.

 Solution

 Represent the statement in symbols as \((P \imp Q) \vee (Q \imp R)\text{,}\) where \(P\) is the statement “you get more doubles than any other player,” \(Q\) is the statement “you will lose,” and \(R\) is the statement “you must have bought the most properties.” Now make a truth table.

 The truth table needs to contain 8 rows in order to account for every possible combination of truth and falsity among the three statements. Here is the full truth table:

 \(P\) \(Q\) \(R\) \(P \imp Q\) \(Q \imp R\) \((P \imp Q) \vee (Q \imp R)\)
 T T T T T T 
 T T F T F T 
 T F T F T T 
 T F F F T T 
 F T T T T T 
 F T F T F T 
 F F T T T T 
 F F F T T T 
 

 The first three columns are simply a systematic listing of all possible combinations of T and F for the three statements (do you see how you would list the 16 possible combinations for four statements?). The next two columns are determined by the values of \(P\text{,}\) \(Q\text{,}\) and \(R\) and the definition of implication. Then, the last column is determined by the values in the previous two columns and the definition of \(\vee\text{.}\) It is this final column we care about.

 Notice that in each of the eight possible cases, the statement in question is true. So our statement about monopoly is true (regardless of how many properties you own, how many doubles you roll, or whether you win or lose).

 The statement about monopoly is an example of a tautology, a statement which is true on the basis of its logical form alone. Tautologies are always true but they don't tell us much about the world. No knowledge about monopoly was required to determine that the statement was true. In fact, it is equally true that “If the moon is made of cheese, then Elvis is still alive, or if Elvis is still alive, then unicorns have 5 legs.”

 Subsection Logical Equivalence

 ¶

 You might have noticed that the final column in the truth table from \(\neg P \vee Q\) is identical to the final column in the truth table for \(P \imp Q\text{:}\)

 \(P\) \(Q\) \(P \imp Q\) \(\neg P \vee Q\)
 T T T T 
 T F F F 
 F T T T 
 F F T T 
 

 This says that no matter what \(P\) and \(Q\) are, the statements \(\neg P \vee Q\) and \(P \imp Q\) either both true or both false. We therefore say these statements are logically equivalent.

 Logical Equivalence

 Two (molecular) statements \(P\) and \(Q\) are logically equivalent provided \(P\) is true precisely when \(Q\) is true. That is, \(P\) and \(Q\) have the same truth value under any assignment of truth values to their atomic parts.

 To verify that two statements are logically equivalent, you can make a truth table for each and check whether the columns for the two statements are identical.

 Recognizing two statements as logically equivalent can be very helpful. Rephrasing a mathematical statement can often lends insight into what it is saying, or how to prove or refute it. Using truth tables we can systematically verify that two statements are indeed logically equivalent.

 Example 3.1.3

 Are the statements, “it will not rain or snow” and “it will not rain and it will not snow” logically equivalent?

 Solution

 We want to know whether \(\neg(P \vee Q)\) is logically equivalent to \(\neg P \wedge \neg Q\text{.}\) Make a truth table which includes both statements:

 \(P\) \(Q\) \(\neg(P \vee Q)\) \(\neg P \wedge \neg Q\)
 T T F F 
 T F F F 
 F T F F 
 F F T T 
 

 Since in every row the truth values for the two statements are equal, the two statements are logically equivalent.

 Notice that this example gives us a way to “distribute” a negation over a disjunction (an “or”). We have a similar rule for distributing over conjunctions (“and”s):

 De Morgan's Laws

 \begin{equation*} \neg(P \wedge Q) \text{ is logically equivalent to } \neg P \vee \neg Q. \end{equation*} \begin{equation*} \neg(P \vee Q) \text{ is logically equivalent to } \neg P \wedge \neg Q. \end{equation*}

 This suggests there might be a sort of “algebra” you could apply to statements (okay, there is: it is called Boolean algebra) to transform one statement into another. We can start collecting useful examples of logical equivalence, and apply them in succession to a statement, instead of writing out a complicated truth table. We will probably also want a way to deal with double negation:

 Double Negation

 \begin{equation*} \neg \neg P \mbox{ is logically equivalent to } P. \end{equation*}

 Example: “It is not the case that \(c\) is not odd” means “\(c\) is odd.”

 Let's see how we can apply the equivalences we have encountered so far.

 Example 3.1.4

 Prove that the statements \(\neg(P \imp Q)\) and \(P\wedge \neg Q\) are logically equivalent without using truth tables.

 Solution

 We want to start with one of the statements, and transform it into the other through a sequence of logically equivalent statements. Start with \(\neg(P \imp Q)\text{.}\) We can rewrite the implication as a disjunction this is logically equivalent to

 \begin{equation*} \neg(\neg P \vee Q). \end{equation*}

 Now apply DeMorgan's law to get

 \begin{equation*} \neg\neg P \wedge \neg Q. \end{equation*}

 Finally, use double negation to arrive at \(P \wedge \neg Q\)

 Notice that the above example illustrates that the negation of an implication is NOT an implication: it is a conjunction!

 To verify that two statements are logically equivalent, you can use truth tables or a sequence of logically equivalent replacements. The truth table method, although cumbersome, has the advantage that it can verify that two statements are NOT logically equivalent.

 Example 3.1.5

 Are the statements \((P \vee Q) \imp R\) and \((P \imp R) \vee (Q \imp R)\) logically equivalent?

 Solution

 Note that while we could start rewriting these statements with logically equivalent replacements in the hopes of transforming one into another, we will never be sure that our failure is due to their lack of logical equivalence rather than our lack of imagination. So instead, let's make a truth table:

 \(P\) \(Q\) \(R\) \((P\vee Q) \imp R\) \((P\imp R) \vee (Q \imp R)\)
 T T T T T 
 T T F F F 
 T F T T T 
 T F F F T 
 F T T T T 
 F T F F T 
 F F T T T 
 F F F T T 
 
 

 Look at the fourth (or sixth) row. In this case, \((P \imp R) \vee (Q \imp R)\) is true, but \((P \vee Q) \imp R\) is false. Therefore the statements are not logically equivalent.

 While we don't have logical equivalence, it is the case that whenever \((P \vee Q) \imp R\) is true, so is \((P \imp R) \vee (Q \imp R)\text{.}\) This tells us that we can deduce \((P \imp R) \vee (Q \imp R)\) from \((P \vee Q) \imp R\text{,}\) just not the reverse direction.

 Subsection Deductions

 ¶

 Investigate! 26

 Holmes owns two suits: one black and one tweed. He always wears either a tweed suit or sandals. Whenever he wears his tweed suit and a purple shirt, he chooses to not wear a tie. He never wears the tweed suit unless he is also wearing either a purple shirt or sandals. Whenever he wears sandals, he also wears a purple shirt. Yesterday, Holmes wore a bow tie. What else did he wear?

 Earlier we claimed that the following was a valid argument:

 If Edith eats her vegetables, then she can have a cookie. Edith ate her vegetables. Therefore Edith gets a cookie.

 How do we know this is valid? Let's look at the form of the statements. Let \(P\) denote “Edith eats her vegetables” and \(Q\) denote “Edith can have a cookie.” The logical form of the argument is then:

 \(P \imp Q\)
 \(P\) 
 \(\therefore\) \(Q\) 
 

 This is an example of a deduction rule, an argument form which is always valid. This one is a particularly famous rule called modus ponens. Are you convinced that it is a valid deduction rule? If not, consider the following truth table:

 \(P\) \(Q\) \(P\imp Q\)
 T T T 
 T F F 
 F T T 
 F F T 
 

 This is just the truth table for \(P \imp Q\text{,}\) but what matters here is that all the lines in the deduction rule have their own column in the truth table. Remember that an argument is valid provided the conclusion must be true given that the premises are true. The premises in this case are \(P \imp Q\) and \(P\text{.}\) Which rows of the truth table correspond to both of these being true? \(P\) is true in the first two rows, and of those, only the first row has \(P \imp Q\) true as well. And lo-and-behold, in this one case, \(Q\) is also true. So if \(P\imp Q\) and \(P\) are both true, we see that \(Q\) must be true as well.

 Here are a few more examples.

 Example 3.1.6

 Show that

 \(P \imp Q\) 
 \(\neg P \imp Q\)
 \(\therefore\) \(Q\) 
 

 is a valid deduction rule.

 Solution

 We make a truth table which contains all the lines of the argument form:

 \(P\) \(Q\) \(P\imp Q\) \(\neg P\) \(\neg P \imp Q\)
 T T T F T 
 T F F F T 
 F T T T T 
 F F T T F 
 

 (we include a column for \(\neg P\) just as a step to help getting the column for \(\neg P \imp Q\)).

 Now look at all the rows for which both \(P \imp Q\) and \(\neg P \imp Q\) are true. This happens only in rows 1 and 3. Hey! In those rows \(Q\) is true as well, so the argument form is valid (it is a valid deduction rule).

 Example 3.1.7

 Decide whether

 \(P \imp R\)
 \(Q \imp R\)
 \(R\) 
 \(\therefore\) \(P \vee Q\)
 

 is a valid deduction rule.

 Solution

 Let's make a truth table containing all four statements.

 \(P\) \(Q\) \(R\) \(P \imp R\) \(Q \imp R\) \(P \vee Q\)
 T T T T T T 
 T T F F F T 
 T F T T T T 
 T F F F T T 
 F T T T T T 
 F T F T F T 
 F F T T T F 
 F F F T T F 
 

 Look at the second to last row. Here all three premises of the argument are true, but the conclusion is false. Thus this is not a valid deduction rule.

 While we have the truth table in front of us, look at rows 1 and 5. These are the only rows in which all of the statements statements \(P \imp R\text{,}\) \(Q \imp R\text{,}\) and \(P\vee Q\) are true. It also happens that \(R\) is true in these rows as well. Thus we have discovered a new deduction rule we know is valid:

 \(P \imp R\)
 \(Q \imp R\)
 \(P \vee Q\)
 \(\therefore\) \(R\) 
 

 Subsection Beyond Propositions

 ¶

 As we saw in Section 0.2, not every statement can be analyzed using logical connectives alone. For example, we might want to work with the statement:

 All primes greater than 2 are odd.

 To write this statement symbolically, we must use quantifiers. We can translate as follows:

 \begin{equation*} \forall x ((P(x) \wedge x \gt 2) \imp O(x)). \end{equation*}

 In this case, we are using \(P(x)\) to denote “\(x\) is prime” and \(O(x)\) to denote “\(x\) is odd.” These are not propositions, since their truth value depends on the input \(x\text{.}\) Better to think of \(P\) and \(O\) as denoting properties of their input. The technical term for these is predicates and when we study them in logic, we need to use predicate logic.

 It is important to stress that predicate logic extends propositional logic (much in the way quantum mechanics extends classical mechanics). You will notice that our statement above still used the (propositional) logical connectives. Everything that we learned about logical equivalence and deductions still applies. However, predicate logic allows us to analyze statements at a higher resolution, digging down into the individual propositions \(P\text{,}\) \(Q\text{,}\) etc.

 A full treatment of predicate logic is beyond the scope of this text. One reason is that there is no systematic procedure for deciding whether two statements in predicate logic are logically equivalent (i.e., there is no analogue to truth tables here). Rather, we end with a couple of examples of logical equivalence and deduction, to pique your interest.

 Example 3.1.8

 Suppose we claim that there is no smallest number. We can translate this into symbols as

 \begin{equation*} \neg \exists x \forall y (x \le y) \end{equation*}

 (literally, “it is not true that there is a number \(x\) such that for all numbers \(y\text{,}\) \(x\) is less than or equal to \(y\)”).

 However, we know how negation interacts with quantifiers: we can pass a negation over a quantifier by switching the quantifier type (between universal and existential). So the statement above should be logically equivalent to

 \begin{equation*} \forall x \exists y (y \lt x). \end{equation*}

 Notice that \(y \lt x\) is the negation of \(x \le y\text{.}\) This literally says, “for every number \(x\) there is a number \(y\) which is smaller than \(x\text{.}\)” We see that this is another way to make our original claim.

 Example 3.1.9

 Can you switch the order of quantifiers? For example, consider the two statements:

 \begin{equation*} \forall x \exists y P(x,y) \qquad \mathrm{ and } \qquad \exists y \forall x P(x,y). \end{equation*}

 Are these logically equivalent?

 Solution

 These statements are NOT logically equivalent. To see this, we should provide an interpretation of the predicate \(P(x,y)\) which makes one of the statements true and the other false.

 Let \(P(x,y)\) be the predicate \(x \lt y\text{.}\) It is true, in the natural numbers, that for all \(x\) there is some \(y\) greater than it (since there are infinitely many numbers). However, there is not a natural number \(y\) which is greater than every number \(x\text{.}\) Thus it is possible for \(\forall x \exists y P(x,y)\) to be true while \(\exists y \forall x P(x,y)\) is false.

 We cannot do the reverse of this though. If there is some \(y\) for which every \(x\) satisfies \(P(x,y)\text{,}\) then certainly for every \(x\) there is some \(y\) which satisfies \(P(x,y)\text{.}\) The first is saying we can find one \(y\) that works for every \(x\text{.}\) The second allows different \(y\)'s to work for different \(x\)'s, but there is nothing preventing us from using the same \(y\) that work for every \(x\text{.}\) In other words, while we don't have logical equivalence between the two statements, we do have a valid deduction rule:

 \(\exists y \forall x P(x,y)\)
 \(\therefore\) \(\forall x \exists y P(x,y)\)
 

 Put yet another way, this says that the single statement

 \begin{equation*} \exists y \forall x P(x,y) \imp \forall x \exists y P(x,y) \end{equation*}

 is always true. This is sort of like a tautology, although we reserve that term for necessary truths in propositional logic. A statement in predicate logic that is necessarily true gets the more prestigious designation of a law of logic (or sometimes logically valid, but that is less fun).

 Subsection Exercises

 ¶

 1

 Consider the statement about a party, “If it's your birthday or there will be cake, then there will be cake.”

 1. Translate the above statement into symbols. Clearly state which statement is \(P\) and which is \(Q\text{.}\)

 2. Make a truth table for the statement.

 3. Assuming the statement is true, what (if anything) can you conclude if there will be cake?

 4. Assuming the statement is true, what (if anything) can you conclude if there will not be cake?

 5. Suppose you found out that the statement was a lie. What can you conclude?

 Solution

 1. \(P\text{:}\) it's your birthday; \(Q\text{:}\) there will be cake. \((P \vee Q) \imp Q\)

 2. Hint: you should get three T's and one F.

 3. Only that there will be cake.

 4. It's NOT your birthday!

 5. It's your birthday, but the cake is a lie.

 2

 Make a truth table for the statement \((P \vee Q) \imp (P \wedge Q)\text{.}\)

 Solution
 \(P\) \(Q\) \((P \vee Q) \imp (P \wedge Q)\)
 T T T 
 T F F 
 F T F 
 F F T 
 

 3

 Make a truth table for the statement \(\neg P \wedge (Q \imp P)\text{.}\) What can you conclude about \(P\) and \(Q\) if you know the statement is true?

 Solution
 \(P\) \(Q\) \(\neg P \wedge (Q \imp P)\)
 T T F 
 T F F 
 F T F 
 F F T 
 

 If the statement is true, then both \(P\) and \(Q\) are false.

 4

 Make a truth table for the statement \(\neg P \imp (Q \wedge R)\text{.}\)

 Hint

 Like above, only now you will need 8 rows instead of just 4.

 5

 Determine whether the following two statements are logically equivalent: \(\neg(P \imp Q)\) and \(P \wedge \neg Q\text{.}\) Explain how you know you are correct.

 Solution

 Make a truth table for each and compare. The statements are logically equivalent.

 6

 Are the statements \(P \imp (Q\vee R)\) and \((P \imp Q) \vee (P \imp R)\) logically equivalent?

 7

 Simplify the following statements (so that negation only appears right before variables).

 1. \(\neg(P \imp \neg Q)\text{.}\)
 2. \((\neg P \vee \neg Q) \imp \neg (\neg Q \wedge R)\text{.}\)
 3. \(\neg((P \imp \neg Q) \vee \neg (R \wedge \neg R))\text{.}\)

 4. It is false that if Sam is not a man then Chris is a woman, and that Chris is not a woman.

 Answer

 1. \(P \wedge Q\text{.}\)
 2. \((\neg P \vee \neg R) \imp (Q \vee \neg R)\) or, replacing the implication with a disjunction first: \((P \wedge Q) \vee (Q \vee \neg R)\text{.}\)

 3. \((P \wedge Q) \wedge (R \wedge \neg R)\text{.}\) This is necessarily false, so it is also equivalent to \(P \wedge \neg P\text{.}\)

 4. Either Sam is a woman and Chris is a man, or Chris is a woman.

 8

 Use De Morgan's Laws, and any other logical equivalence facts you know to simplify the following statements. Show all your steps. Your final statements should have negations only appear directly next to the sentence variables or predicates (\(P\text{,}\) \(Q\text{,}\) \(E(x)\text{,}\) etc.), and no double negations. It would be a good idea to use only conjunctions, disjunctions, and negations.

 1. \(\neg((\neg P \wedge Q) \vee \neg(R \vee \neg S))\text{.}\)
 2. \(\neg((\neg P \imp \neg Q) \wedge (\neg Q \imp R))\) (careful with the implications).

 9

 Tommy Flanagan was telling you what he ate yesterday afternoon. He tells you, “I had either popcorn or raisins. Also, if I had cucumber sandwiches, then I had soda. But I didn't drink soda or tea.” Of course you know that Tommy is the worlds worst liar, and everything he says is false. What did Tommy eat?

 Justify your answer by writing all of Tommy's statements using sentence variables (\(P, Q, R, S, T\)), taking their negations, and using these to deduce what Tommy actually ate.

 10

 Determine if the following deduction rule is valid:

 \(P \vee Q\)
 \(\neg P\) 
 \(\therefore\) \(Q\) 
 
 Solution

 The deduction rule is valid. To see this, make a truth table which contains \(P \vee Q\) and \(\neg P\) (and \(P\) and \(Q\) of course). Look at the truth value of \(Q\) in each of the rows that have \(P \vee Q\) and \(\neg P\) true.

 11

 Determine if the following is a valid deduction rule:

 \(P \imp (Q \vee R)\)
 \(\neg(P \imp Q)\) 
 \(\therefore\) \(R\) 
 

 12

 Determine if the following is a valid deduction rule:

 \((P \wedge Q) \imp R\)
 \(\neg P \vee \neg Q\) 
 \(\therefore\) \(\neg R\) 
 

 13

 Can you chain implications together? That is, if \(P \imp Q\) and \(Q \imp R\text{,}\) does that means the \(P \imp R\text{?}\) Can you chain more implications together? Let's find out:

 1. Prove that the following is a valid deduction rule:

 \(P \imp Q\)
 \(Q \imp R\)
 \(\therefore\) \(P \imp R\)
 

 2. Prove that the following is a valid deduction rule for any \(n \ge 2\text{:}\)

 \(P_1 \imp P_2\) 
 \(P_2 \imp P_3\) 
 \(\vdots\) 
 \(P_{n-1} \imp P_n\) 
 \(\therefore\) \(P_1 \imp P_n\text{.}\)
 

 I suggest you don't go through the trouble of writing out a \(2^n\) row truth table. Instead, you should use part (a) and mathematical induction.

 14

 We can also simplify statements in predicate logic using our rules for passing negations over quantifiers, and then applying propositional logical equivalence to the “inside” propositional part. Simplify the statements below (so negation appears only directly next to predicates).

 1. \(\neg \exists x \forall y (\neg O(x) \vee E(y))\text{.}\)
 2. \(\neg \forall x \neg \forall y \neg(x \lt y \wedge \exists z (x \lt z \vee y \lt z))\text{.}\)

 3. There is a number \(n\) for which no other number is either less \(n\) than or equal to \(n\text{.}\)

 4. It is false that for every number \(n\) there are two other numbers which \(n\) is between.

 Solution

 1. \(\forall x \exists y (O(x) \wedge \neg E(y))\text{.}\)
 2. \(\exists x \forall y (x \ge y \vee \forall z (x \ge z \wedge y \ge z))\text{.}\)

 3. There is a number \(n\) for which every other number is strictly greater than \(n\text{.}\)

 4. There is a number \(n\) which is not between any other two numbers.

 15

 Suppose \(P\) and \(Q\) are (possibly molecular) propositional statements. Prove that \(P\) and \(Q\) are logically equivalent if any only if \(P \iff Q\) is a tautology.

 Hint

 What do these concepts mean in terms of truth tables?

 16

 Suppose \(P_1, P_2, \ldots, P_n\) and \(Q\) are (possibly molecular) propositional statements. Suppose further that

 \(P_1\) 
 \(P_2\) 
 \(\vdots\)
 \(P_n\) 
 \(\therefore\) \(Q\) 
 

 is a valid deduction rule. Prove that the statement

 \begin{equation*} (P_1 \wedge P_2 \wedge \cdots \wedge P_n) \imp Q \end{equation*}

 is a tautology.

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¶   
You stumble upon two trolls playing Stratego®. They tell you:
Troll 1: If we are cousins, then we are both knaves.
Troll 2: We are cousins or we are both knaves.
Could both trolls be knights? Recall that all trolls are either always-truth-telling knights or always-lying knaves.  A proposition is simply a statement. Propositional logic studies the ways statements can interact with each other. It is important to remember that propositional logic does not really care about the content of the statements. For example, in terms of propositional logic, the claims, “if the moon is made of cheese then basketballs are round,” and “if spiders have eight legs then Sam walks with a limp” are exactly the same. They are both implications: statements of the form, \(P \imp Q\text{.}\)   
¶ Here's a question about playing Monopoly:
 If you get more doubles than any other player then you will lose, or if you lose then you must have bought the most properties. 
True or false? We will answer this question, and won't need to know anything about Monopoly. Instead we will look at the logical form of the statement.
We need to decide when the statement \((P \imp Q) \vee (Q \imp R)\) is true. Using the definitions of the connectives in Section 0.2, we see that for this to be true, either \(P \imp Q\) must be true or \(Q \imp R\) must be true (or both). Those are true if either \(P\) is false or \(Q\) is true (in the first case) and \(Q\) is false or \(R\) is true (in the second case). So—yeah, it gets kind of messy. Luckily, we can make a chart to keep track of all the possibilities. Enter truth tables. The idea is this: on each row, we list a possible combination of T's and F's (for true and false) for each of the sentential variables, and then mark down whether the statement in question is true or false in that case. We do this for every possible combination of T's and F's. Then we can clearly see in which cases the statement is true or false. For complicated statements, we will first fill in values for each part of the statement, as a way of breaking up our task into smaller, more manageable pieces.
Since the truth value of a statement is completely determined by the truth values of its parts and how they are connected, all you really need to know is the truth tables for each of the logical connectives. Here they are:
 
 
\(P\)
\(Q\)
\(P\wedge Q\)
T
T
T
T
F
F
F
T
F
F
F
F
  
 
 
\(P\)
\(Q\)
\(P\vee Q\)
T
T
T
T
F
T
F
T
T
F
F
F
  
 
 
\(P\)
\(Q\)
\(P\imp Q\)
T
T
T
T
F
F
F
T
T
F
F
T
  
 
 
\(P\)
\(Q\)
\(P\iff Q\)
T
T
T
T
F
F
F
T
F
F
F
T
  
The truth table for negation looks like this:
None of these truth tables should come as a surprise; they are all just restating the definitions of the connectives. Let's try another one.
 
Make a truth table for the statement \(\neg P \vee Q\text{.}\)
   
  Now let's answer our question about monopoly:
 
Analyze the statement, “if you get more doubles than any other player you will lose, or that if you lose you must have bought the most properties,” using truth tables.
   
  The statement about monopoly is an example of a tautology, a statement which is true on the basis of its logical form alone. Tautologies are always true but they don't tell us much about the world. No knowledge about monopoly was required to determine that the statement was true. In fact, it is equally true that “If the moon is made of cheese, then Elvis is still alive, or if Elvis is still alive, then unicorns have 5 legs.”   
¶ You might have noticed that the final column in the truth table from \(\neg P \vee Q\) is identical to the final column in the truth table for \(P \imp Q\text{:}\)
 
 
\(P\)
\(Q\)
\(P \imp Q\)
\(\neg P \vee Q\)
T
T
T
T
T
F
F
F
F
T
T
T
F
F
T
T
  
This says that no matter what \(P\) and \(Q\) are, the statements \(\neg P \vee Q\) and \(P \imp Q\) either both true or both false. We therefore say these statements are logically equivalent.
  
Two (molecular) statements \(P\) and \(Q\) are logically equivalent provided \(P\) is true precisely when \(Q\) is true. That is, \(P\) and \(Q\) have the same truth value under any assignment of truth values to their atomic parts.
To verify that two statements are logically equivalent, you can make a truth table for each and check whether the columns for the two statements are identical.  Recognizing two statements as logically equivalent can be very helpful. Rephrasing a mathematical statement can often lends insight into what it is saying, or how to prove or refute it. Using truth tables we can systematically verify that two statements are indeed logically equivalent.
 
Are the statements, “it will not rain or snow” and “it will not rain and it will not snow” logically equivalent?
   
  Notice that this example gives us a way to “distribute” a negation over a disjunction (an “or”). We have a similar rule for distributing over conjunctions (“and”s):
  
\begin{equation*}
\neg(P \wedge Q) \text{ is logically equivalent to } \neg P \vee \neg Q.
\end{equation*}
\begin{equation*}
\neg(P \vee Q) \text{ is logically equivalent to } \neg P \wedge \neg Q.
\end{equation*}
 This suggests there might be a sort of “algebra” you could apply to statements (okay, there is: it is called Boolean algebra) to transform one statement into another. We can start collecting useful examples of logical equivalence, and apply them in succession to a statement, instead of writing out a complicated truth table. We will probably also want a way to deal with double negation:
  
\begin{equation*}
\neg \neg P \mbox{ is logically equivalent to } P.
\end{equation*}
Example: “It is not the case that \(c\) is not odd” means “\(c\) is odd.”  Let's see how we can apply the equivalences we have encountered so far.
 
Prove that the statements \(\neg(P \imp Q)\) and \(P\wedge \neg Q\) are logically equivalent without using truth tables.
   
  Notice that the above example illustrates that the negation of an implication is NOT an implication: it is a conjunction!
To verify that two statements are logically equivalent, you can use truth tables or a sequence of logically equivalent replacements. The truth table method, although cumbersome, has the advantage that it can verify that two statements are NOT logically equivalent.
 
Are the statements \((P \vee Q) \imp R\) and \((P \imp R) \vee (Q \imp R)\) logically equivalent?
   
    
¶  
Holmes owns two suits: one black and one tweed. He always wears either a tweed suit or sandals. Whenever he wears his tweed suit and a purple shirt, he chooses to not wear a tie. He never wears the tweed suit unless he is also wearing either a purple shirt or sandals. Whenever he wears sandals, he also wears a purple shirt. Yesterday, Holmes wore a bow tie. What else did he wear?  Earlier we claimed that the following was a valid argument:
 If Edith eats her vegetables, then she can have a cookie. Edith ate her vegetables. Therefore Edith gets a cookie. 
How do we know this is valid? Let's look at the form of the statements. Let \(P\) denote “Edith eats her vegetables” and \(Q\) denote “Edith can have a cookie.” The logical form of the argument is then:
 
 
 
\(P \imp Q\)
 
\(P\)
\(\therefore\)
\(Q\)
  
This is an example of a deduction rule, an argument form which is always valid. This one is a particularly famous rule called modus ponens. Are you convinced that it is a valid deduction rule? If not, consider the following truth table:
 
 
\(P\)
\(Q\)
\(P\imp Q\)
T
T
T
T
F
F
F
T
T
F
F
T
  
This is just the truth table for \(P \imp Q\text{,}\) but what matters here is that all the lines in the deduction rule have their own column in the truth table. Remember that an argument is valid provided the conclusion must be true given that the premises are true. The premises in this case are \(P \imp Q\) and \(P\text{.}\) Which rows of the truth table correspond to both of these being true? \(P\) is true in the first two rows, and of those, only the first row has \(P \imp Q\) true as well. And lo-and-behold, in this one case, \(Q\) is also true. So if \(P\imp Q\) and \(P\) are both true, we see that \(Q\) must be true as well.
Here are a few more examples.
 
Show that
 
 
 
\(P \imp Q\)
 
\(\neg P \imp Q\)
\(\therefore\)
\(Q\)
  
is a valid deduction rule.
   
   
Decide whether
 
 
 
\(P \imp R\)
 
\(Q \imp R\)
 
\(R\)
\(\therefore\)
\(P \vee Q\)
  
is a valid deduction rule.
   
    
¶ As we saw in Section 0.2, not every statement can be analyzed using logical connectives alone. For example, we might want to work with the statement:
 All primes greater than 2 are odd. 
To write this statement symbolically, we must use quantifiers. We can translate as follows:
\begin{equation*}
\forall x ((P(x) \wedge x \gt 2) \imp O(x)).
\end{equation*}
In this case, we are using \(P(x)\) to denote “\(x\) is prime” and \(O(x)\) to denote “\(x\) is odd.” These are not propositions, since their truth value depends on the input \(x\text{.}\) Better to think of \(P\) and \(O\) as denoting properties of their input. The technical term for these is predicates and when we study them in logic, we need to use predicate logic.
It is important to stress that predicate logic extends propositional logic (much in the way quantum mechanics extends classical mechanics). You will notice that our statement above still used the (propositional) logical connectives. Everything that we learned about logical equivalence and deductions still applies. However, predicate logic allows us to analyze statements at a higher resolution, digging down into the individual propositions \(P\text{,}\) \(Q\text{,}\) etc.
A full treatment of predicate logic is beyond the scope of this text. One reason is that there is no systematic procedure for deciding whether two statements in predicate logic are logically equivalent (i.e., there is no analogue to truth tables here). Rather, we end with a couple of examples of logical equivalence and deduction, to pique your interest.
 
Suppose we claim that there is no smallest number. We can translate this into symbols as
\begin{equation*}
\neg \exists x \forall y (x \le y)
\end{equation*}
(literally, “it is not true that there is a number \(x\) such that for all numbers \(y\text{,}\) \(x\) is less than or equal to \(y\)”).
However, we know how negation interacts with quantifiers: we can pass a negation over a quantifier by switching the quantifier type (between universal and existential). So the statement above should be logically equivalent to
\begin{equation*}
\forall x \exists y (y \lt x).
\end{equation*}
Notice that \(y \lt x\) is the negation of \(x \le y\text{.}\) This literally says, “for every number \(x\) there is a number \(y\) which is smaller than \(x\text{.}\)” We see that this is another way to make our original claim.   
Can you switch the order of quantifiers? For example, consider the two statements:
\begin{equation*}
\forall x \exists y P(x,y) \qquad \mathrm{ and } \qquad \exists y \forall x P(x,y).
\end{equation*}
Are these logically equivalent?
   
    
¶   
Consider the statement about a party, “If it's your birthday or there will be cake, then there will be cake.”
 Translate the above statement into symbols. Clearly state which statement is \(P\) and which is \(Q\text{.}\) 
 Make a truth table for the statement. 
 Assuming the statement is true, what (if anything) can you conclude if there will be cake? 
 Assuming the statement is true, what (if anything) can you conclude if there will not be cake? 
 Suppose you found out that the statement was a lie. What can you conclude? 
   
    
Make a truth table for the statement \((P \vee Q) \imp (P \wedge Q)\text{.}\)
   
    
Make a truth table for the statement \(\neg P \wedge (Q \imp P)\text{.}\) What can you conclude about \(P\) and \(Q\) if you know the statement is true?
   
    
Make a truth table for the statement \(\neg P \imp (Q \wedge R)\text{.}\)
   
    
Determine whether the following two statements are logically equivalent: \(\neg(P \imp Q)\) and \(P \wedge \neg Q\text{.}\) Explain how you know you are correct.
   
    
Are the statements \(P \imp (Q\vee R)\) and \((P \imp Q) \vee (P \imp R)\) logically equivalent?    
Simplify the following statements (so that negation only appears right before variables).
\(\neg(P \imp \neg Q)\text{.}\)
\((\neg P \vee \neg Q) \imp \neg (\neg Q \wedge R)\text{.}\)
\(\neg((P \imp \neg Q) \vee \neg (R \wedge \neg R))\text{.}\)
 It is false that if Sam is not a man then Chris is a woman, and that Chris is not a woman. 
   
    
Use De Morgan's Laws, and any other logical equivalence facts you know to simplify the following statements. Show all your steps. Your final statements should have negations only appear directly next to the sentence variables or predicates (\(P\text{,}\) \(Q\text{,}\) \(E(x)\text{,}\) etc.), and no double negations. It would be a good idea to use only conjunctions, disjunctions, and negations.
\(\neg((\neg P \wedge Q) \vee \neg(R \vee \neg S))\text{.}\)
\(\neg((\neg P \imp \neg Q) \wedge (\neg Q \imp R))\) (careful with the implications).
    
Tommy Flanagan was telling you what he ate yesterday afternoon. He tells you, “I had either popcorn or raisins. Also, if I had cucumber sandwiches, then I had soda. But I didn't drink soda or tea.” Of course you know that Tommy is the worlds worst liar, and everything he says is false. What did Tommy eat?
Justify your answer by writing all of Tommy's statements using sentence variables (\(P, Q, R, S, T\)), taking their negations, and using these to deduce what Tommy actually ate.    
Determine if the following deduction rule is valid:
 
 
 
\(P \vee Q\)
 
\(\neg P\)
\(\therefore\)
\(Q\)
  
   
    
Determine if the following is a valid deduction rule:
 
 
 
\(P \imp (Q \vee R)\)
 
\(\neg(P \imp Q)\)
\(\therefore\)
\(R\)
      
Determine if the following is a valid deduction rule:
 
 
 
\((P \wedge Q) \imp R\)
 
\(\neg P \vee \neg Q\)
\(\therefore\)
\(\neg R\)
      
Can you chain implications together? That is, if \(P \imp Q\) and \(Q \imp R\text{,}\) does that means the \(P \imp R\text{?}\) Can you chain more implications together? Let's find out:
Prove that the following is a valid deduction rule:
 
 
 
\(P \imp Q\)
 
\(Q \imp R\)
\(\therefore\)
\(P \imp R\)
  
Prove that the following is a valid deduction rule for any \(n \ge 2\text{:}\)
 
 
 
\(P_1 \imp P_2\)
 
\(P_2 \imp P_3\)
 
\(\vdots\)
 
\(P_{n-1} \imp P_n\)
\(\therefore\)
\(P_1 \imp P_n\text{.}\)
  
I suggest you don't go through the trouble of writing out a \(2^n\) row truth table. Instead, you should use part (a) and mathematical induction.
    
We can also simplify statements in predicate logic using our rules for passing negations over quantifiers, and then applying propositional logical equivalence to the “inside” propositional part. Simplify the statements below (so negation appears only directly next to predicates).
\(\neg \exists x \forall y (\neg O(x) \vee E(y))\text{.}\)
\(\neg \forall x \neg \forall y \neg(x \lt y \wedge \exists z (x \lt z \vee y \lt z))\text{.}\)
 There is a number \(n\) for which no other number is either less \(n\) than or equal to \(n\text{.}\) 
 It is false that for every number \(n\) there are two other numbers which \(n\) is between. 
   
    
Suppose \(P\) and \(Q\) are (possibly molecular) propositional statements. Prove that \(P\) and \(Q\) are logically equivalent if any only if \(P \iff Q\) is a tautology.
   
    
Suppose \(P_1, P_2, \ldots, P_n\) and \(Q\) are (possibly molecular) propositional statements. Suppose further that
 
 
 
\(P_1\)
 
\(P_2\)
 
\(\vdots\)
 
\(P_n\)
\(\therefore\)
\(Q\)
  
is a valid deduction rule. Prove that the statement
\begin{equation*}
(P_1 \wedge P_2 \wedge \cdots \wedge P_n) \imp Q
\end{equation*}
is a tautology.
Propositional Logic
http://discrete.openmathbooks.org/dmoi2/sec_propositional.html
https://en.wikipedia.org/wiki/Propositional_calculus
Propositional calculus - Wikipedia
2001-09-26
Authority control National libraries Germany Czech Republic
<div><div>
<p><b>Propositional calculus</b> is a branch of <a href="/wiki/Logic">logic</a>. It is also called <b>propositional logic</b>, <b>statement logic</b>, <b>sentential calculus</b>, <b>sentential logic</b>, or sometimes <b><a href="/wiki/Zeroth-order_logic">zeroth-order logic</a></b>. It deals with <a href="/wiki/Propositions">propositions</a> (which can be true or false) and relations between propositions, including the construction of arguments based on them. Compound propositions are formed by connecting propositions by <a href="/wiki/Logical_connective">logical connectives</a>. Propositions that contain no logical connectives are called atomic propositions.
</p><p>Unlike <a href="/wiki/First-order_logic">first-order logic</a>, propositional logic does not deal with non-logical objects, predicates about them, or <a href="/wiki/Quantifier_(logic)">quantifiers</a>. However, all the machinery of propositional logic is included in first-order logic and higher-order logics. In this sense, propositional logic is the foundation of first-order logic and higher-order logic.
</p>
<h2>Explanation[<a href="/w/index.php?title=Propositional_calculus&amp;action=edit&amp;section=1">edit</a>]</h2>
<p>Logical connectives are found in natural languages. In English for example, some examples are "and" (<a href="/wiki/Logical_conjunction">conjunction</a>), "or" (<a href="/wiki/Logical_disjunction">disjunction</a>), "not" (<a href="/wiki/Negation">negation</a>) and "if" (but only when used to denote <a href="/wiki/Material_conditional">material conditional</a>).
</p><p>The following is an example of a very simple inference within the scope of propositional logic:
</p>
<dl><dd>Premise 1: If it's raining then it's cloudy.</dd>
<dd>Premise 2: It's raining.</dd>
<dd>Conclusion: It's cloudy.</dd></dl>
<p>Both premises and the conclusion are propositions. The premises are taken for granted, and with the application of <a href="/wiki/Modus_ponens">modus ponens</a> (an <a href="/wiki/Inference_rule">inference rule</a>), the conclusion follows.
</p><p>As propositional logic is not concerned with the structure of propositions beyond the point where they can't be decomposed any more by logical connectives, this inference can be restated replacing those <i>atomic</i> statements with statement letters, which are interpreted as variables representing statements:
</p>
<dl><dd>Premise 1: </dd>
<dd>Premise 2: </dd>
<dd>Conclusion: </dd></dl>
<p>The same can be stated succinctly in the following way:
</p>
<dl><dd></dd></dl>
<p>When P is interpreted as "It's raining" and Q as "it's cloudy" the above symbolic expressions can be seen to correspond exactly with the original expression in natural language. Not only that, but they will also correspond with any other inference of this <i>form</i>, which will be valid on the same basis this inference is.
</p><p>Propositional logic may be studied through a <a href="/wiki/Formal_system">formal system</a> in which <a href="/wiki/Well-formed_formula">formulas</a> of a <a href="/wiki/Formal_language">formal language</a> may be <a href="/wiki/Interpretation_(logic)">interpreted</a> to represent <a href="/wiki/Propositions">propositions</a>. A <a href="/wiki/Deductive_system">system</a> of <a href="/wiki/Axiom">axioms</a> and <a href="/wiki/Rule_of_inference">inference rules</a> allows certain formulas to be derived. These derived formulas are called <a href="/wiki/Theorem">theorems</a> and may be interpreted to be true propositions. A constructed sequence of such formulas is known as a <i><a href="/wiki/Formal_proof">derivation</a></i> or <i>proof</i> and the last formula of the sequence is the theorem. The derivation may be interpreted as proof of the proposition represented by the theorem.
</p><p>When a formal system is used to represent formal logic, only statement letters (usually capital roman letters such as , and ) are represented directly. The natural language propositions that arise when they're interpreted are outside the scope of the system, and the relation between the formal system and its interpretation is likewise outside the formal system itself.
</p><p>In classical <b>truth-functional propositional logic</b>, formulas are interpreted as having precisely one of two possible <a href="/wiki/Truth_value">truth values</a>, the truth value of <i>true</i> or the truth value of <i>false</i>.<sup><a href="#cite_note-1">[1]</a></sup> The <a href="/wiki/Principle_of_bivalence">principle of bivalence</a> and the <a href="/wiki/Law_of_excluded_middle">law of excluded middle</a> are upheld. Truth-functional propositional logic defined as such and systems <a href="/wiki/Isomorphism">isomorphic</a> to it are considered to be <b><a href="/wiki/Zeroth-order_logic">zeroth-order logic</a></b>. However, alternative propositional logics are also possible. For more, see <a href="#Alternative_calculus">Other logical calculi</a> below.
</p>
<h2>History[<a href="/w/index.php?title=Propositional_calculus&amp;action=edit&amp;section=2">edit</a>]</h2>
<p>Although propositional logic (which is interchangeable with propositional calculus) had been hinted by earlier philosophers, it was developed into a formal logic (<a href="/wiki/Stoic_logic">Stoic logic</a>) by <a href="/wiki/Chrysippus">Chrysippus</a> in the 3rd century BC<sup><a href="#cite_note-2">[2]</a></sup> and expanded by his successor <a href="/wiki/Stoics">Stoics</a>. The logic was focused on <a href="/wiki/Proposition">propositions</a>. This advancement was different from the traditional <a href="/wiki/Syllogism">syllogistic logic</a>, which was focused on <a href="/wiki/Syllogisms#Terms_in_syllogism">terms</a>. However, most of the original writings were lost<sup><a href="#cite_note-3">[3]</a></sup> and the propositional logic developed by the Stoics was no longer understood later in antiquity.<sup>[<i><a href="/wiki/Wikipedia:Citation_needed">citation needed</a></i>]</sup> Consequently, the system was essentially reinvented by <a href="/wiki/Peter_Abelard">Peter Abelard</a> in the 12th century.<sup><a href="#cite_note-4">[4]</a></sup>
</p><p>Propositional logic was eventually refined using <a href="/wiki/Symbolic_logic">symbolic logic</a>. The 17th/18th-century mathematician <a href="/wiki/Gottfried_Leibniz">Gottfried Leibniz</a> has been credited with being the founder of symbolic logic for his work with the <a href="/wiki/Calculus_ratiocinator">calculus ratiocinator</a>. Although his work was the first of its kind, it was unknown to the larger logical community. Consequently, many of the advances achieved by Leibniz were recreated by logicians like <a href="/wiki/George_Boole">George Boole</a> and <a href="/wiki/Augustus_De_Morgan">Augustus De Morgan</a>—completely independent of Leibniz.<sup><a href="#cite_note-5">[5]</a></sup>
</p><p>Just as propositional logic can be considered an advancement from the earlier syllogistic logic, <a href="/wiki/Gottlob_Frege">Gottlob Frege's</a> <a href="/wiki/Predicate_logic">predicate logic</a> can be also considered an advancement from the earlier propositional logic. One author describes predicate logic as combining "the distinctive features of syllogistic logic and propositional logic."<sup><a href="#cite_note-6">[6]</a></sup> Consequently, predicate logic ushered in a new era in logic's history; however, advances in propositional logic were still made after Frege, including <a href="/wiki/Natural_deduction">natural deduction</a>, <a href="/wiki/Method_of_analytic_tableaux">truth trees</a> and <a href="/wiki/Truth-table">truth tables</a>. Natural deduction was invented by <a href="/wiki/Gerhard_Gentzen">Gerhard Gentzen</a> and <a href="/wiki/Jan_%C5%81ukasiewicz">Jan Łukasiewicz</a>. Truth trees were invented by <a href="/wiki/Evert_Willem_Beth">Evert Willem Beth</a>.<sup><a href="#cite_note-7">[7]</a></sup> The invention of truth tables, however, is of uncertain attribution.
</p><p>Within works by Frege<sup><a href="#cite_note-Truth_in_Frege-8">[8]</a></sup> and <a href="/wiki/Bertrand_Russell">Bertrand Russell</a>,<sup><a href="#cite_note-Russell_Truth-Tables-9">[9]</a></sup> are ideas influential to the invention of truth tables. The actual tabular structure (being formatted as a table), itself, is generally credited to either <a href="/wiki/Ludwig_Wittgenstein">Ludwig Wittgenstein</a> or <a href="/wiki/Emil_Post">Emil Post</a> (or both, independently).<sup><a href="#cite_note-Truth_in_Frege-8">[8]</a></sup> Besides Frege and Russell, others credited with having ideas preceding truth tables include Philo, Boole, <a href="/wiki/Charles_Sanders_Peirce">Charles Sanders Peirce</a>,<sup><a href="#cite_note-10">[10]</a></sup> and <a href="/wiki/Ernst_Schr%C3%B6der_(mathematician)">Ernst Schröder</a>. Others credited with the tabular structure include <a href="/wiki/Jan_%C5%81ukasiewicz">Jan Łukasiewicz</a>, <a href="/wiki/Alfred_North_Whitehead">Alfred North Whitehead</a>, <a href="/wiki/William_Stanley_Jevons">William Stanley Jevons</a>, <a href="/wiki/John_Venn">John Venn</a>, and <a href="/wiki/Clarence_Irving_Lewis">Clarence Irving Lewis</a>.<sup><a href="#cite_note-Russell_Truth-Tables-9">[9]</a></sup> Ultimately, some have concluded, like John Shosky, that "It is far from clear that any one person should be given the title of 'inventor' of truth-tables.".<sup><a href="#cite_note-Russell_Truth-Tables-9">[9]</a></sup>
</p>
<h2>Terminology[<a href="/w/index.php?title=Propositional_calculus&amp;action=edit&amp;section=3">edit</a>]</h2>
<p>In general terms, a calculus is a <a href="/wiki/Formal_system">formal system</a> that consists of a set of syntactic expressions (<i><a href="/wiki/Well-formed_formula">well-formed formulas</a></i>), a distinguished subset of these expressions (axioms), plus a set of formal rules that define a specific <a href="/wiki/Binary_relation">binary relation</a>, intended to be interpreted as <a href="/wiki/Logical_equivalence">logical equivalence</a>, on the space of expressions.
</p><p>When the formal system is intended to be a <a href="/wiki/Logical_system">logical system</a>, the expressions are meant to be interpreted as statements, and the rules, known to be <i>inference rules</i>, are typically intended to be truth-preserving. In this setting, the rules, which may include <a href="/wiki/Axioms">axioms</a>, can then be used to derive ("infer") formulas representing true statements—from given formulas representing true statements.
</p><p>The set of axioms may be empty, a nonempty finite set, or a countably infinite set (see <a href="/wiki/Axiom_schema">axiom schema</a>). A <a href="/wiki/Formal_grammar">formal grammar</a> recursively defines the expressions and well-formed formulas of the <a href="/wiki/Formal_language">language</a>. In addition a <a href="/wiki/Semantics">semantics</a> may be given which defines truth and <a href="/wiki/Valuation_(logic)">valuations</a> (or <a href="/wiki/Interpretation_(logic)">interpretations</a>).
</p><p>The <a href="/wiki/Formal_language">language</a> of a propositional calculus consists of
</p>
<ol><li>a set of primitive symbols, variously referred to as <i><a href="/wiki/Atomic_formula">atomic formulas</a></i>, <i>placeholders</i>, <i>proposition letters</i>, or <i>variables</i>, and</li>
<li>a set of operator symbols, variously interpreted as <i><a href="/wiki/Logical_operator">logical operators</a></i> or <i>logical connectives</i>.</li></ol>
<p>A <i><a href="/wiki/Well-formed_formula">well-formed formula</a></i> is any atomic formula, or any formula that can be built up from atomic formulas by means of operator symbols according to the rules of the grammar.
</p><p>Mathematicians sometimes distinguish between propositional constants, propositional variables, and schemata. Propositional constants represent some particular proposition, while propositional variables range over the set of all atomic propositions. Schemata, however, range over all propositions. It is common to represent propositional constants by A, B, and C, propositional variables by P, Q, and R, and schematic letters are often Greek letters, most often φ, ψ, and χ.
</p>
<h2>Basic concepts[<a href="/w/index.php?title=Propositional_calculus&amp;action=edit&amp;section=4">edit</a>]</h2>
<p>The following outlines a standard propositional calculus. Many different formulations exist which are all more or less equivalent, but differ in the details of:
</p>
<ol><li>their language (i.e., the particular collection of primitive symbols and operator symbols),</li>
<li>the set of axioms, or distinguished formulas, and</li>
<li>the set of inference rules.</li></ol>
<p>Any given proposition may be represented with a letter called a 'propositional constant', analogous to representing a number by a letter in mathematics (e.g., <i>a</i> = 5). All propositions require exactly one of two truth-values: true or false. For example, let P be the proposition that it is raining outside. This will be true (P) if it is raining outside, and false otherwise (¬<i>P</i>).
</p>
<ul><li>We then define <a href="/wiki/Truth-functional">truth-functional</a> operators, beginning with negation. ¬<i>P</i> represents the negation of P, which can be thought of as the denial of P. In the example above, ¬<i>P</i> expresses that it is not raining outside, or by a more standard reading: "It is not the case that it is raining outside." When P is true, ¬<i>P</i> is false; and when P is false, ¬<i>P</i> is true. As a result, ¬ ¬<i>P</i> always has the same truth-value as P.</li>
<li>Conjunction is a truth-functional connective which forms a proposition out of two simpler propositions, for example, P and Q. The conjunction of P and Q is written <i>P</i> ∧ <i>Q</i>, and expresses that each are true. We read <i>P</i> ∧ <i>Q</i> as "P and Q". For any two propositions, there are four possible assignments of truth values:
<ol><li>P is true and Q is true</li>
<li>P is true and Q is false</li>
<li>P is false and Q is true</li>
<li>P is false and Q is false</li></ol></li></ul>
<dl><dd>The conjunction of P and Q is true in case 1, and is false otherwise. Where P is the proposition that it is raining outside and Q is the proposition that a cold-front is over Kansas, <i>P</i> ∧ <i>Q</i> is true when it is raining outside <i>and</i> there is a cold-front over Kansas. If it is not raining outside, then P ∧ Q is false; and if there is no cold-front over Kansas, then <i>P</i> ∧ <i>Q</i> is also false.</dd></dl>
<ul><li>Disjunction resembles conjunction in that it forms a proposition out of two simpler propositions. We write it <i>P</i> ∨ <i>Q</i>, and it is read "P or Q". It expresses that either P or Q is true. Thus, in the cases listed above, the disjunction of P with Q is true in all cases—except case 4. Using the example above, the disjunction expresses that it is either raining outside, or there is a cold front over Kansas. (Note, this use of disjunction is supposed to resemble the use of the English word "or". However, it is most like the English <a href="/wiki/Inclusive_disjunction">inclusive</a> "or", which can be used to express the truth of at least one of two propositions. It is not like the English <a href="/wiki/Exclusive_disjunction">exclusive</a> "or", which expresses the truth of exactly one of two propositions. In other words, the exclusive "or" is false when both P and Q are true (case 1), and similarly is false when both P and Q are false (case 4). An example of the exclusive or is: You may have a bagel or a pastry, but not both. Often in natural language, given the appropriate context, the addendum "but not both" is omitted—but implied. In mathematics, however, "or" is always inclusive or; if exclusive or is meant it will be specified, possibly by "xor".)</li>
<li>Material conditional also joins two simpler propositions, and we write <i>P</i> → <i>Q</i>, which is read "if P then Q". The proposition to the left of the arrow is called the antecedent, and the proposition to the right is called the consequent. (There is no such designation for conjunction or disjunction, since they are <a href="/wiki/Commutative_property">commutative</a> operations.) It expresses that Q is true whenever P is true. Thus <i>P</i> → <i>Q</i> is true in every case above except case 2, because this is the only case when P is true but Q is not. Using the example, if P then Q expresses that if it is raining outside, then there is a cold-front over Kansas. The material conditional is often confused with physical causation. The material conditional, however, only relates two propositions by their truth-values—which is not the relation of cause and effect. It is contentious in the literature whether the material implication represents logical causation.</li>
<li>Biconditional joins two simpler propositions, and we write <i>P</i> ↔ <i>Q</i>, which is read "P if and only if Q". It expresses that P and Q have the same truth-value, and in cases 1 and 4. 'P is true if and only if Q' is true, and is false otherwise.</li></ul>
<p>It is very helpful to look at the <a href="/wiki/Truth_table">truth tables</a> for these different operators, as well as the <a href="/wiki/Method_of_analytic_tableaux">method of analytic tableaux</a>.
</p>
<h3>Closure under operations[<a href="/w/index.php?title=Propositional_calculus&amp;action=edit&amp;section=5">edit</a>]</h3>
<p>Propositional logic is <a href="/wiki/Closure_(mathematics)">closed</a> under truth-functional connectives. That is to say, for any proposition φ, ¬<i>φ</i> is also a proposition. Likewise, for any propositions φ and ψ, <i>φ</i> ∧ <i>ψ</i> is a proposition, and similarly for disjunction, conditional, and biconditional. This implies that, for instance, <i>φ</i> ∧ <i>ψ</i> is a proposition, and so it can be conjoined with another proposition. In order to represent this, we need to use parentheses to indicate which proposition is conjoined with which. For instance, <i>P</i> ∧ <i>Q</i> ∧ <i>R</i> is not a well-formed formula, because we do not know if we are conjoining <i>P</i> ∧ <i>Q</i> with R or if we are conjoining P with <i>Q</i> ∧ <i>R</i>. Thus we must write either (<i>P</i> ∧ <i>Q</i>) ∧ <i>R</i> to represent the former, or <i>P</i> ∧ (<i>Q</i> ∧ <i>R</i>) to represent the latter. By evaluating the truth conditions, we see that both expressions have the same truth conditions (will be true in the same cases), and moreover that any proposition formed by arbitrary conjunctions will have the same truth conditions, regardless of the location of the parentheses. This means that conjunction is <a href="/wiki/Associative_property">associative</a>, however, one should not assume that parentheses never serve a purpose. For instance, the sentence <i>P</i> ∧ (<i>Q</i> ∨ <i>R</i>) does not have the same truth conditions of (<i>P</i> ∧ <i>Q</i>) ∨ <i>R</i>, so they are different sentences distinguished only by the parentheses. One can verify this by the truth-table method referenced above.
</p><p>Note: For any arbitrary number of propositional constants, we can form a finite number of cases which list their possible truth-values. A simple way to generate this is by truth-tables, in which one writes P, Q, ..., Z, for any list of k propositional constants—that is to say, any list of propositional constants with k entries. Below this list, one writes 2<sup><i>k</i></sup> rows, and below P one fills in the first half of the rows with true (or T) and the second half with false (or F). Below Q one fills in one-quarter of the rows with T, then one-quarter with F, then one-quarter with T and the last quarter with F. The next column alternates between true and false for each eighth of the rows, then sixteenths, and so on, until the last propositional constant varies between T and F for each row. This will give a complete listing of cases or truth-value assignments possible for those propositional constants.
</p>
<h3>Argument[<a href="/w/index.php?title=Propositional_calculus&amp;action=edit&amp;section=6">edit</a>]</h3>
<p>The propositional calculus then defines an <i><a href="/wiki/Argument">argument</a></i> to be a list of propositions. A valid argument is a list of propositions, the last of which follows from—or is implied by—the rest. All other arguments are invalid. The simplest valid argument is <a href="/wiki/Modus_ponens">modus ponens</a>, one instance of which is the following list of propositions:
</p>
<dl><dd></dd></dl>
<p>This is a list of three propositions, each line is a proposition, and the last follows from the rest. The first two lines are called premises, and the last line the conclusion. We say that any proposition C follows from any set of propositions , if C must be true whenever every member of the set is true. In the argument above, for any P and Q, whenever <i>P</i> → <i>Q</i> and P are true, necessarily Q is true. Notice that, when P is true, we cannot consider cases 3 and 4 (from the truth table). When <i>P</i> → <i>Q</i> is true, we cannot consider case 2. This leaves only case 1, in which Q is also true. Thus Q is implied by the premises.
</p><p>This generalizes schematically. Thus, where φ and ψ may be any propositions at all,
</p>
<dl><dd></dd></dl>
<p>Other argument forms are convenient, but not necessary. Given a complete set of axioms (see below for one such set), modus ponens is sufficient to prove all other argument forms in propositional logic, thus they may be considered to be a derivative. Note, this is not true of the extension of propositional logic to other logics like <a href="/wiki/First-order_logic">first-order logic</a>. First-order logic requires at least one additional rule of inference in order to obtain <a href="/wiki/Completeness_(logic)">completeness</a>.
</p><p>The significance of argument in formal logic is that one may obtain new truths from established truths. In the first example above, given the two premises, the truth of Q is not yet known or stated. After the argument is made, Q is deduced. In this way, we define a deduction system to be a set of all propositions that may be deduced from another set of propositions. For instance, given the set of propositions , we can define a deduction system, Γ, which is the set of all propositions which follow from A. <a href="/wiki/Deduction_theorem#Virtual_rules_of_inference">Reiteration</a> is always assumed, so . Also, from the first element of A, last element, as well as modus ponens, R is a consequence, and so . Because we have not included sufficiently complete axioms, though, nothing else may be deduced. Thus, even though most deduction systems studied in propositional logic are able to deduce , this one is too weak to prove such a proposition.
</p>
<h2>Generic description of a propositional calculus[<a href="/w/index.php?title=Propositional_calculus&amp;action=edit&amp;section=7">edit</a>]</h2>
<p>A <b>propositional calculus</b> is a <a href="/wiki/Formal_system">formal system</a> , where:
</p>
<div><ul><li>The <i>alpha set</i> is a countably infinite set of elements called <i>proposition symbols</i> or <i><a href="/wiki/Propositional_variable">propositional variables</a></i>. Syntactically speaking, these are the most basic elements of the formal language , otherwise referred to as <i><a href="/wiki/Atomic_formula">atomic formulas</a></i> or <i>terminal elements</i>. In the examples to follow, the elements of are typically the letters p, q, r, and so on.</li><li>The <i>omega set</i> Ω is a finite set of elements called <i><a href="/wiki/Operation_(mathematics)">operator symbols</a></i> or <i><a href="/wiki/Logical_connective">logical connectives</a></i>. The set Ω is <a href="/wiki/Partition_of_a_set">partitioned</a> into disjoint subsets as follows:
<dl><dd></dd></dl>
<p>In this partition, is the set of operator symbols of <i><a href="/wiki/Arity">arity</a></i> j.
</p><p>In the more familiar propositional calculi, Ω is typically partitioned as follows:
</p>
<dl><dd></dd>
<dd></dd></dl>
<p>A frequently adopted convention treats the constant <a href="/wiki/Logical_value">logical values</a> as operators of arity zero, thus:
</p>
<dl><dd></dd></dl>
Some writers use the <a href="/wiki/Tilde">tilde</a> (~), or N, instead of ¬; and some use v instead of as well as the <a href="/wiki/Ampersand">ampersand</a> (&amp;), the prefixed K, or instead of . Notation varies even more for the set of logical values, with symbols like {false, true}, {F, T}, or {0, 1} all being seen in various contexts instead of .</li><li>The <i>zeta set</i> is a finite set of <i>transformation rules</i> that are called <i><a href="/wiki/Inference_rule">inference rules</a></i> when they acquire logical applications.</li><li>The <i>iota set</i> is a countable set of <i>initial points</i> that are called <i><a href="/wiki/Axiom">axioms</a></i> when they receive logical interpretations.</li></ul></div>
<p>The <i>language</i> of , also known as its set of <i>formulas</i>, <i><a href="/wiki/Well-formed_formula">well-formed formulas</a></i>, is <a href="/wiki/Inductive_definition">inductively defined</a> by the following rules:
</p>
<ol><li>Base: Any element of the alpha set is a formula of .</li>
<li>If are formulas and is in , then is a formula.</li>
<li>Closed: Nothing else is a formula of .</li></ol>
<p>Repeated applications of these rules permits the construction of complex formulas. For example:
</p>
<h2>Example 1. Simple axiom system[<a href="/w/index.php?title=Propositional_calculus&amp;action=edit&amp;section=8">edit</a>]</h2>
<p>Let , where , , , are defined as follows:
</p>
<p>Then is defined as , and is defined as .
</p>
<p>This system is used in <a href="/wiki/Metamath">Metamath</a> <a href="http://us.metamath.org/mpegif/mmset.html#scaxioms">set.mm</a> formal proof database.
</p>
<h2>Example 2. Natural deduction system[<a href="/w/index.php?title=Propositional_calculus&amp;action=edit&amp;section=9">edit</a>]</h2>
<p>Let , where , , , are defined as follows:
</p>
<p>In the following example of a propositional calculus, the transformation rules are intended to be interpreted as the inference rules of a so-called <i><a href="/wiki/Natural_deduction_system">natural deduction system</a></i>. The particular system presented here has no initial points, which means that its interpretation for logical applications derives its <a href="/wiki/Theorem">theorems</a> from an empty axiom set.
</p>
<p>Our propositional calculus has eleven inference rules. These rules allow us to derive other true formulas given a set of formulas that are assumed to be true. The first ten simply state that we can infer certain well-formed formulas from other well-formed formulas. The last rule however uses hypothetical reasoning in the sense that in the premise of the rule we temporarily assume an (unproven) hypothesis to be part of the set of inferred formulas to see if we can infer a certain other formula. Since the first ten rules don't do this they are usually described as <i>non-hypothetical</i> rules, and the last one as a <i>hypothetical</i> rule.
</p><p>In describing the transformation rules, we may introduce a metalanguage symbol . It is basically a convenient shorthand for saying "infer that". The format is , in which Γ is a (possibly empty) set of formulas called premises, and ψ is a formula called conclusion. The transformation rule means that if every proposition in Γ is a theorem (or has the same truth value as the axioms), then ψ is also a theorem. Note that considering the following rule <a href="/wiki/Conjunction_introduction">Conjunction introduction</a>, we will know whenever Γ has more than one formula, we can always safely reduce it into one formula using conjunction. So for short, from that time on we may represent Γ as one formula instead of a set. Another omission for convenience is when Γ is an empty set, in which case Γ may not appear.
</p>
<dl><dt><a href="/wiki/Negation_introduction">Negation introduction</a></dt>
<dd>From and , infer .</dd>
<dd>That is, .</dd>
<dt><a href="/wiki/Negation_elimination">Negation elimination</a></dt>
<dd>From , infer .</dd>
<dd>That is, .</dd>
<dt><a href="/wiki/Double_negation_elimination">Double negation elimination</a></dt>
<dd>From , infer p.</dd>
<dd>That is, .</dd>
<dt><a href="/wiki/Conjunction_introduction">Conjunction introduction</a></dt>
<dd>From p and q, infer .</dd>
<dd>That is, .</dd>
<dt><a href="/wiki/Conjunction_elimination">Conjunction elimination</a></dt>
<dd>From , infer p.</dd>
<dd>From , infer q.</dd>
<dd>That is, and .</dd>
<dt><a href="/wiki/Disjunction_introduction">Disjunction introduction</a></dt>
<dd>From p, infer .</dd>
<dd>From q, infer .</dd>
<dd>That is, and .</dd>
<dt><a href="/wiki/Disjunction_elimination">Disjunction elimination</a></dt>
<dd>From and and , infer r.</dd>
<dd>That is, .</dd>
<dt><a href="/wiki/Biconditional_introduction">Biconditional introduction</a></dt>
<dd>From and , infer .</dd>
<dd>That is, .</dd>
<dt><a href="/wiki/Biconditional_elimination">Biconditional elimination</a></dt>
<dd>From , infer .</dd>
<dd>From , infer .</dd>
<dd>That is, and .</dd>
<dt><a href="/wiki/Modus_ponens">Modus ponens</a> (conditional elimination)</dt>
<dd>From p and , infer q.</dd>
<dd>That is, .</dd>
<dt><a href="/wiki/Conditional_proof">Conditional proof</a> (conditional introduction)</dt>
<dd>From [accepting p allows a proof of q], infer .</dd>
<dd>That is, .</dd></dl>
<h2>Basic and derived argument forms[<a href="/w/index.php?title=Propositional_calculus&amp;action=edit&amp;section=10">edit</a>]</h2>
<table>
<tbody><tr>
<th>Name
</th>
<th>Sequent
</th>
<th>Description
</th></tr>
<tr>
<td><a href="/wiki/Modus_Ponens">Modus Ponens</a>
</td>
<td>
</td>
<td>If p then q; p; therefore q
</td></tr>
<tr>
<td><a href="/wiki/Modus_Tollens">Modus Tollens</a>
</td>
<td>
</td>
<td>If p then q; not q; therefore not p
</td></tr>
<tr>
<td><a href="/wiki/Hypothetical_Syllogism">Hypothetical Syllogism</a>
</td>
<td>
</td>
<td>If p then q; if q then r; therefore, if p then r
</td></tr>
<tr>
<td><a href="/wiki/Disjunctive_syllogism">Disjunctive Syllogism</a>
</td>
<td>
</td>
<td>Either p or q, or both; not p; therefore, q
</td></tr>
<tr>
<td><a href="/wiki/Constructive_dilemma">Constructive Dilemma</a>
</td>
<td>
</td>
<td>If p then q; and if r then s; but p or r; therefore q or s
</td></tr>
<tr>
<td><a href="/wiki/Destructive_dilemma">Destructive Dilemma</a>
</td>
<td>
</td>
<td>If p then q; and if r then s; but not q or not s; therefore not p or not r
</td></tr>
<tr>
<td>Bidirectional Dilemma
</td>
<td>
</td>
<td>If p then q; and if r then s; but p or not s; therefore q or not r
</td></tr>
<tr>
<td><a href="/wiki/Conjunction_elimination">Simplification</a>
</td>
<td>
</td>
<td>p and q are true; therefore p is true
</td></tr>
<tr>
<td><a href="/wiki/Logical_conjunction">Conjunction</a>
</td>
<td>
</td>
<td>p and q are true separately; therefore they are true conjointly
</td></tr>
<tr>
<td><a href="/wiki/Logical_disjunction">Addition</a>
</td>
<td>
</td>
<td>p is true; therefore the disjunction (p or q) is true
</td></tr>
<tr>
<td><a href="/wiki/Distributive_property">Composition</a>
</td>
<td>
</td>
<td>If p then q; and if p then r; therefore if p is true then q and r are true
</td></tr>
<tr>
<td><a href="/wiki/De_Morgan%27s_laws">De Morgan's Theorem</a> (1)
</td>
<td>
</td>
<td>The negation of (p and q) is equiv. to (not p or not q)
</td></tr>
<tr>
<td><a href="/wiki/De_Morgan%27s_laws">De Morgan's Theorem</a> (2)
</td>
<td>
</td>
<td>The negation of (p or q) is equiv. to (not p and not q)
</td></tr>
<tr>
<td><a href="/wiki/Commutative_property">Commutation</a> (1)
</td>
<td>
</td>
<td>(p or q) is equiv. to (q or p)
</td></tr>
<tr>
<td><a href="/wiki/Commutative_property">Commutation</a> (2)
</td>
<td>
</td>
<td>(p and q) is equiv. to (q and p)
</td></tr>
<tr>
<td><a href="/wiki/Commutative_property">Commutation</a> (3)
</td>
<td>
</td>
<td>(p is equiv. to q) is equiv. to (q is equiv. to p)
</td></tr>
<tr>
<td><a href="/wiki/Associative_property">Association</a> (1)
</td>
<td>
</td>
<td>p or (q or r) is equiv. to (p or q) or r
</td></tr>
<tr>
<td><a href="/wiki/Associative_property">Association</a> (2)
</td>
<td>
</td>
<td>p and (q and r) is equiv. to (p and q) and r
</td></tr>
<tr>
<td><a href="/wiki/Distributive_property">Distribution</a> (1)
</td>
<td>
</td>
<td>p and (q or r) is equiv. to (p and q) or (p and r)
</td></tr>
<tr>
<td><a href="/wiki/Distributive_property">Distribution</a> (2)
</td>
<td>
</td>
<td>p or (q and r) is equiv. to (p or q) and (p or r)
</td></tr>
<tr>
<td><a href="/wiki/Double_negation_elimination">Double Negation</a>
</td>
<td> and 
</td>
<td>p is equivalent to the negation of not p
</td></tr>
<tr>
<td><a href="/wiki/Transposition_(logic)">Transposition</a>
</td>
<td>
</td>
<td>If p then q is equiv. to if not q then not p
</td></tr>
<tr>
<td><a href="/wiki/Material_implication_(rule_of_inference)">Material Implication</a>
</td>
<td>
</td>
<td>If p then q is equiv. to not p or q
</td></tr>
<tr>
<td><a href="/wiki/Material_equivalence">Material Equivalence</a> (1)
</td>
<td>
</td>
<td>(p iff q) is equiv. to (if p is true then q is true) and (if q is true then p is true)
</td></tr>
<tr>
<td><a href="/wiki/Material_equivalence">Material Equivalence</a> (2)
</td>
<td>
</td>
<td>(p iff q) is equiv. to either (p and q are true) or (both p and q are false)
</td></tr>
<tr>
<td><a href="/wiki/Material_equivalence">Material Equivalence</a> (3)
</td>
<td>
</td>
<td>(p iff q) is equiv to., both (p or not q is true) and (not p or q is true)
</td></tr>
<tr>
<td><a href="/wiki/Exportation_(logic)">Exportation</a><sup><a href="#cite_note-12">[12]</a></sup>
</td>
<td>
</td>
<td>from (if p and q are true then r is true) we can prove (if q is true then r is true, if p is true)
</td></tr>
<tr>
<td><a href="/wiki/Exportation_(logic)">Importation</a>
</td>
<td>
</td>
<td>If p then (if q then r) is equivalent to if p and q then r
</td></tr>
<tr>
<td><a href="/wiki/Tautology_(rule_of_inference)">Tautology</a> (1)
</td>
<td>
</td>
<td>p is true is equiv. to p is true or p is true
</td></tr>
<tr>
<td><a href="/wiki/Tautology_(rule_of_inference)">Tautology</a> (2)
</td>
<td>
</td>
<td>p is true is equiv. to p is true and p is true
</td></tr>
<tr>
<td><a href="/wiki/Law_of_excluded_middle">Tertium non datur (Law of Excluded Middle)</a>
</td>
<td>
</td>
<td>p or not p is true
</td></tr>
<tr>
<td><a href="/wiki/Law_of_noncontradiction">Law of Non-Contradiction</a>
</td>
<td>
</td>
<td>p and not p is false, is a true statement
</td></tr></tbody></table>
<h2>Proofs in propositional calculus[<a href="/w/index.php?title=Propositional_calculus&amp;action=edit&amp;section=11">edit</a>]</h2>
<p>One of the main uses of a propositional calculus, when interpreted for logical applications, is to determine relations of logical equivalence between propositional formulas. These relationships are determined by means of the available transformation rules, sequences of which are called <i>derivations</i> or <i>proofs</i>.
</p><p>In the discussion to follow, a proof is presented as a sequence of numbered lines, with each line consisting of a single formula followed by a <i>reason</i> or <i>justification</i> for introducing that formula. Each premise of the argument, that is, an assumption introduced as an hypothesis of the argument, is listed at the beginning of the sequence and is marked as a "premise" in lieu of other justification. The conclusion is listed on the last line. A proof is complete if every line follows from the previous ones by the correct application of a transformation rule. (For a contrasting approach, see <a href="/wiki/Method_of_analytic_tableaux">proof-trees</a>).
</p>
<h3>Example of a proof in natural deduction system[<a href="/w/index.php?title=Propositional_calculus&amp;action=edit&amp;section=12">edit</a>]</h3>
<ul><li>To be shown that <i>A</i> → <i>A</i>.</li>
<li>One possible proof of this (which, though valid, happens to contain more steps than are necessary) may be arranged as follows:</li></ul>
<table>
<caption>Example of a proof
</caption>
<tbody><tr>
<th>Number
</th>
<th>Formula
</th>
<th>Reason
</th></tr>
<tr>
<td>1</td>
<td></td>
<td>premise
</td></tr>
<tr>
<td>2</td>
<td></td>
<td>From (<b><a href="#math_1">1</a></b>) by disjunction introduction
</td></tr>
<tr>
<td>3</td>
<td></td>
<td>From (<b><a href="#math_1">1</a></b>) and (<b><a href="#math_2">2</a></b>) by conjunction introduction
</td></tr>
<tr>
<td>4</td>
<td></td>
<td>From (<b><a href="#math_3">3</a></b>) by conjunction elimination
</td></tr>
<tr>
<td>5</td>
<td></td>
<td>Summary of (<b><a href="#math_1">1</a></b>) through (<b><a href="#math_4">4</a></b>)
</td></tr>
<tr>
<td>6</td>
<td></td>
<td>From (<b><a href="#math_5">5</a></b>) by conditional proof
</td></tr></tbody></table>
<p>Interpret as "Assuming A, infer A". Read as "Assuming nothing, infer that A implies A", or "It is a tautology that A implies A", or "It is always true that A implies A".
</p>
<h3>Example of a proof in a classical propositional calculus system[<a href="/w/index.php?title=Propositional_calculus&amp;action=edit&amp;section=13">edit</a>]</h3>
<p>We now prove the same theorem in the axiomatic system by <a href="/wiki/Jan_%C5%81ukasiewicz">Jan Łukasiewicz</a> described above, which is an example of a <a href="/wiki/Hilbert-style_deductive_system">Hilbert-style deductive system</a> for the classical propositional calculus.
</p><p>The axioms are:
</p>
<dl><dd>(A1) </dd>
<dd>(A2) </dd>
<dd>(A3) </dd></dl>
<p>And the proof is as follows:
</p>
<ol><li>       (instance of (A1))</li>
<li>       (instance of (A2))</li>
<li>       (from (1) and (2) by <a href="/wiki/Modus_ponens">modus ponens</a>)</li>
<li>       (instance of (A1))</li>
<li>       (from (4) and (3) by modus ponens)</li></ol>
<h2>Soundness and completeness of the rules[<a href="/w/index.php?title=Propositional_calculus&amp;action=edit&amp;section=14">edit</a>]</h2>
<p>The crucial properties of this set of rules are that they are <i><a href="/wiki/Soundness">sound</a></i> and <i>complete</i>. Informally this means that the rules are correct and that no other rules are required. These claims can be made more formal as follows.
Note that the proofs for the soundness and completeness of the propositional logic are not themselves proofs in propositional logic ; these are theorems in <a href="/wiki/ZFC">ZFC</a> used as a <a href="/wiki/Metatheory#In_mathematics">metatheory</a> to prove properties of propositional logic.
</p><p>We define a <i>truth assignment</i> as a <a href="/wiki/Function_(mathematics)">function</a> that maps propositional variables to <b>true</b> or <b>false</b>. Informally such a truth assignment can be understood as the description of a possible <a href="/wiki/State_of_affairs_(philosophy)">state of affairs</a> (or <a href="/wiki/Possible_world">possible world</a>) where certain statements are true and others are not. The semantics of formulas can then be formalized by defining for which "state of affairs" they are considered to be true, which is what is done by the following definition.
</p><p>We define when such a truth assignment A satisfies a certain <a href="/wiki/Well-formed_formula">well-formed formula</a> with the following rules:
</p>
<ul><li>A satisfies the propositional variable P <a href="/wiki/If_and_only_if">if and only if</a> <i>A</i>(<i>P</i>) = true</li>
<li>A satisfies ¬<i>φ</i> if and only if A does not satisfy φ</li>
<li>A satisfies (<i>φ</i> ∧ <i>ψ</i>) if and only if A satisfies both φ and ψ</li>
<li>A satisfies (<i>φ</i> ∨ <i>ψ</i>) if and only if A satisfies at least one of either φ or ψ</li>
<li>A satisfies (<i>φ</i> → <i>ψ</i>) if and only if it is not the case that A satisfies φ but not ψ</li>
<li>A satisfies (<i>φ</i> ↔ <i>ψ</i>) if and only if A satisfies both φ and ψ or satisfies neither one of them</li></ul>
<p>With this definition we can now formalize what it means for a formula φ to be implied by a certain set S of formulas. Informally this is true if in all worlds that are possible given the set of formulas S the formula φ also holds. This leads to the following formal definition: We say that a set S of well-formed formulas <i>semantically entails</i> (or <i>implies</i>) a certain well-formed formula φ if all truth assignments that satisfy all the formulas in S also satisfy φ.
</p><p>Finally we define <i>syntactical entailment</i> such that φ is syntactically entailed by S if and only if we can derive it with the inference rules that were presented above in a finite number of steps. This allows us to formulate exactly what it means for the set of inference rules to be sound and complete:
</p><p><b>Soundness:</b> If the set of well-formed formulas S <i>syntactically</i> entails the well-formed formula φ then S <i>semantically</i> entails φ.
</p><p><b>Completeness:</b> If the set of well-formed formulas S <i>semantically</i> entails the well-formed formula φ then S <i>syntactically</i> entails φ.
</p><p>For the above set of rules this is indeed the case.
</p>
<h3>Sketch of a soundness proof[<a href="/w/index.php?title=Propositional_calculus&amp;action=edit&amp;section=15">edit</a>]</h3>
<p>(For most <a href="/wiki/Logical_system">logical systems</a>, this is the comparatively "simple" direction of proof)
</p><p>Notational conventions: Let G be a variable ranging over sets of sentences. Let A, B and C range over sentences. For "G syntactically entails A" we write "G proves A". For "G semantically entails A" we write "G implies A".
</p><p>We want to show: (<i>A</i>)(<i>G</i>) (if G proves A, then G implies A).
</p><p>We note that "G proves A" has an inductive definition, and that gives us the immediate resources for demonstrating claims of the form "If G proves A, then ...". So our proof proceeds by induction.
</p>
<div><ol><li>Basis. Show: If A is a member of G, then G implies A.</li><li>Basis. Show: If A is an axiom, then G implies A.</li><li>Inductive step (induction on n, the length of the proof):
<div><ol><li>Assume for arbitrary G and A that if G proves A in n or fewer steps, then G implies A.</li><li>For each possible application of a rule of inference at step <i>n</i> + 1, leading to a new theorem B, show that G implies B.</li></ol></div></li></ol></div>
<p>Notice that Basis Step II can be omitted for <a href="/wiki/Natural_deduction">natural deduction</a> systems because they have no axioms. When used, Step II involves showing that each of the axioms is a (semantic) <a href="/wiki/Logical_truth">logical truth</a>.
</p><p>The Basis steps demonstrate that the simplest provable sentences from G are also implied by G, for any G. (The proof is simple, since the semantic fact that a set implies any of its members, is also trivial.) The Inductive step will systematically cover all the further sentences that might be provable—by considering each case where we might reach a logical conclusion using an inference rule—and shows that if a new sentence is provable, it is also logically implied. (For example, we might have a rule telling us that from "A" we can derive "A or B". In III.a We assume that if A is provable it is implied. We also know that if A is provable then "A or B" is provable. We have to show that then "A or B" too is implied. We do so by appeal to the semantic definition and the assumption we just made. A is provable from G, we assume. So it is also implied by G. So any semantic valuation making all of G true makes A true. But any valuation making A true makes "A or B" true, by the defined semantics for "or". So any valuation which makes all of G true makes "A or B" true. So "A or B" is implied.) Generally, the Inductive step will consist of a lengthy but simple <a href="/wiki/Proof_by_cases">case-by-case analysis</a> of all the rules of inference, showing that each "preserves" semantic implication.
</p><p>By the definition of provability, there are no sentences provable other than by being a member of G, an axiom, or following by a rule; so if all of those are semantically implied, the deduction calculus is sound.
</p>
<h3>Sketch of completeness proof[<a href="/w/index.php?title=Propositional_calculus&amp;action=edit&amp;section=16">edit</a>]</h3>
<p>(This is usually the much harder direction of proof.)
</p><p>We adopt the same notational conventions as above.
</p><p>We want to show: If G implies A, then G proves A. We proceed by <a href="/wiki/Contraposition">contraposition</a>: We show instead that if G does <b>not</b> prove A then G does <b>not</b> imply A. If we show that there is a <a href="/wiki/Mathematical_model">model</a> where A does not hold despite G being true, then obviously G does not imply A. The idea is to build such a model out of our very assumption that G does not prove A.
</p>
<div><ol><li>G does not prove A. (Assumption)</li><li>If G does not prove A, then we can construct an (infinite) <b>Maximal Set</b>, <i>G</i><sup>∗</sup>, which is a superset of G and which also does not prove A.
 <div><ol><li>Place an ordering (with <a href="/wiki/Order_type">order type</a> ω) on all the sentences in the language (e.g., shortest first, and equally long ones in extended alphabetical ordering), and number them (<i>E</i><sub>1</sub>, <i>E</i><sub>2</sub>, ...)</li><li>Define a series G<sub>n</sub> of sets (<i>G</i><sub>0</sub>, <i>G</i><sub>1</sub>, ...) inductively:
 </li><li>Define <i>G</i><sup>∗</sup> as the union of all the G<sub>n</sub>. (That is, <i>G</i><sup>∗</sup> is the set of all the sentences that are in any G<sub>n</sub>.)</li><li>It can be easily shown that
 <div><ol><li><i>G</i><sup>∗</sup> contains (is a superset of) G (by (b.i));</li><li><i>G</i><sup>∗</sup> does not prove A (because the proof would contain only finitely many sentences and when the last of them is introduced in some G<sub>n</sub>, that G<sub>n</sub> would prove A contrary to the definition of G<sub>n</sub>); and</li><li><i>G</i><sup>∗</sup> is a Maximal Set with respect to A: If any more sentences whatever were added to <i>G</i><sup>∗</sup>, it would prove A. (Because if it were possible to add any more sentences, they should have been added when they were encountered during the construction of the G<sub>n</sub>, again by definition)</li></ol></div></li></ol></div></li><li>If <i>G</i><sup>∗</sup> is a Maximal Set with respect to A, then it is <b>truth-like</b>. This means that it contains C if and only if it does <b>not</b> contain ¬C; If it contains C and contains "If C then B" then it also contains B; and so forth. In order to show this, one has to show the axiomatic system is strong enough for the following:
<ul><li>For any formulas C and D, if it proves both C and ¬C, then it proves D. From this it follows, that a Maximal Set with respect to A cannot prove both C and ¬C, as otherwise it would prove A.</li>
<li>For any formulas C and D, if it proves both C→D and ¬C→D, then it proves D. This is used, together with the <a href="/wiki/Deduction_theorem">deduction theorem</a>, to show that for any formula, either it or its negation is in <i>G</i><sup>∗</sup>: Let B be a formula not in <i>G</i><sup>∗</sup>; then <i>G</i><sup>∗</sup> with the addition of B proves A. Thus from the deduction theorem it follows that <i>G</i><sup>∗</sup> proves B→A. But suppose ¬B were also not in <i>G</i><sup>∗</sup>, then by the same logic <i>G</i><sup>∗</sup> also proves ¬B→A; but then <i>G</i><sup>∗</sup> proves A, which we have already shown to be false.</li>
<li>For any formulas C and D, if it proves C and D, then it proves C→D.</li>
<li>For any formulas C and D, if it proves C and ¬D, then it proves ¬(C→D).</li>
<li>For any formulas C and D, if it proves ¬C, then it proves C→D.</li></ul>
If additional logical operation (such as conjunction and/or disjunction) are part of the vocabulary as well, then there are additional requirement on the axiomatic system (e.g. that if it proves C and D, it would also prove their conjunction).</li><li>If <i>G</i><sup>∗</sup> is truth-like there is a <i>G</i><sup>∗</sup>-Canonical valuation of the language: one that makes every sentence in <i>G</i><sup>∗</sup> true and everything outside <i>G</i><sup>∗</sup> false while still obeying the laws of semantic composition in the language. Note that the requirement that it is truth-like is needed to guarantee that the laws of semantic composition in the language will be satisfied by this truth assignment.</li><li>A <i>G</i><sup>∗</sup>-canonical valuation will make our original set G all true, and make A false.</li><li>If there is a valuation on which G are true and A is false, then G does not (semantically) imply A.</li></ol></div>
<p>Thus every system that has modus ponens as an inference rule, and proves the following theorems (including substitutions thereof) is complete:
</p>
<p>The first five are used for the satisfaction of the five conditions in stage III above, and the last three for proving the deduction theorem.
</p>
<h4>Example[<a href="/w/index.php?title=Propositional_calculus&amp;action=edit&amp;section=17">edit</a>]</h4>
<p>As an example, it can be shown that as any other tautology, the three axioms of the classical propositional calculus system described earlier can be proven in any system that satisfies the above, namely that has modus ponens as an inference rule, and proves the above eight theorems (including substitutions thereof). Out of the eight theorems, the last two are two of the three axioms; the third axiom, , can be proven as well, as we now show.
</p><p>For the proof we may use the <a href="/wiki/Hypothetical_syllogism#Proof_2">hypothetical syllogism theorem</a> (in the form relevant for this axiomatic system), since it only relies on the two axioms that are already in the above set of eight theorems.
The proof then is as follows:
</p>
<ol><li>       (instance of the 7th theorem)</li>
<li>       (instance of the 7th theorem)</li>
<li>       (from (1) and (2) by modus ponens)</li>
<li>       (instance of the hypothetical syllogism theorem)</li>
<li>       (instance of the 5th theorem)</li>
<li>       (from (5) and (4) by modus ponens)</li>
<li>       (instance of the 2nd theorem)</li>
<li>       (instance of the 7th theorem)</li>
<li>       (from (7) and (8) by modus ponens)</li>
<li>
<dl><dd><dl><dd>       (instance of the 8th theorem)</dd></dl></dd></dl></li>
<li>       (from (9) and (10) by modus ponens)</li>
<li>       (from (3) and (11) by modus ponens)</li>
<li>       (instance of the 8th theorem)</li>
<li>       (from (12) and (13) by modus ponens)</li>
<li>       (from (6) and (14) by modus ponens)</li></ol>
<h4>Verifying completeness for the classical propositional calculus system[<a href="/w/index.php?title=Propositional_calculus&amp;action=edit&amp;section=18">edit</a>]</h4>
<p>We now verify that the classical propositional calculus system described earlier can indeed prove the required eight theorems mentioned above. We use several lemmas proven <a href="/wiki/Hilbert_system#Some_useful_theorems_and_their_proofs">here</a>:
</p>
<dl><dd>(DN1) - <a href="/wiki/Double_negation#In_classical_propositional_calculus_system">Double negation</a> (one direction)</dd>
<dd>(DN2) - Double negation (another direction)</dd>
<dd>(HS1) - one form of <a href="/wiki/Hypothetical_syllogism#Alternative_forms">Hypothetical syllogism</a></dd>
<dd>(HS2) - another form of Hypothetical syllogism</dd>
<dd>(TR1) - <a href="/wiki/Transposition_(logic)#In_classical_propositional_calculus_system">Transposition</a></dd>
<dd>(TR2) - another form of transposition.</dd>
<dd>(L1) </dd>
<dd>(L3) </dd></dl>
<p>We also use the method of the <a href="/wiki/Hypothetical_syllogism#As_a_metatheorem">hypothetical syllogism metatheorem</a> as a shorthand for several proof steps.
</p>
<h3>Another outline for a completeness proof[<a href="/w/index.php?title=Propositional_calculus&amp;action=edit&amp;section=19">edit</a>]</h3>
<p>If a formula is a <a href="/wiki/Tautology_(logic)">tautology</a>, then there is a <a href="/wiki/Truth_table">truth table</a> for it which shows that each valuation yields the value true for the formula. Consider such a valuation. By mathematical induction on the length of the subformulas, show that the truth or falsity of the subformula follows from the truth or falsity (as appropriate for the valuation) of each propositional variable in the subformula. Then combine the lines of the truth table together two at a time by using "(P is true implies S) implies ((P is false implies S) implies S)". Keep repeating this until all dependencies on propositional variables have been eliminated. The result is that we have proved the given tautology. Since every tautology is provable, the logic is complete.
</p>
<h2>Interpretation of a truth-functional propositional calculus[<a href="/w/index.php?title=Propositional_calculus&amp;action=edit&amp;section=20">edit</a>]</h2>
<p>An <b>interpretation of a truth-functional propositional calculus</b> is an <a href="/wiki/Assignment_(mathematical_logic)">assignment</a> to each <a href="/wiki/Propositional_variable">propositional symbol</a> of of one or the other (but not both) of the <a href="/wiki/Truth_value">truth values</a> <a href="/wiki/Truth">truth</a> (<b>T</b>) and <a href="/wiki/False_(logic)">falsity</a> (<b>F</b>), and an assignment to the <a href="/wiki/Logical_connective">connective symbols</a> of of their usual truth-functional meanings. An interpretation of a truth-functional propositional calculus may also be expressed in terms of <a href="/wiki/Truth_tables">truth tables</a>.<sup><a href="#cite_note-metalogic-13">[13]</a></sup>
</p><p>For distinct propositional symbols there are distinct possible interpretations. For any particular symbol , for example, there are possible interpretations:
</p>
<ol><li> is assigned <b>T</b>, or</li>
<li> is assigned <b>F</b>.</li></ol>
<p>For the pair , there are possible interpretations:
</p>
<ol><li>both are assigned <b>T</b>,</li>
<li>both are assigned <b>F</b>,</li>
<li> is assigned <b>T</b> and is assigned <b>F</b>, or</li>
<li> is assigned <b>F</b> and is assigned <b>T</b>.<sup><a href="#cite_note-metalogic-13">[13]</a></sup></li></ol>
<p>Since has , that is, <a href="/wiki/Denumerably_infinite">denumerably</a> many propositional symbols, there are , and therefore <a href="/wiki/Cardinality_of_the_continuum">uncountably many</a> distinct possible interpretations of .<sup><a href="#cite_note-metalogic-13">[13]</a></sup>
</p>
<h3>Interpretation of a sentence of truth-functional propositional logic[<a href="/w/index.php?title=Propositional_calculus&amp;action=edit&amp;section=21">edit</a>]</h3>
<p>If φ and ψ are <a href="/wiki/Formula_(mathematical_logic)">formulas</a> of and is an interpretation of then the following definitions apply:
</p>
<p>Some consequences of these definitions:
</p>
<h2>Alternative calculus[<a href="/w/index.php?title=Propositional_calculus&amp;action=edit&amp;section=22">edit</a>]</h2>
<p>It is possible to define another version of propositional calculus, which defines most of the syntax of the logical operators by means of axioms, and which uses only one inference rule.
</p>
<h3>Axioms[<a href="/w/index.php?title=Propositional_calculus&amp;action=edit&amp;section=23">edit</a>]</h3>
<p>Let φ, χ, and ψ stand for well-formed formulas. (The well-formed formulas themselves would not contain any Greek letters, but only capital Roman letters, connective operators, and parentheses.) Then the axioms are as follows:
</p>
<table>
<caption>Axioms
</caption>
<tbody><tr>
<th>Name
</th>
<th>Axiom Schema
</th>
<th>Description
</th></tr>
<tr>
<td>THEN-1
</td>
<td>
</td>
<td>Add hypothesis χ, implication introduction
</td></tr>
<tr>
<td>THEN-2
</td>
<td>
</td>
<td>Distribute hypothesis over implication
</td></tr>
<tr>
<td>AND-1
</td>
<td>
</td>
<td>Eliminate conjunction
</td></tr>
<tr>
<td>AND-2
</td>
<td>
</td>
<td> 
</td></tr>
<tr>
<td>AND-3
</td>
<td>
</td>
<td>Introduce conjunction
</td></tr>
<tr>
<td>OR-1
</td>
<td>
</td>
<td>Introduce disjunction
</td></tr>
<tr>
<td>OR-2
</td>
<td>
</td>
<td> 
</td></tr>
<tr>
<td>OR-3
</td>
<td>
</td>
<td>Eliminate disjunction
</td></tr>
<tr>
<td>NOT-1
</td>
<td>
</td>
<td>Introduce negation
</td></tr>
<tr>
<td>NOT-2
</td>
<td>
</td>
<td>Eliminate negation
</td></tr>
<tr>
<td>NOT-3
</td>
<td>
</td>
<td>Excluded middle, classical logic
</td></tr>
<tr>
<td>IFF-1
</td>
<td>
</td>
<td>Eliminate equivalence
</td></tr>
<tr>
<td>IFF-2
</td>
<td>
</td>
<td> 
</td></tr>
<tr>
<td>IFF-3
</td>
<td>
</td>
<td>Introduce equivalence
</td></tr></tbody></table>
<ul><li>Axiom <b><a href="#math_THEN-2">THEN-2</a></b> may be considered to be a "distributive property of implication with respect to implication."</li>
<li>Axioms <b><a href="#math_AND-1">AND-1</a></b> and <b><a href="#math_AND-2">AND-2</a></b> correspond to "conjunction elimination". The relation between <b><a href="#math_AND-1">AND-1</a></b> and <b><a href="#math_AND-2">AND-2</a></b> reflects the commutativity of the conjunction operator.</li>
<li>Axiom <b><a href="#math_AND-3">AND-3</a></b> corresponds to "conjunction introduction."</li>
<li>Axioms <b><a href="#math_OR-1">OR-1</a></b> and <b><a href="#math_OR-2">OR-2</a></b> correspond to "disjunction introduction." The relation between <b><a href="#math_OR-1">OR-1</a></b> and <b><a href="#math_OR-2">OR-2</a></b> reflects the commutativity of the disjunction operator.</li>
<li>Axiom <b><a href="#math_NOT-1">NOT-1</a></b> corresponds to "reductio ad absurdum."</li>
<li>Axiom <b><a href="#math_NOT-2">NOT-2</a></b> says that "anything can be deduced from a contradiction."</li>
<li>Axiom <b><a href="#math_NOT-3">NOT-3</a></b> is called "<a href="/wiki/Law_of_excluded_middle">tertium non-datur</a>" (<a href="/wiki/Latin">Latin</a>: "a third is not given") and reflects the semantic valuation of propositional formulas: a formula can have a truth-value of either true or false. There is no third truth-value, at least not in classical logic. <a href="/wiki/Intuitionistic_logic">Intuitionistic logicians</a> do not accept the axiom <b><a href="#math_NOT-3">NOT-3</a></b>.</li></ul>
<h3>Inference rule[<a href="/w/index.php?title=Propositional_calculus&amp;action=edit&amp;section=24">edit</a>]</h3>
<p>The inference rule is <a href="/wiki/Modus_ponens">modus ponens</a>:
</p>
<dl><dd>.</dd></dl>
<h3>Meta-inference rule[<a href="/w/index.php?title=Propositional_calculus&amp;action=edit&amp;section=25">edit</a>]</h3>
<p>Let a demonstration be represented by a sequence, with hypotheses to the left of the <a href="/wiki/Turnstile_(symbol)">turnstile</a> and the conclusion to the right of the turnstile. Then the <a href="/wiki/Deduction_theorem">deduction theorem</a> can be stated as follows:
</p>
<dl><dd><i>If the sequence</i>
<dl><dd></dd></dl></dd>
<dd><i>has been demonstrated, then it is also possible to demonstrate the sequence</i>
<dl><dd>.</dd></dl></dd></dl>
<p>This deduction theorem (DT) is not itself formulated with propositional calculus: it is not a theorem of propositional calculus, but a theorem about propositional calculus. In this sense, it is a <a href="/wiki/Meta-theorem">meta-theorem</a>, comparable to theorems about the soundness or completeness of propositional calculus.
</p><p>On the other hand, DT is so useful for simplifying the syntactical proof process that it can be considered and used as another inference rule, accompanying modus ponens. In this sense, DT corresponds to the natural <a href="/wiki/Conditional_proof">conditional proof</a> inference rule which is part of the first version of propositional calculus introduced in this article.
</p><p>The converse of DT is also valid:
</p>
<dl><dd><i>If the sequence</i>
<dl><dd></dd></dl></dd>
<dd><i>has been demonstrated, then it is also possible to demonstrate the sequence</i>
<dl><dd></dd></dl></dd></dl>
<p>in fact, the validity of the converse of DT is almost trivial compared to that of DT:
</p>
<dl><dd><i>If</i>
<dl><dd></dd></dl></dd>
<dd><i>then</i>
<dl><dd>1: </dd>
<dd>2: </dd></dl></dd>
<dd><i>and from (1) and (2) can be deduced</i>
<dl><dd>3: </dd></dl></dd>
<dd><i>by means of modus ponens, Q.E.D.</i></dd></dl>
<p>The converse of DT has powerful implications: it can be used to convert an axiom into an inference rule. For example, by axiom AND-1 we have,
</p>
<dl><dd></dd></dl>
<p>which can be transformed by means of the converse of the deduction theorem into
</p>
<dl><dd></dd></dl>
<p>which tells us that the inference rule
</p>
<dl><dd></dd></dl>
<p>is <a href="/wiki/Admissible_rule">admissible</a>. This inference rule is <a href="/wiki/Conjunction_elimination">conjunction elimination</a>, one of the ten inference rules used in the first version (in this article) of the propositional calculus.
</p>
<h3>Example of a proof[<a href="/w/index.php?title=Propositional_calculus&amp;action=edit&amp;section=26">edit</a>]</h3>
<p>The following is an example of a (syntactical) demonstration, involving only axioms <b><a href="#math_THEN-1">THEN-1</a></b> and <b><a href="#math_THEN-2">THEN-2</a></b>:
</p><p><b>Prove:</b> (Reflexivity of implication).
</p><p><b>Proof:</b>
</p>
<ol><li>
<dl><dd>Axiom <b><a href="#math_THEN-2">THEN-2</a></b> with </dd></dl></li>
<li>
<dl><dd>Axiom <b><a href="#math_THEN-1">THEN-1</a></b> with </dd></dl></li>
<li>
<dl><dd>From (1) and (2) by modus ponens.</dd></dl></li>
<li>
<dl><dd>Axiom <b><a href="#math_THEN-1">THEN-1</a></b> with </dd></dl></li>
<li>
<dl><dd>From (3) and (4) by modus ponens.</dd></dl></li></ol>
<h2>Equivalence to equational logics[<a href="/w/index.php?title=Propositional_calculus&amp;action=edit&amp;section=27">edit</a>]</h2>
<p>The preceding alternative calculus is an example of a <a href="/wiki/Hilbert-style_deduction_system">Hilbert-style deduction system</a>. In the case of propositional systems the axioms are terms built with logical connectives and the only inference rule is modus ponens. Equational logic as standardly used informally in high school algebra is a different kind of calculus from Hilbert systems. Its theorems are equations and its inference rules express the properties of equality, namely that it is a congruence on terms that admits substitution.
</p><p>Classical propositional calculus as described above is equivalent to <a href="/wiki/Boolean_algebra_(logic)">Boolean algebra</a>, while <a href="/wiki/Intuitionistic_logic">intuitionistic propositional calculus</a> is equivalent to <a href="/wiki/Heyting_algebra">Heyting algebra</a>. The equivalence is shown by translation in each direction of the theorems of the respective systems. Theorems of classical or intuitionistic propositional calculus are translated as equations of Boolean or Heyting algebra respectively. Conversely theorems of Boolean or Heyting algebra are translated as theorems of classical or intuitionistic calculus respectively, for which is a standard abbreviation. In the case of Boolean algebra can also be translated as , but this translation is incorrect intuitionistically.
</p><p>In both Boolean and Heyting algebra, inequality can be used in place of equality. The equality is expressible as a pair of inequalities and . Conversely the inequality is expressible as the equality , or as . The significance of inequality for Hilbert-style systems is that it corresponds to the latter's deduction or <a href="/wiki/Entailment">entailment</a> symbol . An entailment
</p>
<dl><dd><dl><dd></dd></dl></dd></dl>
<p>is translated in the inequality version of the algebraic framework as
</p>
<dl><dd><dl><dd></dd></dl></dd></dl>
<p>Conversely the algebraic inequality is translated as the entailment
</p>
<dl><dd><dl><dd>.</dd></dl></dd></dl>
<p>The difference between implication and inequality or <a href="/wiki/Entailment">entailment</a> or is that the former is internal to the logic while the latter is external. Internal implication between two terms is another term of the same kind. Entailment as external implication between two terms expresses a metatruth outside the language of the logic, and is considered part of the <a href="/wiki/Metalanguage">metalanguage</a>. Even when the logic under study is intuitionistic, entailment is ordinarily understood classically as two-valued: either the left side entails, or is less-or-equal to, the right side, or it is not.
</p><p>Similar but more complex translations to and from algebraic logics are possible for natural deduction systems as described above and for the <a href="/wiki/Sequent_calculus">sequent calculus</a>. The entailments of the latter can be interpreted as two-valued, but a more insightful interpretation is as a set, the elements of which can be understood as abstract proofs organized as the morphisms of a <a href="/wiki/Category_(mathematics)">category</a>. In this interpretation the cut rule of the sequent calculus corresponds to composition in the category. Boolean and Heyting algebras enter this picture as special categories having at most one morphism per homset, i.e., one proof per entailment, corresponding to the idea that existence of proofs is all that matters: any proof will do and there is no point in distinguishing them.
</p>
<h2>Graphical calculi[<a href="/w/index.php?title=Propositional_calculus&amp;action=edit&amp;section=28">edit</a>]</h2>
<p>It is possible to generalize the definition of a formal language from a set of finite sequences over a finite basis to include many other sets of mathematical structures, so long as they are built up by finitary means from finite materials. What's more, many of these families of formal structures are especially well-suited for use in logic.
</p><p>For example, there are many families of <a href="/wiki/Graph_(discrete_mathematics)">graphs</a> that are close enough analogues of formal languages that the concept of a calculus is quite easily and naturally extended to them. Many species of graphs arise as <i><a href="/wiki/Parse_graph">parse graphs</a></i> in the syntactic analysis of the corresponding families of text structures. The exigencies of practical computation on formal languages frequently demand that text strings be converted into <a href="/wiki/Pointer_structure">pointer structure</a> renditions of parse graphs, simply as a matter of checking whether strings are well-formed formulas or not. Once this is done, there are many advantages to be gained from developing the graphical analogue of the calculus on strings. The mapping from strings to parse graphs is called <i><a href="/wiki/Parsing">parsing</a></i> and the inverse mapping from parse graphs to strings is achieved by an operation that is called <i><a href="/wiki/Graph_traversal">traversing</a></i> the graph.
</p>
<h2>Other logical calculi[<a href="/w/index.php?title=Propositional_calculus&amp;action=edit&amp;section=29">edit</a>]</h2>
<p>Propositional calculus is about the simplest kind of logical calculus in current use. It can be extended in several ways. (<a href="/wiki/Term_logic">Aristotelian "syllogistic" calculus</a>, which is largely supplanted in modern logic, is in <i>some</i> ways simpler – but in other ways more complex – than propositional calculus.) The most immediate way to develop a more complex logical calculus is to introduce rules that are sensitive to more fine-grained details of the sentences being used.
</p><p><a href="/wiki/First-order_logic">First-order logic</a> (a.k.a. first-order predicate logic) results when the "atomic sentences" of propositional logic are broken up into <a href="/wiki/Singular_term">terms</a>, <a href="/wiki/Variable_(mathematics)">variables</a>, <a href="/wiki/Predicate_(logic)">predicates</a>, and <a href="/wiki/Quantifier_(logic)">quantifiers</a>, all keeping the rules of propositional logic with some new ones introduced. (For example, from "All dogs are mammals" we may infer "If Rover is a dog then Rover is a mammal".) With the tools of first-order logic it is possible to formulate a number of theories, either with explicit axioms or by rules of inference, that can themselves be treated as logical calculi. <a href="/wiki/Arithmetic">Arithmetic</a> is the best known of these; others include <a href="/wiki/Set_theory">set theory</a> and <a href="/wiki/Mereology">mereology</a>. <a href="/wiki/Second-order_logic">Second-order logic</a> and other <a href="/wiki/Higher-order_logic">higher-order logics</a> are formal extensions of first-order logic. Thus, it makes sense to refer to propositional logic as <i>"zeroth-order logic"</i>, when comparing it with these logics.
</p><p><a href="/wiki/Modal_logic">Modal logic</a> also offers a variety of inferences that cannot be captured in propositional calculus. For example, from "Necessarily p" we may infer that p. From p we may infer "It is possible that p". The translation between modal logics and algebraic logics concerns classical and intuitionistic logics but with the introduction of a unary operator on Boolean or Heyting algebras, different from the Boolean operations, interpreting the possibility modality, and in the case of Heyting algebra a second operator interpreting necessity (for Boolean algebra this is redundant since necessity is the De Morgan dual of possibility). The first operator preserves 0 and disjunction while the second preserves 1 and conjunction.
</p><p><a href="/wiki/Many-valued_logic">Many-valued logics</a> are those allowing sentences to have values other than <i>true</i> and <i>false</i>. (For example, <i>neither</i> and <i>both</i> are standard "extra values"; "continuum logic" allows each sentence to have any of an infinite number of "degrees of truth" between <i>true</i> and <i>false</i>.) These logics often require calculational devices quite distinct from propositional calculus. When the values form a Boolean algebra (which may have more than two or even infinitely many values), many-valued logic reduces to classical logic; many-valued logics are therefore only of independent interest when the values form an algebra that is not Boolean.
</p>
<h2>Solvers[<a href="/w/index.php?title=Propositional_calculus&amp;action=edit&amp;section=30">edit</a>]</h2>
<p>Deciding satisfiability of propositional logic formulas is an <a href="/wiki/NP-complete">NP-complete</a> problem. However, practical methods exist (e.g., <a href="/wiki/DPLL_algorithm">DPLL algorithm</a>, 1962; <a href="/wiki/Chaff_algorithm">Chaff algorithm</a>, 2001) that are very fast for many useful cases. Recent work has extended the <a href="/wiki/SAT_solver">SAT solver</a> algorithms to work with propositions containing <a href="/wiki/Arithmetic_expression">arithmetic expressions</a>; these are the <a href="/wiki/SMT_solver">SMT solvers</a>.
</p>
<h2>See also[<a href="/w/index.php?title=Propositional_calculus&amp;action=edit&amp;section=31">edit</a>]</h2>
<h3>Higher logical levels[<a href="/w/index.php?title=Propositional_calculus&amp;action=edit&amp;section=32">edit</a>]</h3>
<ul><li><a href="/wiki/First-order_logic">First-order logic</a></li>
<li><a href="/wiki/Second-order_propositional_logic">Second-order propositional logic</a></li>
<li><a href="/wiki/Second-order_logic">Second-order logic</a></li>
<li><a href="/wiki/Higher-order_logic">Higher-order logic</a></li></ul>
<h3>[<a href="/w/index.php?title=Propositional_calculus&amp;action=edit&amp;section=33">edit</a>]</h3>
<div>
<ul><li><a href="/wiki/Boolean_algebra_(logic)">Boolean algebra (logic)</a></li>
<li><a href="/wiki/Boolean_algebra_(structure)">Boolean algebra (structure)</a></li>
<li><a href="/wiki/Boolean_algebra_topics">Boolean algebra topics</a></li>
<li><a href="/wiki/Boolean_domain">Boolean domain</a></li>
<li><a href="/wiki/Boolean_function">Boolean function</a></li>
<li><a href="/wiki/Boolean-valued_function">Boolean-valued function</a></li>
<li><a href="/wiki/Categorical_logic">Categorical logic</a></li>
<li><a href="/wiki/Combinational_logic">Combinational logic</a></li>
<li><a href="/wiki/Combinatory_logic">Combinatory logic</a></li>
<li><a href="/wiki/Conceptual_graph">Conceptual graph</a></li>
<li><a href="/wiki/Disjunctive_syllogism">Disjunctive syllogism</a></li>
<li><a href="/wiki/Entitative_graph">Entitative graph</a></li>
<li><a href="/wiki/Equational_logic">Equational logic</a></li>
<li><a href="/wiki/Existential_graph">Existential graph</a></li>
<li><a href="/wiki/Frege%27s_propositional_calculus">Frege's propositional calculus</a></li>
<li><a href="/wiki/Implicational_propositional_calculus">Implicational propositional calculus</a></li>
<li><a href="/wiki/Intuitionistic_propositional_calculus">Intuitionistic propositional calculus</a></li>
<li><a href="/wiki/Jean_Buridan">Jean Buridan</a></li>
<li><i><a href="/wiki/Laws_of_Form">Laws of Form</a></i></li>
<li><a href="/wiki/List_of_logic_symbols">List of logic symbols</a></li>
<li><a href="/wiki/Logical_graph">Logical graph</a></li>
<li><a href="/wiki/Logical_NOR">Logical NOR</a></li>
<li><a href="/wiki/Logical_value">Logical value</a></li>
<li><a href="/wiki/Mathematical_logic">Mathematical logic</a></li>
<li><a href="/wiki/Operation_(mathematics)">Operation (mathematics)</a></li>
<li><a href="/wiki/Paul_of_Venice">Paul of Venice</a></li>
<li><a href="/wiki/Peirce%27s_law">Peirce's law</a></li>
<li><a href="/wiki/Peter_of_Spain_(author)">Peter of Spain (author)</a></li>
<li><a href="/wiki/Propositional_formula">Propositional formula</a></li>
<li><a href="/wiki/Symmetric_difference">Symmetric difference</a></li>
<li><a href="/wiki/Tautology_(rule_of_inference)">Tautology (rule of inference)</a></li>
<li><a href="/wiki/Truth_function">Truth function</a></li>
<li><a href="/wiki/Truth_table">Truth table</a></li>
<li><a href="/wiki/Walter_Burley">Walter Burley</a></li>
<li><a href="/wiki/William_of_Sherwood">William of Sherwood</a></li></ul>
</div>
<h2>References[<a href="/w/index.php?title=Propositional_calculus&amp;action=edit&amp;section=34">edit</a>]</h2>
<div>
<ol>
<li><b><a href="#cite_ref-1">^</a></b> <cite><a href="https://brilliant.org/wiki/propositional-logic/">"Propositional Logic | Brilliant Math &amp; Science Wiki"</a>. <i>brilliant.org</i>. Retrieved 20 August 2020.</cite>
</li>
<li><b><a href="#cite_ref-2">^</a></b> <cite>Bobzien, Susanne (1 January 2016). Zalta, Edward N. (ed.). <a href="http://plato.stanford.edu/archives/spr2016/entries/logic-ancient/"><i>The Stanford Encyclopedia of Philosophy</i></a> – via Stanford Encyclopedia of Philosophy.</cite>
</li>
<li><b><a href="#cite_ref-3">^</a></b> <cite><a href="https://iep.utm.edu/prop-log/">"Propositional Logic | Internet Encyclopedia of Philosophy"</a>. Retrieved 20 August 2020.</cite>
</li>
<li><b><a href="#cite_ref-4">^</a></b> <cite>Marenbon, John (2007). <a href="https://archive.org/details/introductiontome00mare"><i>Medieval philosophy: an historical and philosophical introduction</i></a>. Routledge. p. <a href="https://archive.org/details/introductiontome00mare/page/n151">137</a>.</cite>
</li>
<li><b><a href="#cite_ref-5">^</a></b> <cite>Peckhaus, Volker (1 January 2014). Zalta, Edward N. (ed.). <a href="http://plato.stanford.edu/archives/spr2014/entries/leibniz-logic-influence/"><i>The Stanford Encyclopedia of Philosophy</i></a> – via Stanford Encyclopedia of Philosophy.</cite>
</li>
<li><b><a href="#cite_ref-6">^</a></b> <cite>Hurley, Patrick (2007). <i>A Concise Introduction to Logic 10th edition</i>. Wadsworth Publishing. p. 392.</cite>
</li>
<li><b><a href="#cite_ref-7">^</a></b> Beth, Evert W.; "Semantic entailment and formal derivability", series: Mededlingen van de Koninklijke Nederlandse Akademie van Wetenschappen, Afdeling Letterkunde, Nieuwe Reeks, vol. 18, no. 13, Noord-Hollandsche Uitg. Mij., Amsterdam, 1955, pp. 309–42. Reprinted in Jaakko Intikka (ed.) <i>The Philosophy of Mathematics</i>, Oxford University Press, 1969
</li>
<li>^ <a href="#cite_ref-Truth_in_Frege_8-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Truth_in_Frege_8-1"><sup><i><b>b</b></i></sup></a> <a href="https://web.archive.org/web/20120807235445/http://frege.brown.edu/heck/pdf/unpublished/TruthInFrege.pdf">Truth in Frege</a>
</li>
<li>^ <a href="#cite_ref-Russell_Truth-Tables_9-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Russell_Truth-Tables_9-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-Russell_Truth-Tables_9-2"><sup><i><b>c</b></i></sup></a> <cite><a href="http://digitalcommons.mcmaster.ca/cgi/viewcontent.cgi?article=1219&amp;context=russelljournal">"Russell: the Journal of Bertrand Russell Studies"</a>.</cite>
</li>
<li><b><a href="#cite_ref-10">^</a></b> <cite><a href="/wiki/Irving_Anellis">Anellis, Irving H.</a> (2012). "Peirce's Truth-functional Analysis and the Origin of the Truth Table". <i>History and Philosophy of Logic</i>. <b>33</b>: 87–97. <a href="/wiki/Doi_(identifier)">doi</a>:<a href="https://doi.org/10.1080%2F01445340.2011.621702">10.1080/01445340.2011.621702</a>. <a href="/wiki/S2CID_(identifier)">S2CID</a> <a href="https://api.semanticscholar.org/CorpusID:170654885">170654885</a>.</cite>
</li>
<li><b><a href="#cite_ref-Wernick_11-0">^</a></b> Wernick, William (1942) "Complete Sets of Logical Functions," <i>Transactions of the American Mathematical Society</i> <b>51</b>, pp. 117–132.
</li>
<li><b><a href="#cite_ref-12">^</a></b> <cite>Toida, Shunichi (2 August 2009). <a href="http://www.cs.odu.edu/~toida/nerzic/content/logic/prop_logic/implications/implication_proof.html">"Proof of Implications"</a>. <i>CS381 Discrete Structures/Discrete Mathematics Web Course Material</i>. Department of Computer Science, <a href="/wiki/Old_Dominion_University">Old Dominion University</a>. Retrieved 10 March 2010.</cite>
</li>
<li>^ <a href="#cite_ref-metalogic_13-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-metalogic_13-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-metalogic_13-2"><sup><i><b>c</b></i></sup></a> <a href="#cite_ref-metalogic_13-3"><sup><i><b>d</b></i></sup></a> <a href="#cite_ref-metalogic_13-4"><sup><i><b>e</b></i></sup></a> <a href="#cite_ref-metalogic_13-5"><sup><i><b>f</b></i></sup></a> <a href="#cite_ref-metalogic_13-6"><sup><i><b>g</b></i></sup></a> <a href="#cite_ref-metalogic_13-7"><sup><i><b>h</b></i></sup></a> <a href="#cite_ref-metalogic_13-8"><sup><i><b>i</b></i></sup></a> <a href="#cite_ref-metalogic_13-9"><sup><i><b>j</b></i></sup></a> <a href="#cite_ref-metalogic_13-10"><sup><i><b>k</b></i></sup></a> <cite>Hunter, Geoffrey (1971). <i>Metalogic: An Introduction to the Metatheory of Standard First-Order Logic</i>. University of California Press. <a href="/wiki/ISBN_(identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-520-02356-0"><bdi>0-520-02356-0</bdi></a>.</cite>
</li>
</ol></div>
<h2>Further reading[<a href="/w/index.php?title=Propositional_calculus&amp;action=edit&amp;section=35">edit</a>]</h2>
<ul><li>Brown, Frank Markham (2003), <i>Boolean Reasoning: The Logic of Boolean Equations</i>, 1st edition, Kluwer Academic Publishers, Norwell, MA. 2nd edition, Dover Publications, Mineola, NY.</li>
<li><a href="/wiki/Chen_Chung_Chang">Chang, C.C.</a> and <a href="/wiki/Howard_Jerome_Keisler">Keisler, H.J.</a> (1973), <i>Model Theory</i>, North-Holland, Amsterdam, Netherlands.</li>
<li>Kohavi, Zvi (1978), <i>Switching and Finite Automata Theory</i>, 1st edition, McGraw–Hill, 1970. 2nd edition, McGraw–Hill, 1978.</li>
<li><a href="/wiki/Robert_R._Korfhage">Korfhage, Robert R.</a> (1974), <i>Discrete Computational Structures</i>, Academic Press, New York, NY.</li>
<li><a href="/wiki/Joachim_Lambek">Lambek, J.</a> and Scott, P.J. (1986), <i>Introduction to Higher Order Categorical Logic</i>, Cambridge University Press, Cambridge, UK.</li>
<li>Mendelson, Elliot (1964), <i>Introduction to Mathematical Logic</i>, D. Van Nostrand Company.</li></ul>
<h3>[<a href="/w/index.php?title=Propositional_calculus&amp;action=edit&amp;section=36">edit</a>]</h3>
<ul><li><cite><a href="/wiki/Douglas_Hofstadter">Hofstadter, Douglas</a> (1979). <i><a href="/wiki/G%C3%B6del,_Escher,_Bach">Gödel, Escher, Bach: An Eternal Golden Braid</a></i>. <a href="/wiki/Basic_Books">Basic Books</a>. <a href="/wiki/ISBN_(identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-465-02656-2"><bdi>978-0-465-02656-2</bdi></a>.</cite></li></ul>
<h2>External links[<a href="/w/index.php?title=Propositional_calculus&amp;action=edit&amp;section=37">edit</a>]</h2>
<ul><li><a href="/w/index.php?title=Kevin_C._Klement&amp;action=edit&amp;redlink=1">Klement, Kevin C.</a> (2006), "Propositional Logic", in James Fieser and Bradley Dowden (eds.), <i><a href="/wiki/Internet_Encyclopedia_of_Philosophy">Internet Encyclopedia of Philosophy</a></i>, <a href="http://www.iep.utm.edu/p/prop-log.htm">Eprint</a>.</li>
<li><a href="http://www.qedeq.org/current/doc/math/qedeq_formal_logic_v1_en.pdf">Formal Predicate Calculus</a>, contains a systematic formal development along the lines of <a href="#Alternative_calculus">Alternative calculus</a></li>
<li><i><a href="http://www.fecundity.com/logic/">forall x: an introduction to formal logic</a></i>, by <a href="/w/index.php?title=P.D._Magnus&amp;action=edit&amp;redlink=1">P.D. Magnus</a>, covers formal semantics and <a href="/wiki/Proof_theory">proof theory</a> for sentential logic.</li>
<li><a href="http://logicinaction.org/docs/ch2.pdf">Chapter 2 / Propositional Logic</a> from <a href="http://logicinaction.org/">Logic In Action</a></li>
<li><a href="https://www.nayuki.io/page/propositional-sequent-calculus-prover">Propositional sequent calculus prover</a> on Project Nayuki. (<i>note</i>: implication can be input in the form <code>!X|Y</code>, and a sequent can be a single formula prefixed with <code>&gt;</code> and having no commas)</li>
<li><a href="https://docs.google.com/document/d/1DhtRAPcMwJmiQnbdmFcHWaOddQ7kuqqDnWp2LZcGlnY/edit?usp=sharing">Propositional Logic - A Generative Grammar</a></li></ul>
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 Contents

 move to sidebar hide
 * (Top)
 * 1 Explanation
 * 2 History
 * 3 Terminology
 * 4 Basic concepts
 + 4.1 Closure under operations
 + 4.2 Argument
 * 5 Generic description of a propositional calculus
 * 6 Example 1. Simple axiom system
 * 7 Example 2. Natural deduction system
 * 8 Basic and derived argument forms
 * 9 Proofs in propositional calculus
 + 9.1 Example of a proof in natural deduction system
 + 9.2 Example of a proof in a classical propositional calculus system
 * 10 Soundness and completeness of the rules
 + 10.1 Sketch of a soundness proof
 + 10.2 Sketch of completeness proof
 o 10.2.1 Example
 o 10.2.2 Verifying completeness for the classical propositional calculus system
 + 10.3 Another outline for a completeness proof
 * 11 Interpretation of a truth-functional propositional calculus
 + 11.1 Interpretation of a sentence of truth-functional propositional logic
 * 12 Alternative calculus
 + 12.1 Axioms
 + 12.2 Inference rule
 + 12.3 Meta-inference rule
 + 12.4 Example of a proof
 * 13 Equivalence to equational logics
 * 14 Graphical calculi
 * 15 Other logical calculi
 * 16 Solvers
 * 17 See also
 + 17.1 Higher logical levels
 + 17.2 Related topics
 * 18 References
 * 19 Further reading
 + 19.1 Related works
 * 20 External links
 Toggle the table of contents

 Propositional calculus

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 From Wikipedia, the free encyclopedia
 Not to be confused with Propositional analysis.

 Transformation rules 
 Propositional calculus 
 Rules of inference 
 * Implication introduction / elimination ( modus ponens )
 * Biconditional introduction / elimination 
 * Conjunction introduction / elimination 
 * Disjunction introduction / elimination 
 * Disjunctive / hypothetical syllogism 
 * Constructive / destructive dilemma 
 * Absorption / modus tollens / modus ponendo tollens 
 * Negation introduction 
 Rules of replacement 
 * Associativity 
 * Commutativity 
 * Distributivity 
 * Double negation 
 * De Morgan's laws 
 * Transposition 
 * Material implication 
 * Exportation 
 * Tautology 
 Predicate logic 
 Rules of inference 
 * Universal generalization / instantiation 
 * Existential generalization / instantiation 
 

 Propositional calculus is a branch of logic. It is also called propositional logic, statement logic, sentential calculus, sentential logic, or sometimes zeroth-order logic. It deals with propositions (which can be true or false) and relations between propositions, including the construction of arguments based on them. Compound propositions are formed by connecting propositions by logical connectives. Propositions that contain no logical connectives are called atomic propositions.

 Unlike first-order logic, propositional logic does not deal with non-logical objects, predicates about them, or quantifiers. However, all the machinery of propositional logic is included in first-order logic and higher-order logics. In this sense, propositional logic is the foundation of first-order logic and higher-order logic.

 Explanation [ edit ]

 Logical connectives are found in natural languages. In English for example, some examples are "and" (conjunction), "or" (disjunction), "not" (negation) and "if" (but only when used to denote material conditional).

 The following is an example of a very simple inference within the scope of propositional logic:

 Premise 1: If it's raining then it's cloudy. Premise 2: It's raining. Conclusion: It's cloudy.

 Both premises and the conclusion are propositions. The premises are taken for granted, and with the application of modus ponens (an inference rule), the conclusion follows.

 As propositional logic is not concerned with the structure of propositions beyond the point where they can't be decomposed any more by logical connectives, this inference can be restated replacing those atomic statements with statement letters, which are interpreted as variables representing statements:

 Premise 1: Premise 2: Conclusion:

 The same can be stated succinctly in the following way:

 When P is interpreted as "It's raining" and Q as "it's cloudy" the above symbolic expressions can be seen to correspond exactly with the original expression in natural language. Not only that, but they will also correspond with any other inference of this form, which will be valid on the same basis this inference is.

 Propositional logic may be studied through a formal system in which formulas of a formal language may be interpreted to represent propositions. A system of axioms and inference rules allows certain formulas to be derived. These derived formulas are called theorems and may be interpreted to be true propositions. A constructed sequence of such formulas is known as a derivation or proof and the last formula of the sequence is the theorem. The derivation may be interpreted as proof of the proposition represented by the theorem.

 When a formal system is used to represent formal logic, only statement letters (usually capital roman letters such as , and ) are represented directly. The natural language propositions that arise when they're interpreted are outside the scope of the system, and the relation between the formal system and its interpretation is likewise outside the formal system itself.

 In classical truth-functional propositional logic, formulas are interpreted as having precisely one of two possible truth values, the truth value of true or the truth value of false.[1] The principle of bivalence and the law of excluded middle are upheld. Truth-functional propositional logic defined as such and systems isomorphic to it are considered to be zeroth-order logic. However, alternative propositional logics are also possible. For more, see Other logical calculi below.

 History [ edit ]

 Main article: History of logic

 Although propositional logic (which is interchangeable with propositional calculus) had been hinted by earlier philosophers, it was developed into a formal logic (Stoic logic) by Chrysippus in the 3rd century BC[2] and expanded by his successor Stoics. The logic was focused on propositions. This advancement was different from the traditional syllogistic logic, which was focused on terms. However, most of the original writings were lost[3] and the propositional logic developed by the Stoics was no longer understood later in antiquity.[ citation needed ] Consequently, the system was essentially reinvented by Peter Abelard in the 12th century.[4]

 Propositional logic was eventually refined using symbolic logic. The 17th/18th-century mathematician Gottfried Leibniz has been credited with being the founder of symbolic logic for his work with the calculus ratiocinator. Although his work was the first of its kind, it was unknown to the larger logical community. Consequently, many of the advances achieved by Leibniz were recreated by logicians like George Boole and Augustus De Morgan—completely independent of Leibniz.[5]

 Just as propositional logic can be considered an advancement from the earlier syllogistic logic, Gottlob Frege's predicate logic can be also considered an advancement from the earlier propositional logic. One author describes predicate logic as combining "the distinctive features of syllogistic logic and propositional logic."[6] Consequently, predicate logic ushered in a new era in logic's history; however, advances in propositional logic were still made after Frege, including natural deduction, truth trees and truth tables. Natural deduction was invented by Gerhard Gentzen and Jan Łukasiewicz. Truth trees were invented by Evert Willem Beth.[7] The invention of truth tables, however, is of uncertain attribution.

 Within works by Frege[8] and Bertrand Russell,[9] are ideas influential to the invention of truth tables. The actual tabular structure (being formatted as a table), itself, is generally credited to either Ludwig Wittgenstein or Emil Post (or both, independently).[8] Besides Frege and Russell, others credited with having ideas preceding truth tables include Philo, Boole, Charles Sanders Peirce,[10] and Ernst Schröder. Others credited with the tabular structure include Jan Łukasiewicz, Alfred North Whitehead, William Stanley Jevons, John Venn, and Clarence Irving Lewis.[9] Ultimately, some have concluded, like John Shosky, that "It is far from clear that any one person should be given the title of 'inventor' of truth-tables.".[9]

 Terminology [ edit ]

 In general terms, a calculus is a formal system that consists of a set of syntactic expressions (well-formed formulas), a distinguished subset of these expressions (axioms), plus a set of formal rules that define a specific binary relation, intended to be interpreted as logical equivalence, on the space of expressions.

 When the formal system is intended to be a logical system, the expressions are meant to be interpreted as statements, and the rules, known to be inference rules, are typically intended to be truth-preserving. In this setting, the rules, which may include axioms, can then be used to derive ("infer") formulas representing true statements—from given formulas representing true statements.

 The set of axioms may be empty, a nonempty finite set, or a countably infinite set (see axiom schema). A formal grammar recursively defines the expressions and well-formed formulas of the language. In addition a semantics may be given which defines truth and valuations (or interpretations).

 The language of a propositional calculus consists of

 1. a set of primitive symbols, variously referred to as atomic formulas, placeholders, proposition letters, or variables, and
 2. a set of operator symbols, variously interpreted as logical operators or logical connectives.

 A well-formed formula is any atomic formula, or any formula that can be built up from atomic formulas by means of operator symbols according to the rules of the grammar.

 Mathematicians sometimes distinguish between propositional constants, propositional variables, and schemata. Propositional constants represent some particular proposition, while propositional variables range over the set of all atomic propositions. Schemata, however, range over all propositions. It is common to represent propositional constants by A , B , and C , propositional variables by P , Q , and R , and schematic letters are often Greek letters, most often φ , ψ , and χ .

 Basic concepts [ edit ]

 The following outlines a standard propositional calculus. Many different formulations exist which are all more or less equivalent, but differ in the details of:

 1. their language (i.e., the particular collection of primitive symbols and operator symbols),
 2. the set of axioms, or distinguished formulas, and
 3. the set of inference rules.

 Any given proposition may be represented with a letter called a 'propositional constant', analogous to representing a number by a letter in mathematics (e.g., a = 5 ). All propositions require exactly one of two truth-values: true or false. For example, let P be the proposition that it is raining outside. This will be true ( P ) if it is raining outside, and false otherwise ( ¬ P).

 * We then define truth-functional operators, beginning with negation. ¬ P represents the negation of P , which can be thought of as the denial of P . In the example above, ¬ P expresses that it is not raining outside, or by a more standard reading: "It is not the case that it is raining outside." When P is true, ¬ P is false; and when P is false, ¬ P is true. As a result, ¬ ¬ P always has the same truth-value as P .
 * Conjunction is a truth-functional connective which forms a proposition out of two simpler propositions, for example, P and Q . The conjunction of P and Q is written P ∧ Q, and expresses that each are true. We read P ∧ Q as " P and Q ". For any two propositions, there are four possible assignments of truth values:
 1. P is true and Q is true
 2. P is true and Q is false
 3. P is false and Q is true
 4. P is false and Q is false
 The conjunction of P and Q is true in case 1, and is false otherwise. Where P is the proposition that it is raining outside and Q is the proposition that a cold-front is over Kansas, P ∧ Q is true when it is raining outside and there is a cold-front over Kansas. If it is not raining outside, then P ∧ Q is false; and if there is no cold-front over Kansas, then P ∧ Q is also false.
 * Disjunction resembles conjunction in that it forms a proposition out of two simpler propositions. We write it P ∨ Q, and it is read " P or Q ". It expresses that either P or Q is true. Thus, in the cases listed above, the disjunction of P with Q is true in all cases—except case 4. Using the example above, the disjunction expresses that it is either raining outside, or there is a cold front over Kansas. (Note, this use of disjunction is supposed to resemble the use of the English word "or". However, it is most like the English inclusive "or", which can be used to express the truth of at least one of two propositions. It is not like the English exclusive "or", which expresses the truth of exactly one of two propositions. In other words, the exclusive "or" is false when both P and Q are true (case 1), and similarly is false when both P and Q are false (case 4). An example of the exclusive or is: You may have a bagel or a pastry, but not both. Often in natural language, given the appropriate context, the addendum "but not both" is omitted—but implied. In mathematics, however, "or" is always inclusive or; if exclusive or is meant it will be specified, possibly by "xor".)
 * Material conditional also joins two simpler propositions, and we write P → Q, which is read "if P then Q ". The proposition to the left of the arrow is called the antecedent, and the proposition to the right is called the consequent. (There is no such designation for conjunction or disjunction, since they are commutative operations.) It expresses that Q is true whenever P is true. Thus P → Q is true in every case above except case 2, because this is the only case when P is true but Q is not. Using the example, if P then Q expresses that if it is raining outside, then there is a cold-front over Kansas. The material conditional is often confused with physical causation. The material conditional, however, only relates two propositions by their truth-values—which is not the relation of cause and effect. It is contentious in the literature whether the material implication represents logical causation.
 * Biconditional joins two simpler propositions, and we write P ↔ Q, which is read " P if and only if Q ". It expresses that P and Q have the same truth-value, and in cases 1 and 4. ' P is true if and only if Q ' is true, and is false otherwise.

 It is very helpful to look at the truth tables for these different operators, as well as the method of analytic tableaux.

 Closure under operations [ edit ]

 Propositional logic is closed under truth-functional connectives. That is to say, for any proposition φ , ¬ φ is also a proposition. Likewise, for any propositions φ and ψ , φ ∧ ψ is a proposition, and similarly for disjunction, conditional, and biconditional. This implies that, for instance, φ ∧ ψ is a proposition, and so it can be conjoined with another proposition. In order to represent this, we need to use parentheses to indicate which proposition is conjoined with which. For instance, P ∧ Q ∧ R is not a well-formed formula, because we do not know if we are conjoining P ∧ Q with R or if we are conjoining P with Q ∧ R. Thus we must write either ( P ∧ Q ) ∧ R to represent the former, or P ∧ ( Q ∧ R ) to represent the latter. By evaluating the truth conditions, we see that both expressions have the same truth conditions (will be true in the same cases), and moreover that any proposition formed by arbitrary conjunctions will have the same truth conditions, regardless of the location of the parentheses. This means that conjunction is associative, however, one should not assume that parentheses never serve a purpose. For instance, the sentence P ∧ ( Q ∨ R ) does not have the same truth conditions of ( P ∧ Q ) ∨ R, so they are different sentences distinguished only by the parentheses. One can verify this by the truth-table method referenced above.

 Note: For any arbitrary number of propositional constants, we can form a finite number of cases which list their possible truth-values. A simple way to generate this is by truth-tables, in which one writes P , Q , ..., Z , for any list of k propositional constants—that is to say, any list of propositional constants with k entries. Below this list, one writes 2 k rows, and below P one fills in the first half of the rows with true (or T) and the second half with false (or F). Below Q one fills in one-quarter of the rows with T, then one-quarter with F, then one-quarter with T and the last quarter with F. The next column alternates between true and false for each eighth of the rows, then sixteenths, and so on, until the last propositional constant varies between T and F for each row. This will give a complete listing of cases or truth-value assignments possible for those propositional constants.

 Argument [ edit ]

 The propositional calculus then defines an argument to be a list of propositions. A valid argument is a list of propositions, the last of which follows from—or is implied by—the rest. All other arguments are invalid. The simplest valid argument is modus ponens, one instance of which is the following list of propositions:

 This is a list of three propositions, each line is a proposition, and the last follows from the rest. The first two lines are called premises, and the last line the conclusion. We say that any proposition C follows from any set of propositions , if C must be true whenever every member of the set is true. In the argument above, for any P and Q , whenever P → Q and P are true, necessarily Q is true. Notice that, when P is true, we cannot consider cases 3 and 4 (from the truth table). When P → Q is true, we cannot consider case 2. This leaves only case 1, in which Q is also true. Thus Q is implied by the premises.

 This generalizes schematically. Thus, where φ and ψ may be any propositions at all,

 Other argument forms are convenient, but not necessary. Given a complete set of axioms (see below for one such set), modus ponens is sufficient to prove all other argument forms in propositional logic, thus they may be considered to be a derivative. Note, this is not true of the extension of propositional logic to other logics like first-order logic. First-order logic requires at least one additional rule of inference in order to obtain completeness.

 The significance of argument in formal logic is that one may obtain new truths from established truths. In the first example above, given the two premises, the truth of Q is not yet known or stated. After the argument is made, Q is deduced. In this way, we define a deduction system to be a set of all propositions that may be deduced from another set of propositions. For instance, given the set of propositions , we can define a deduction system, Γ , which is the set of all propositions which follow from A . Reiteration is always assumed, so . Also, from the first element of A , last element, as well as modus ponens, R is a consequence, and so . Because we have not included sufficiently complete axioms, though, nothing else may be deduced. Thus, even though most deduction systems studied in propositional logic are able to deduce , this one is too weak to prove such a proposition.

 Generic description of a propositional calculus [ edit ]

 A propositional calculus is a formal system , where:

 * The alpha set is a countably infinite set of elements called proposition symbols or propositional variables. Syntactically speaking, these are the most basic elements of the formal language , otherwise referred to as atomic formulas or terminal elements. In the examples to follow, the elements of are typically the letters p , q , r , and so on.
 * The omega set Ω is a finite set of elements called operator symbols or logical connectives. The set Ω is partitioned into disjoint subsets as follows:

 In this partition, is the set of operator symbols of arity j .

 In the more familiar propositional calculi, Ω is typically partitioned as follows:

 A frequently adopted convention treats the constant logical values as operators of arity zero, thus:

 Some writers use the tilde (~), or N, instead of ¬ ; and some use v instead of as well as the ampersand (&), the prefixed K, or instead of . Notation varies even more for the set of logical values, with symbols like {false, true}, {F, T}, or {0, 1} all being seen in various contexts instead of .
 * The zeta set is a finite set of transformation rules that are called inference rules when they acquire logical applications.
 * The iota set is a countable set of initial points that are called axioms when they receive logical interpretations.

 The language of , also known as its set of formulas, well-formed formulas, is inductively defined by the following rules:

 1. Base: Any element of the alpha set is a formula of .
 2. If are formulas and is in , then is a formula.
 3. Closed: Nothing else is a formula of .

 Repeated applications of these rules permits the construction of complex formulas. For example:

 * By rule 1, p is a formula.
 * By rule 2, is a formula.
 * By rule 1, q is a formula.
 * By rule 2, is a formula.

 Example 1. Simple axiom system [ edit ]

 Let , where , , , are defined as follows:

 * The set , the countably infinite set of symbols that serve to represent logical propositions:
 * The functionally complete set of logical operators (logical connectives and negation) is as follows. Of the three connectives for conjunction, disjunction, and implication (, and → ), one can be taken as primitive and the other two can be defined in terms of it and negation ( ¬ ).[11] Alternatively, all of the logical operators may be defined in terms of a sole sufficient operator, such as the Sheffer stroke (nand). The biconditional () can of course be defined in terms of conjunction and implication as .
 Adopting negation and implication as the two primitive operations of a propositional calculus is tantamount to having the omega set partition as follows:

 Then is defined as , and is defined as .

 * The set (the set of initial points of logical deduction, i.e., logical axioms) is the axiom system proposed by Jan Łukasiewicz, and used as the propositional-calculus part of a Hilbert system. The axioms are all substitution instances of:
 + 
 + 
 + 
 * The set of transformation rules (rules of inference) is the sole rule modus ponens (i.e., from any formulas of the form and , infer ).

 This system is used in Metamath set.mm formal proof database.

 Example 2. Natural deduction system [ edit ]

 Let , where , , , are defined as follows:

 * The alpha set , is a countably infinite set of symbols, for example:
 * The omega set partitions as follows:

 In the following example of a propositional calculus, the transformation rules are intended to be interpreted as the inference rules of a so-called natural deduction system. The particular system presented here has no initial points, which means that its interpretation for logical applications derives its theorems from an empty axiom set.

 * The set of initial points is empty, that is, .
 * The set of transformation rules, , is described as follows:

 Our propositional calculus has eleven inference rules. These rules allow us to derive other true formulas given a set of formulas that are assumed to be true. The first ten simply state that we can infer certain well-formed formulas from other well-formed formulas. The last rule however uses hypothetical reasoning in the sense that in the premise of the rule we temporarily assume an (unproven) hypothesis to be part of the set of inferred formulas to see if we can infer a certain other formula. Since the first ten rules don't do this they are usually described as non-hypothetical rules, and the last one as a hypothetical rule.

 In describing the transformation rules, we may introduce a metalanguage symbol . It is basically a convenient shorthand for saying "infer that". The format is , in which Γ is a (possibly empty) set of formulas called premises, and ψ is a formula called conclusion. The transformation rule means that if every proposition in Γ is a theorem (or has the same truth value as the axioms), then ψ is also a theorem. Note that considering the following rule Conjunction introduction, we will know whenever Γ has more than one formula, we can always safely reduce it into one formula using conjunction. So for short, from that time on we may represent Γ as one formula instead of a set. Another omission for convenience is when Γ is an empty set, in which case Γ may not appear.

 Negation introduction From and , infer . That is, . Negation elimination From , infer . That is, . Double negation elimination From , infer p . That is, . Conjunction introduction From p and q , infer . That is, . Conjunction elimination From , infer p . From , infer q . That is, and . Disjunction introduction From p , infer . From q , infer . That is, and . Disjunction elimination From and and , infer r . That is, . Biconditional introduction From and , infer . That is, . Biconditional elimination From , infer . From , infer . That is, and . Modus ponens (conditional elimination) From p and , infer q . That is, . Conditional proof (conditional introduction) From [accepting p allows a proof of q ], infer . That is, .

 Basic and derived argument forms [ edit ]

 Name Sequent Description  
 Modus Ponens If p then q ; p ; therefore q  
 Modus Tollens If p then q ; not q ; therefore not p 
 Hypothetical Syllogism If p then q ; if q then r ; therefore, if p then r 
 Disjunctive Syllogism Either p or q , or both; not p ; therefore, q 
 Constructive Dilemma If p then q ; and if r then s ; but p or r ; therefore q or s 
 Destructive Dilemma If p then q ; and if r then s ; but not q or not s ; therefore not p or not r 
 Bidirectional Dilemma If p then q ; and if r then s ; but p or not s ; therefore q or not r 
 Simplification p and q are true; therefore p is true 
 Conjunction p and q are true separately; therefore they are true conjointly 
 Addition p is true; therefore the disjunction ( p or q ) is true 
 Composition If p then q ; and if p then r ; therefore if p is true then q and r are true 
 De Morgan's Theorem (1) The negation of ( p and q ) is equiv. to (not p or not q ) 
 De Morgan's Theorem (2) The negation of ( p or q ) is equiv. to (not p and not q ) 
 Commutation (1) ( p or q ) is equiv. to ( q or p ) 
 Commutation (2) ( p and q ) is equiv. to ( q and p ) 
 Commutation (3) ( p is equiv. to q ) is equiv. to ( q is equiv. to p ) 
 Association (1) p or ( q or r ) is equiv. to ( p or q ) or r 
 Association (2) p and ( q and r ) is equiv. to ( p and q ) and r 
 Distribution (1) p and ( q or r ) is equiv. to ( p and q ) or ( p and r ) 
 Distribution (2) p or ( q and r ) is equiv. to ( p or q ) and ( p or r ) 
 Double Negation and p is equivalent to the negation of not p 
 Transposition If p then q is equiv. to if not q then not p 
 Material Implication If p then q is equiv. to not p or q 
 Material Equivalence (1) ( p iff q ) is equiv. to (if p is true then q is true) and (if q is true then p is true) 
 Material Equivalence (2) ( p iff q ) is equiv. to either ( p and q are true) or (both p and q are false) 
 Material Equivalence (3) ( p iff q ) is equiv to., both ( p or not q is true) and (not p or q is true) 
 Exportation[12] from (if p and q are true then r is true) we can prove (if q is true then r is true, if p is true)
 Importation If p then (if q then r ) is equivalent to if p and q then r 
 Tautology (1) p is true is equiv. to p is true or p is true 
 Tautology (2) p is true is equiv. to p is true and p is true 
 Tertium non datur (Law of Excluded Middle) p or not p is true  
 Law of Non-Contradiction p and not p is false, is a true statement 
 

 Proofs in propositional calculus [ edit ]

 One of the main uses of a propositional calculus, when interpreted for logical applications, is to determine relations of logical equivalence between propositional formulas. These relationships are determined by means of the available transformation rules, sequences of which are called derivations or proofs.

 In the discussion to follow, a proof is presented as a sequence of numbered lines, with each line consisting of a single formula followed by a reason or justification for introducing that formula. Each premise of the argument, that is, an assumption introduced as an hypothesis of the argument, is listed at the beginning of the sequence and is marked as a "premise" in lieu of other justification. The conclusion is listed on the last line. A proof is complete if every line follows from the previous ones by the correct application of a transformation rule. (For a contrasting approach, see proof-trees).

 Example of a proof in natural deduction system [ edit ]

 * To be shown that A → A.
 * One possible proof of this (which, though valid, happens to contain more steps than are necessary) may be arranged as follows:
 Example of a proof
 Number Formula Reason 
 1 premise 
 2 From (1) by disjunction introduction 
 3 From (1) and (2) by conjunction introduction
 4 From (3) by conjunction elimination 
 5 Summary of (1) through (4) 
 6 From (5) by conditional proof 
 

 Interpret as "Assuming A , infer A ". Read as "Assuming nothing, infer that A implies A ", or "It is a tautology that A implies A ", or "It is always true that A implies A ".

 Example of a proof in a classical propositional calculus system [ edit ]

 We now prove the same theorem in the axiomatic system by Jan Łukasiewicz described above, which is an example of a Hilbert-style deductive system for the classical propositional calculus.

 The axioms are:

 (A1) (A2) (A3)

 And the proof is as follows:

 1. (instance of (A1))
 2. (instance of (A2))
 3. (from (1) and (2) by modus ponens)
 4. (instance of (A1))
 5. (from (4) and (3) by modus ponens)

 Soundness and completeness of the rules [ edit ]

 The crucial properties of this set of rules are that they are sound and complete. Informally this means that the rules are correct and that no other rules are required. These claims can be made more formal as follows. Note that the proofs for the soundness and completeness of the propositional logic are not themselves proofs in propositional logic ; these are theorems in ZFC used as a metatheory to prove properties of propositional logic.

 We define a truth assignment as a function that maps propositional variables to true or false. Informally such a truth assignment can be understood as the description of a possible state of affairs (or possible world) where certain statements are true and others are not. The semantics of formulas can then be formalized by defining for which "state of affairs" they are considered to be true, which is what is done by the following definition.

 We define when such a truth assignment A satisfies a certain well-formed formula with the following rules:

 * A satisfies the propositional variable P if and only if A ( P ) = true
 * A satisfies ¬ φ if and only if A does not satisfy φ
 * A satisfies ( φ ∧ ψ ) if and only if A satisfies both φ and ψ
 * A satisfies ( φ ∨ ψ ) if and only if A satisfies at least one of either φ or ψ
 * A satisfies ( φ → ψ ) if and only if it is not the case that A satisfies φ but not ψ
 * A satisfies ( φ ↔ ψ ) if and only if A satisfies both φ and ψ or satisfies neither one of them

 With this definition we can now formalize what it means for a formula φ to be implied by a certain set S of formulas. Informally this is true if in all worlds that are possible given the set of formulas S the formula φ also holds. This leads to the following formal definition: We say that a set S of well-formed formulas semantically entails (or implies) a certain well-formed formula φ if all truth assignments that satisfy all the formulas in S also satisfy φ .

 Finally we define syntactical entailment such that φ is syntactically entailed by S if and only if we can derive it with the inference rules that were presented above in a finite number of steps. This allows us to formulate exactly what it means for the set of inference rules to be sound and complete:

 Soundness: If the set of well-formed formulas S syntactically entails the well-formed formula φ then S semantically entails φ .

 Completeness: If the set of well-formed formulas S semantically entails the well-formed formula φ then S syntactically entails φ .

 For the above set of rules this is indeed the case.

 Sketch of a soundness proof [ edit ]

 (For most logical systems, this is the comparatively "simple" direction of proof)

 Notational conventions: Let G be a variable ranging over sets of sentences. Let A, B and C range over sentences. For " G syntactically entails A " we write " G proves A ". For " G semantically entails A " we write " G implies A ".

 We want to show: ( A )( G ) (if G proves A , then G implies A ).

 We note that " G proves A " has an inductive definition, and that gives us the immediate resources for demonstrating claims of the form "If G proves A , then ...". So our proof proceeds by induction.

 1. Basis. Show: If A is a member of G , then G implies A .
 2. Basis. Show: If A is an axiom, then G implies A .
 3. Inductive step (induction on n , the length of the proof):
 1. Assume for arbitrary G and A that if G proves A in n or fewer steps, then G implies A .
 2. For each possible application of a rule of inference at step n + 1 , leading to a new theorem B , show that G implies B .

 Notice that Basis Step II can be omitted for natural deduction systems because they have no axioms. When used, Step II involves showing that each of the axioms is a (semantic) logical truth.

 The Basis steps demonstrate that the simplest provable sentences from G are also implied by G , for any G . (The proof is simple, since the semantic fact that a set implies any of its members, is also trivial.) The Inductive step will systematically cover all the further sentences that might be provable—by considering each case where we might reach a logical conclusion using an inference rule—and shows that if a new sentence is provable, it is also logically implied. (For example, we might have a rule telling us that from " A " we can derive " A or B ". In III.a We assume that if A is provable it is implied. We also know that if A is provable then " A or B " is provable. We have to show that then " A or B " too is implied. We do so by appeal to the semantic definition and the assumption we just made. A is provable from G , we assume. So it is also implied by G . So any semantic valuation making all of G true makes A true. But any valuation making A true makes " A or B " true, by the defined semantics for "or". So any valuation which makes all of G true makes " A or B " true. So " A or B " is implied.) Generally, the Inductive step will consist of a lengthy but simple case-by-case analysis of all the rules of inference, showing that each "preserves" semantic implication.

 By the definition of provability, there are no sentences provable other than by being a member of G , an axiom, or following by a rule; so if all of those are semantically implied, the deduction calculus is sound.

 Sketch of completeness proof [ edit ]

 (This is usually the much harder direction of proof.)

 We adopt the same notational conventions as above.

 We want to show: If G implies A , then G proves A . We proceed by contraposition: We show instead that if G does not prove A then G does not imply A . If we show that there is a model where A does not hold despite G being true, then obviously G does not imply A . The idea is to build such a model out of our very assumption that G does not prove A .

 1. G does not prove A . (Assumption)
 2. If G does not prove A , then we can construct an (infinite) Maximal Set, G∗, which is a superset of G and which also does not prove A .
 1. Place an ordering (with order type ω) on all the sentences in the language (e.g., shortest first, and equally long ones in extended alphabetical ordering), and number them ( E1 , E2 , ...)
 2. Define a series G n of sets ( G0 , G1 , ...) inductively:
 1. 
 2. If proves A , then
 3. If does not prove A , then
 3. Define G∗ as the union of all the G n. (That is, G∗ is the set of all the sentences that are in any G n.)
 4. It can be easily shown that
 1. G∗ contains (is a superset of) G (by (b.i));
 2. G∗ does not prove A (because the proof would contain only finitely many sentences and when the last of them is introduced in some G n, that G n would prove A contrary to the definition of G n); and
 3. G∗ is a Maximal Set with respect to A : If any more sentences whatever were added to G∗, it would prove A . (Because if it were possible to add any more sentences, they should have been added when they were encountered during the construction of the G n, again by definition)
 3. If G∗ is a Maximal Set with respect to A , then it is truth-like. This means that it contains C if and only if it does not contain ¬C ; If it contains C and contains "If C then B " then it also contains B ; and so forth. In order to show this, one has to show the axiomatic system is strong enough for the following:
 + For any formulas C and D , if it proves both C and ¬C , then it proves D . From this it follows, that a Maximal Set with respect to A cannot prove both C and ¬C , as otherwise it would prove A .
 + For any formulas C and D , if it proves both C → D and ¬C → D , then it proves D . This is used, together with the deduction theorem, to show that for any formula, either it or its negation is in G∗: Let B be a formula not in G∗; then G∗ with the addition of B proves A . Thus from the deduction theorem it follows that G∗ proves B → A . But suppose ¬B were also not in G∗, then by the same logic G∗ also proves ¬B → A ; but then G∗ proves A , which we have already shown to be false.
 + For any formulas C and D , if it proves C and D , then it proves C → D .
 + For any formulas C and D , if it proves C and ¬D , then it proves ¬( C → D ).
 + For any formulas C and D , if it proves ¬C , then it proves C → D .
 If additional logical operation (such as conjunction and/or disjunction) are part of the vocabulary as well, then there are additional requirement on the axiomatic system (e.g. that if it proves C and D , it would also prove their conjunction).
 4. If G∗ is truth-like there is a G∗-Canonical valuation of the language: one that makes every sentence in G∗ true and everything outside G∗ false while still obeying the laws of semantic composition in the language. Note that the requirement that it is truth-like is needed to guarantee that the laws of semantic composition in the language will be satisfied by this truth assignment.
 5. A G∗-canonical valuation will make our original set G all true, and make A false.
 6. If there is a valuation on which G are true and A is false, then G does not (semantically) imply A .

 Thus every system that has modus ponens as an inference rule, and proves the following theorems (including substitutions thereof) is complete:

 * 
 * 
 * 
 * 
 * 
 * 
 * 
 * 

 The first five are used for the satisfaction of the five conditions in stage III above, and the last three for proving the deduction theorem.

 Example [ edit ]

 As an example, it can be shown that as any other tautology, the three axioms of the classical propositional calculus system described earlier can be proven in any system that satisfies the above, namely that has modus ponens as an inference rule, and proves the above eight theorems (including substitutions thereof). Out of the eight theorems, the last two are two of the three axioms; the third axiom, , can be proven as well, as we now show.

 For the proof we may use the hypothetical syllogism theorem (in the form relevant for this axiomatic system), since it only relies on the two axioms that are already in the above set of eight theorems. The proof then is as follows:

 1. (instance of the 7th theorem)
 2. (instance of the 7th theorem)
 3. (from (1) and (2) by modus ponens)
 4. (instance of the hypothetical syllogism theorem)
 5. (instance of the 5th theorem)
 6. (from (5) and (4) by modus ponens)
 7. (instance of the 2nd theorem)
 8. (instance of the 7th theorem)
 9. (from (7) and (8) by modus ponens)
 10. (instance of the 8th theorem)
 11. (from (9) and (10) by modus ponens)
 12. (from (3) and (11) by modus ponens)
 13. (instance of the 8th theorem)
 14. (from (12) and (13) by modus ponens)
 15. (from (6) and (14) by modus ponens)

 Verifying completeness for the classical propositional calculus system [ edit ]

 We now verify that the classical propositional calculus system described earlier can indeed prove the required eight theorems mentioned above. We use several lemmas proven here:

 (DN1) - Double negation (one direction) (DN2) - Double negation (another direction) (HS1) - one form of Hypothetical syllogism (HS2) - another form of Hypothetical syllogism (TR1) - Transposition (TR2) - another form of transposition. (L1) (L3)

 We also use the method of the hypothetical syllogism metatheorem as a shorthand for several proof steps.

 * - proof:
 1. (instance of (A1))
 2. (instance of (TR1))
 3. (from (1) and (2) using the hypothetical syllogism metatheorem)
 4. (instance of (DN1))
 5. (instance of (HS1))
 6. (from (4) and (5) using modus ponens)
 7. (from (3) and (6) using the hypothetical syllogism metatheorem)
 * - proof:
 1. (instance of (HS1))
 2. (instance of (L3))
 3. (instance of (HS1))
 4. (from (2) and (3) by modus ponens)
 5. (from (1) and (4) using the hypothetical syllogism metatheorem)
 6. (instance of (TR2))
 7. (instance of (HS2))
 8. (from (6) and (7) using modus ponens)
 9. (from (5) and (8) using the hypothetical syllogism metatheorem)
 * - proof:
 1. (instance of (A1))
 2. (instance of (A1))
 3. (from (1) and (2) using modus ponens)
 * - proof:
 1. (instance of (L1))
 2. (instance of (TR1))
 3. (from (1) and (2) using the hypothetical syllogism metatheorem)
 * - proof:
 1. (instance of (A1))
 2. (instance of (A3))
 3. (from (1) and (2) using the hypothetical syllogism metatheorem)
 * - proof given in the proof example above
 * - axiom (A1)
 * - axiom (A2)

 Another outline for a completeness proof [ edit ]

 If a formula is a tautology, then there is a truth table for it which shows that each valuation yields the value true for the formula. Consider such a valuation. By mathematical induction on the length of the subformulas, show that the truth or falsity of the subformula follows from the truth or falsity (as appropriate for the valuation) of each propositional variable in the subformula. Then combine the lines of the truth table together two at a time by using "( P is true implies S ) implies (( P is false implies S ) implies S )". Keep repeating this until all dependencies on propositional variables have been eliminated. The result is that we have proved the given tautology. Since every tautology is provable, the logic is complete.

 Interpretation of a truth-functional propositional calculus [ edit ]

 An interpretation of a truth-functional propositional calculus is an assignment to each propositional symbol of of one or the other (but not both) of the truth values truth (T) and falsity (F), and an assignment to the connective symbols of of their usual truth-functional meanings. An interpretation of a truth-functional propositional calculus may also be expressed in terms of truth tables.[13]

 For distinct propositional symbols there are distinct possible interpretations. For any particular symbol , for example, there are possible interpretations:

 1. is assigned T, or
 2. is assigned F.

 For the pair , there are possible interpretations:

 1. both are assigned T,
 2. both are assigned F,
 3. is assigned T and is assigned F, or
 4. is assigned F and is assigned T.[13]

 Since has , that is, denumerably many propositional symbols, there are , and therefore uncountably many distinct possible interpretations of .[13]

 Interpretation of a sentence of truth-functional propositional logic [ edit ]

 Main article: Interpretation (logic)

 If φ and ψ are formulas of and is an interpretation of then the following definitions apply:

 * A sentence of propositional logic is true under an interpretation if assigns the truth value T to that sentence. If a sentence is true under an interpretation, then that interpretation is called a model of that sentence.
 * φ is false under an interpretation if φ is not true under .[13]
 * A sentence of propositional logic is logically valid if it is true under every interpretation. φ means that φ is logically valid.
 * A sentence ψ of propositional logic is a semantic consequence of a sentence φ if there is no interpretation under which φ is true and ψ is false.
 * A sentence of propositional logic is consistent if it is true under at least one interpretation. It is inconsistent if it is not consistent.

 Some consequences of these definitions:

 * For any given interpretation a given formula is either true or false.[13]
 * No formula is both true and false under the same interpretation.[13]
 * φ is false for a given interpretation iff is true for that interpretation; and φ is true under an interpretation iff is false under that interpretation.[13]
 * If φ and are both true under a given interpretation, then ψ is true under that interpretation.[13]
 * If and , then .[13]
 * is true under iff φ is not true under .
 * is true under iff either φ is not true under or ψ is true under .[13]
 * A sentence ψ of propositional logic is a semantic consequence of a sentence φ iff is logically valid, that is, iff .[13]

 Alternative calculus [ edit ]

 It is possible to define another version of propositional calculus, which defines most of the syntax of the logical operators by means of axioms, and which uses only one inference rule.

 Axioms [ edit ]

 Let φ , χ , and ψ stand for well-formed formulas. (The well-formed formulas themselves would not contain any Greek letters, but only capital Roman letters, connective operators, and parentheses.) Then the axioms are as follows:

 Axioms
 Name Axiom Schema Description 
 THEN-1 Add hypothesis χ , implication introduction
 THEN-2 Distribute hypothesis over implication 
 AND-1 Eliminate conjunction 
 AND-2 
 AND-3 Introduce conjunction 
 OR-1 Introduce disjunction 
 OR-2 
 OR-3 Eliminate disjunction 
 NOT-1 Introduce negation 
 NOT-2 Eliminate negation 
 NOT-3 Excluded middle, classical logic 
 IFF-1 Eliminate equivalence 
 IFF-2 
 IFF-3 Introduce equivalence 
 
 * Axiom THEN-2 may be considered to be a "distributive property of implication with respect to implication."
 * Axioms AND-1 and AND-2 correspond to "conjunction elimination". The relation between AND-1 and AND-2 reflects the commutativity of the conjunction operator.
 * Axiom AND-3 corresponds to "conjunction introduction."
 * Axioms OR-1 and OR-2 correspond to "disjunction introduction." The relation between OR-1 and OR-2 reflects the commutativity of the disjunction operator.
 * Axiom NOT-1 corresponds to "reductio ad absurdum."
 * Axiom NOT-2 says that "anything can be deduced from a contradiction."
 * Axiom NOT-3 is called "tertium non-datur" (Latin: "a third is not given") and reflects the semantic valuation of propositional formulas: a formula can have a truth-value of either true or false. There is no third truth-value, at least not in classical logic. Intuitionistic logicians do not accept the axiom NOT-3.

 Inference rule [ edit ]

 The inference rule is modus ponens:

 .

 Meta-inference rule [ edit ]

 Let a demonstration be represented by a sequence, with hypotheses to the left of the turnstile and the conclusion to the right of the turnstile. Then the deduction theorem can be stated as follows:

 If the sequence has been demonstrated, then it is also possible to demonstrate the sequence .

 This deduction theorem (DT) is not itself formulated with propositional calculus: it is not a theorem of propositional calculus, but a theorem about propositional calculus. In this sense, it is a meta-theorem, comparable to theorems about the soundness or completeness of propositional calculus.

 On the other hand, DT is so useful for simplifying the syntactical proof process that it can be considered and used as another inference rule, accompanying modus ponens. In this sense, DT corresponds to the natural conditional proof inference rule which is part of the first version of propositional calculus introduced in this article.

 The converse of DT is also valid:

 If the sequence has been demonstrated, then it is also possible to demonstrate the sequence

 in fact, the validity of the converse of DT is almost trivial compared to that of DT:

 If then 1: 2: and from (1) and (2) can be deduced 3: by means of modus ponens, Q.E.D.

 The converse of DT has powerful implications: it can be used to convert an axiom into an inference rule. For example, by axiom AND-1 we have,

 which can be transformed by means of the converse of the deduction theorem into

 which tells us that the inference rule

 is admissible. This inference rule is conjunction elimination, one of the ten inference rules used in the first version (in this article) of the propositional calculus.

 Example of a proof [ edit ]

 The following is an example of a (syntactical) demonstration, involving only axioms THEN-1 and THEN-2:

 Prove: (Reflexivity of implication).

 Proof:

 1. Axiom THEN-2 with
 2. Axiom THEN-1 with
 3. From (1) and (2) by modus ponens.
 4. Axiom THEN-1 with
 5. From (3) and (4) by modus ponens.

 Equivalence to equational logics [ edit ]

 The preceding alternative calculus is an example of a Hilbert-style deduction system. In the case of propositional systems the axioms are terms built with logical connectives and the only inference rule is modus ponens. Equational logic as standardly used informally in high school algebra is a different kind of calculus from Hilbert systems. Its theorems are equations and its inference rules express the properties of equality, namely that it is a congruence on terms that admits substitution.

 Classical propositional calculus as described above is equivalent to Boolean algebra, while intuitionistic propositional calculus is equivalent to Heyting algebra. The equivalence is shown by translation in each direction of the theorems of the respective systems. Theorems of classical or intuitionistic propositional calculus are translated as equations of Boolean or Heyting algebra respectively. Conversely theorems of Boolean or Heyting algebra are translated as theorems of classical or intuitionistic calculus respectively, for which is a standard abbreviation. In the case of Boolean algebra can also be translated as , but this translation is incorrect intuitionistically.

 In both Boolean and Heyting algebra, inequality can be used in place of equality. The equality is expressible as a pair of inequalities and . Conversely the inequality is expressible as the equality , or as . The significance of inequality for Hilbert-style systems is that it corresponds to the latter's deduction or entailment symbol . An entailment

 is translated in the inequality version of the algebraic framework as

 Conversely the algebraic inequality is translated as the entailment

 .

 The difference between implication and inequality or entailment or is that the former is internal to the logic while the latter is external. Internal implication between two terms is another term of the same kind. Entailment as external implication between two terms expresses a metatruth outside the language of the logic, and is considered part of the metalanguage. Even when the logic under study is intuitionistic, entailment is ordinarily understood classically as two-valued: either the left side entails, or is less-or-equal to, the right side, or it is not.

 Similar but more complex translations to and from algebraic logics are possible for natural deduction systems as described above and for the sequent calculus. The entailments of the latter can be interpreted as two-valued, but a more insightful interpretation is as a set, the elements of which can be understood as abstract proofs organized as the morphisms of a category. In this interpretation the cut rule of the sequent calculus corresponds to composition in the category. Boolean and Heyting algebras enter this picture as special categories having at most one morphism per homset, i.e., one proof per entailment, corresponding to the idea that existence of proofs is all that matters: any proof will do and there is no point in distinguishing them.

 Graphical calculi [ edit ]

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 It is possible to generalize the definition of a formal language from a set of finite sequences over a finite basis to include many other sets of mathematical structures, so long as they are built up by finitary means from finite materials. What's more, many of these families of formal structures are especially well-suited for use in logic.

 For example, there are many families of graphs that are close enough analogues of formal languages that the concept of a calculus is quite easily and naturally extended to them. Many species of graphs arise as parse graphs in the syntactic analysis of the corresponding families of text structures. The exigencies of practical computation on formal languages frequently demand that text strings be converted into pointer structure renditions of parse graphs, simply as a matter of checking whether strings are well-formed formulas or not. Once this is done, there are many advantages to be gained from developing the graphical analogue of the calculus on strings. The mapping from strings to parse graphs is called parsing and the inverse mapping from parse graphs to strings is achieved by an operation that is called traversing the graph.

 Other logical calculi [ edit ]

 Propositional calculus is about the simplest kind of logical calculus in current use. It can be extended in several ways. (Aristotelian "syllogistic" calculus, which is largely supplanted in modern logic, is in some ways simpler – but in other ways more complex – than propositional calculus.) The most immediate way to develop a more complex logical calculus is to introduce rules that are sensitive to more fine-grained details of the sentences being used.

 First-order logic (a.k.a. first-order predicate logic) results when the "atomic sentences" of propositional logic are broken up into terms, variables, predicates, and quantifiers, all keeping the rules of propositional logic with some new ones introduced. (For example, from "All dogs are mammals" we may infer "If Rover is a dog then Rover is a mammal".) With the tools of first-order logic it is possible to formulate a number of theories, either with explicit axioms or by rules of inference, that can themselves be treated as logical calculi. Arithmetic is the best known of these; others include set theory and mereology. Second-order logic and other higher-order logics are formal extensions of first-order logic. Thus, it makes sense to refer to propositional logic as "zeroth-order logic", when comparing it with these logics.

 Modal logic also offers a variety of inferences that cannot be captured in propositional calculus. For example, from "Necessarily p " we may infer that p . From p we may infer "It is possible that p ". The translation between modal logics and algebraic logics concerns classical and intuitionistic logics but with the introduction of a unary operator on Boolean or Heyting algebras, different from the Boolean operations, interpreting the possibility modality, and in the case of Heyting algebra a second operator interpreting necessity (for Boolean algebra this is redundant since necessity is the De Morgan dual of possibility). The first operator preserves 0 and disjunction while the second preserves 1 and conjunction.

 Many-valued logics are those allowing sentences to have values other than true and false. (For example, neither and both are standard "extra values"; "continuum logic" allows each sentence to have any of an infinite number of "degrees of truth" between true and false.) These logics often require calculational devices quite distinct from propositional calculus. When the values form a Boolean algebra (which may have more than two or even infinitely many values), many-valued logic reduces to classical logic; many-valued logics are therefore only of independent interest when the values form an algebra that is not Boolean.

 Solvers [ edit ]

 Deciding satisfiability of propositional logic formulas is an NP-complete problem. However, practical methods exist (e.g., DPLL algorithm, 1962; Chaff algorithm, 2001) that are very fast for many useful cases. Recent work has extended the SAT solver algorithms to work with propositions containing arithmetic expressions; these are the SMT solvers.

 See also [ edit ]

 * Philosophy portal

 Higher logical levels [ edit ]

 * First-order logic
 * Second-order propositional logic
 * Second-order logic
 * Higher-order logic

 Related topics [ edit ]

 * Boolean algebra (logic)
 * Boolean algebra (structure)
 * Boolean algebra topics
 * Boolean domain
 * Boolean function
 * Boolean-valued function
 * Categorical logic
 * Combinational logic
 * Combinatory logic
 * Conceptual graph
 * Disjunctive syllogism
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 * Equational logic
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 * Frege's propositional calculus
 * Implicational propositional calculus
 * Intuitionistic propositional calculus
 * Jean Buridan
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 * Mathematical logic
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 * Paul of Venice
 * Peirce's law
 * Peter of Spain (author)
 * Propositional formula
 * Symmetric difference
 * Tautology (rule of inference)
 * Truth function
 * Truth table
 * Walter Burley
 * William of Sherwood

 References [ edit ]

 1. ^ "Propositional Logic | Brilliant Math & Science Wiki". brilliant.org . Retrieved 20 August 2020 .
 2. ^ Bobzien, Susanne (1 January 2016). Zalta, Edward N. (ed.). The Stanford Encyclopedia of Philosophy – via Stanford Encyclopedia of Philosophy.
 3. ^ "Propositional Logic | Internet Encyclopedia of Philosophy" . Retrieved 20 August 2020 .
 4. ^ Marenbon, John (2007). Medieval philosophy: an historical and philosophical introduction. Routledge. p. 137.
 5. ^ Peckhaus, Volker (1 January 2014). Zalta, Edward N. (ed.). The Stanford Encyclopedia of Philosophy – via Stanford Encyclopedia of Philosophy.
 6. ^ Hurley, Patrick (2007). A Concise Introduction to Logic 10th edition. Wadsworth Publishing. p. 392.
 7. ^ Beth, Evert W.; "Semantic entailment and formal derivability", series: Mededlingen van de Koninklijke Nederlandse Akademie van Wetenschappen, Afdeling Letterkunde, Nieuwe Reeks, vol. 18, no. 13, Noord-Hollandsche Uitg. Mij., Amsterdam, 1955, pp. 309–42. Reprinted in Jaakko Intikka (ed.) The Philosophy of Mathematics , Oxford University Press, 1969
 8. ^ a b Truth in Frege
 9. ^ a b c "Russell: the Journal of Bertrand Russell Studies".
 10. ^ Anellis, Irving H. (2012). "Peirce's Truth-functional Analysis and the Origin of the Truth Table". History and Philosophy of Logic. 33: 87–97. doi:10.1080/01445340.2011.621702. S2CID 170654885.
 11. ^ Wernick, William (1942) "Complete Sets of Logical Functions," Transactions of the American Mathematical Society 51 , pp. 117–132.
 12. ^ Toida, Shunichi (2 August 2009). "Proof of Implications". CS381 Discrete Structures/Discrete Mathematics Web Course Material. Department of Computer Science, Old Dominion University . Retrieved 10 March 2010 .
 13. ^ a b c d e f g h i j k Hunter, Geoffrey (1971). Metalogic: An Introduction to the Metatheory of Standard First-Order Logic. University of California Press. ISBN 0-520-02356-0.

 Further reading [ edit ]

 * Brown, Frank Markham (2003), Boolean Reasoning: The Logic of Boolean Equations, 1st edition, Kluwer Academic Publishers, Norwell, MA. 2nd edition, Dover Publications, Mineola, NY.
 * Chang, C.C. and Keisler, H.J. (1973), Model Theory, North-Holland, Amsterdam, Netherlands.
 * Kohavi, Zvi (1978), Switching and Finite Automata Theory, 1st edition, McGraw–Hill, 1970. 2nd edition, McGraw–Hill, 1978.
 * Korfhage, Robert R. (1974), Discrete Computational Structures, Academic Press, New York, NY.
 * Lambek, J. and Scott, P.J. (1986), Introduction to Higher Order Categorical Logic, Cambridge University Press, Cambridge, UK.
 * Mendelson, Elliot (1964), Introduction to Mathematical Logic, D. Van Nostrand Company.

 Related works [ edit ]

 * Hofstadter, Douglas (1979). Gödel, Escher, Bach: An Eternal Golden Braid. Basic Books. ISBN 978-0-465-02656-2.

 External links [ edit ]

 Wikimedia Commons has media related to Propositional logic.
 * Klement, Kevin C. (2006), "Propositional Logic", in James Fieser and Bradley Dowden (eds.), Internet Encyclopedia of Philosophy, Eprint.
 * Formal Predicate Calculus, contains a systematic formal development along the lines of Alternative calculus
 * forall x: an introduction to formal logic, by P.D. Magnus, covers formal semantics and proof theory for sentential logic.
 * Chapter 2 / Propositional Logic from Logic In Action
 * Propositional sequent calculus prover on Project Nayuki. (note: implication can be input in the form !X|Y, and a sequent can be a single formula prefixed with > and having no commas)
 * Propositional Logic - A Generative Grammar
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Propositional calculus is a branch of logic. It is also called propositional logic, statement logic, sentential calculus, sentential logic, or sometimes  zeroth-order logic . It deals with propositions (which can be true or false) and relations between propositions, including the construction of arguments based on them. Compound propositions are formed by connecting propositions by logical connectives. Propositions that contain no logical connectives are called atomic propositions.
 Unlike first-order logic, propositional logic does not deal with non-logical objects, predicates about them, or quantifiers. However, all the machinery of propositional logic is included in first-order logic and higher-order logics. In this sense, propositional logic is the foundation of first-order logic and higher-order logic.
Explanation[edit]
Logical connectives are found in natural languages. In English for example, some examples are "and" (conjunction), "or" (disjunction), "not" (negation) and "if" (but only when used to denote material conditional).
 The following is an example of a very simple inference within the scope of propositional logic:
 Premise 1: If it's raining then it's cloudy.
Premise 2: It's raining.
Conclusion: It's cloudy. 
Both premises and the conclusion are propositions. The premises are taken for granted, and with the application of modus ponens (an inference rule), the conclusion follows.
 As propositional logic is not concerned with the structure of propositions beyond the point where they can't be decomposed any more by logical connectives, this inference can be restated replacing those atomic statements with statement letters, which are interpreted as variables representing statements:
 Premise 1: 
Premise 2: 
Conclusion:  
The same can be stated succinctly in the following way:
   
When P is interpreted as "It's raining" and Q as "it's cloudy" the above symbolic expressions can be seen to correspond exactly with the original expression in natural language. Not only that, but they will also correspond with any other inference of this form, which will be valid on the same basis this inference is.
 Propositional logic may be studied through a formal system in which formulas of a formal language may be interpreted to represent propositions. A system of axioms and inference rules allows certain formulas to be derived. These derived formulas are called theorems and may be interpreted to be true propositions. A constructed sequence of such formulas is known as a  derivation  or proof and the last formula of the sequence is the theorem. The derivation may be interpreted as proof of the proposition represented by the theorem.
 When a formal system is used to represent formal logic, only statement letters (usually capital roman letters such as , and ) are represented directly. The natural language propositions that arise when they're interpreted are outside the scope of the system, and the relation between the formal system and its interpretation is likewise outside the formal system itself.
 In classical truth-functional propositional logic, formulas are interpreted as having precisely one of two possible truth values, the truth value of true or the truth value of false. [1]  The principle of bivalence and the law of excluded middle are upheld. Truth-functional propositional logic defined as such and systems isomorphic to it are considered to be  zeroth-order logic . However, alternative propositional logics are also possible. For more, see Other logical calculi below.
History[edit]
Although propositional logic (which is interchangeable with propositional calculus) had been hinted by earlier philosophers, it was developed into a formal logic (Stoic logic) by Chrysippus in the 3rd century BC [2]  and expanded by his successor Stoics. The logic was focused on propositions. This advancement was different from the traditional syllogistic logic, which was focused on terms. However, most of the original writings were lost [3]  and the propositional logic developed by the Stoics was no longer understood later in antiquity.[ citation needed ] Consequently, the system was essentially reinvented by Peter Abelard in the 12th century. [4] 
 Propositional logic was eventually refined using symbolic logic. The 17th/18th-century mathematician Gottfried Leibniz has been credited with being the founder of symbolic logic for his work with the calculus ratiocinator. Although his work was the first of its kind, it was unknown to the larger logical community. Consequently, many of the advances achieved by Leibniz were recreated by logicians like George Boole and Augustus De Morgan—completely independent of Leibniz. [5] 
 Just as propositional logic can be considered an advancement from the earlier syllogistic logic, Gottlob Frege's predicate logic can be also considered an advancement from the earlier propositional logic. One author describes predicate logic as combining "the distinctive features of syllogistic logic and propositional logic." [6]  Consequently, predicate logic ushered in a new era in logic's history; however, advances in propositional logic were still made after Frege, including natural deduction, truth trees and truth tables. Natural deduction was invented by Gerhard Gentzen and Jan Łukasiewicz. Truth trees were invented by Evert Willem Beth. [7]  The invention of truth tables, however, is of uncertain attribution.
 Within works by Frege [8]  and Bertrand Russell, [9]  are ideas influential to the invention of truth tables. The actual tabular structure (being formatted as a table), itself, is generally credited to either Ludwig Wittgenstein or Emil Post (or both, independently). [8]  Besides Frege and Russell, others credited with having ideas preceding truth tables include Philo, Boole, Charles Sanders Peirce, [10]  and Ernst Schröder. Others credited with the tabular structure include Jan Łukasiewicz, Alfred North Whitehead, William Stanley Jevons, John Venn, and Clarence Irving Lewis. [9]  Ultimately, some have concluded, like John Shosky, that "It is far from clear that any one person should be given the title of 'inventor' of truth-tables.". [9] 
Terminology[edit]
In general terms, a calculus is a formal system that consists of a set of syntactic expressions ( well-formed formulas ), a distinguished subset of these expressions (axioms), plus a set of formal rules that define a specific binary relation, intended to be interpreted as logical equivalence, on the space of expressions.
 When the formal system is intended to be a logical system, the expressions are meant to be interpreted as statements, and the rules, known to be inference rules, are typically intended to be truth-preserving. In this setting, the rules, which may include axioms, can then be used to derive ("infer") formulas representing true statements—from given formulas representing true statements.
 The set of axioms may be empty, a nonempty finite set, or a countably infinite set (see axiom schema). A formal grammar recursively defines the expressions and well-formed formulas of the language. In addition a semantics may be given which defines truth and valuations (or interpretations).
 The language of a propositional calculus consists of
 a set of primitive symbols, variously referred to as  atomic formulas , placeholders, proposition letters, or variables, and
a set of operator symbols, variously interpreted as  logical operators  or logical connectives. 
A  well-formed formula  is any atomic formula, or any formula that can be built up from atomic formulas by means of operator symbols according to the rules of the grammar.
 Mathematicians sometimes distinguish between propositional constants, propositional variables, and schemata. Propositional constants represent some particular proposition, while propositional variables range over the set of all atomic propositions. Schemata, however, range over all propositions. It is common to represent propositional constants by A, B, and C, propositional variables by P, Q, and R, and schematic letters are often Greek letters, most often φ, ψ, and χ.
Basic concepts[edit]
The following outlines a standard propositional calculus. Many different formulations exist which are all more or less equivalent, but differ in the details of:
 their language (i.e., the particular collection of primitive symbols and operator symbols),
the set of axioms, or distinguished formulas, and
the set of inference rules. 
Any given proposition may be represented with a letter called a 'propositional constant', analogous to representing a number by a letter in mathematics (e.g., a = 5). All propositions require exactly one of two truth-values: true or false. For example, let P be the proposition that it is raining outside. This will be true (P) if it is raining outside, and false otherwise (¬P).
 We then define truth-functional operators, beginning with negation. ¬P represents the negation of P, which can be thought of as the denial of P. In the example above, ¬P expresses that it is not raining outside, or by a more standard reading: "It is not the case that it is raining outside." When P is true, ¬P is false; and when P is false, ¬P is true. As a result, ¬ ¬P always has the same truth-value as P.
Conjunction is a truth-functional connective which forms a proposition out of two simpler propositions, for example, P and Q. The conjunction of P and Q is written P ∧ Q, and expresses that each are true. We read P ∧ Q as "P and Q". For any two propositions, there are four possible assignments of truth values:
 P is true and Q is true
P is true and Q is false
P is false and Q is true
P is false and Q is false   
 The conjunction of P and Q is true in case 1, and is false otherwise. Where P is the proposition that it is raining outside and Q is the proposition that a cold-front is over Kansas, P ∧ Q is true when it is raining outside and there is a cold-front over Kansas. If it is not raining outside, then P ∧ Q is false; and if there is no cold-front over Kansas, then P ∧ Q is also false. 
 Disjunction resembles conjunction in that it forms a proposition out of two simpler propositions. We write it P ∨ Q, and it is read "P or Q". It expresses that either P or Q is true. Thus, in the cases listed above, the disjunction of P with Q is true in all cases—except case 4. Using the example above, the disjunction expresses that it is either raining outside, or there is a cold front over Kansas. (Note, this use of disjunction is supposed to resemble the use of the English word "or". However, it is most like the English inclusive "or", which can be used to express the truth of at least one of two propositions. It is not like the English exclusive "or", which expresses the truth of exactly one of two propositions. In other words, the exclusive "or" is false when both P and Q are true (case 1), and similarly is false when both P and Q are false (case 4). An example of the exclusive or is: You may have a bagel or a pastry, but not both. Often in natural language, given the appropriate context, the addendum "but not both" is omitted—but implied. In mathematics, however, "or" is always inclusive or; if exclusive or is meant it will be specified, possibly by "xor".)
Material conditional also joins two simpler propositions, and we write P → Q, which is read "if P then Q". The proposition to the left of the arrow is called the antecedent, and the proposition to the right is called the consequent. (There is no such designation for conjunction or disjunction, since they are commutative operations.) It expresses that Q is true whenever P is true. Thus P → Q is true in every case above except case 2, because this is the only case when P is true but Q is not. Using the example, if P then Q expresses that if it is raining outside, then there is a cold-front over Kansas. The material conditional is often confused with physical causation. The material conditional, however, only relates two propositions by their truth-values—which is not the relation of cause and effect. It is contentious in the literature whether the material implication represents logical causation.
Biconditional joins two simpler propositions, and we write P ↔ Q, which is read "P if and only if Q". It expresses that P and Q have the same truth-value, and in cases 1 and 4. 'P is true if and only if Q' is true, and is false otherwise. 
It is very helpful to look at the truth tables for these different operators, as well as the method of analytic tableaux.
Closure under operations[edit]
Propositional logic is closed under truth-functional connectives. That is to say, for any proposition φ, ¬φ is also a proposition. Likewise, for any propositions φ and ψ, φ ∧ ψ is a proposition, and similarly for disjunction, conditional, and biconditional. This implies that, for instance, φ ∧ ψ is a proposition, and so it can be conjoined with another proposition. In order to represent this, we need to use parentheses to indicate which proposition is conjoined with which. For instance, P ∧ Q ∧ R is not a well-formed formula, because we do not know if we are conjoining P ∧ Q with R or if we are conjoining P with Q ∧ R. Thus we must write either (P ∧ Q) ∧ R to represent the former, or P ∧ (Q ∧ R) to represent the latter. By evaluating the truth conditions, we see that both expressions have the same truth conditions (will be true in the same cases), and moreover that any proposition formed by arbitrary conjunctions will have the same truth conditions, regardless of the location of the parentheses. This means that conjunction is associative, however, one should not assume that parentheses never serve a purpose. For instance, the sentence P ∧ (Q ∨ R) does not have the same truth conditions of (P ∧ Q) ∨ R, so they are different sentences distinguished only by the parentheses. One can verify this by the truth-table method referenced above.
 Note: For any arbitrary number of propositional constants, we can form a finite number of cases which list their possible truth-values. A simple way to generate this is by truth-tables, in which one writes P, Q, ..., Z, for any list of k propositional constants—that is to say, any list of propositional constants with k entries. Below this list, one writes 2 k  rows, and below P one fills in the first half of the rows with true (or T) and the second half with false (or F). Below Q one fills in one-quarter of the rows with T, then one-quarter with F, then one-quarter with T and the last quarter with F. The next column alternates between true and false for each eighth of the rows, then sixteenths, and so on, until the last propositional constant varies between T and F for each row. This will give a complete listing of cases or truth-value assignments possible for those propositional constants.
Argument[edit]
The propositional calculus then defines an  argument  to be a list of propositions. A valid argument is a list of propositions, the last of which follows from—or is implied by—the rest. All other arguments are invalid. The simplest valid argument is modus ponens, one instance of which is the following list of propositions:
   
This is a list of three propositions, each line is a proposition, and the last follows from the rest. The first two lines are called premises, and the last line the conclusion. We say that any proposition C follows from any set of propositions , if C must be true whenever every member of the set is true. In the argument above, for any P and Q, whenever P → Q and P are true, necessarily Q is true. Notice that, when P is true, we cannot consider cases 3 and 4 (from the truth table). When P → Q is true, we cannot consider case 2. This leaves only case 1, in which Q is also true. Thus Q is implied by the premises.
 This generalizes schematically. Thus, where φ and ψ may be any propositions at all,
   
Other argument forms are convenient, but not necessary. Given a complete set of axioms (see below for one such set), modus ponens is sufficient to prove all other argument forms in propositional logic, thus they may be considered to be a derivative. Note, this is not true of the extension of propositional logic to other logics like first-order logic. First-order logic requires at least one additional rule of inference in order to obtain completeness.
 The significance of argument in formal logic is that one may obtain new truths from established truths. In the first example above, given the two premises, the truth of Q is not yet known or stated. After the argument is made, Q is deduced. In this way, we define a deduction system to be a set of all propositions that may be deduced from another set of propositions. For instance, given the set of propositions , we can define a deduction system, Γ, which is the set of all propositions which follow from A. Reiteration is always assumed, so . Also, from the first element of A, last element, as well as modus ponens, R is a consequence, and so . Because we have not included sufficiently complete axioms, though, nothing else may be deduced. Thus, even though most deduction systems studied in propositional logic are able to deduce , this one is too weak to prove such a proposition.
Generic description of a propositional calculus[edit]
A propositional calculus is a formal system , where:
  The alpha set is a countably infinite set of elements called proposition symbols or  propositional variables . Syntactically speaking, these are the most basic elements of the formal language , otherwise referred to as  atomic formulas  or terminal elements. In the examples to follow, the elements of are typically the letters p, q, r, and so on. The omega set Ω is a finite set of elements called  operator symbols  or  logical connectives . The set Ω is partitioned into disjoint subsets as follows:
   
In this partition, is the set of operator symbols of  arity  j.
 In the more familiar propositional calculi, Ω is typically partitioned as follows:
  
  
A frequently adopted convention treats the constant logical values as operators of arity zero, thus:
   
Some writers use the tilde (~), or N, instead of ¬; and some use v instead of as well as the ampersand (&amp;), the prefixed K, or instead of . Notation varies even more for the set of logical values, with symbols like {false, true}, {F, T}, or {0, 1} all being seen in various contexts instead of . The zeta set is a finite set of transformation rules that are called  inference rules  when they acquire logical applications. The iota set is a countable set of initial points that are called  axioms  when they receive logical interpretations.  
The language of , also known as its set of formulas,  well-formed formulas , is inductively defined by the following rules:
 Base: Any element of the alpha set is a formula of .
If are formulas and is in , then is a formula.
Closed: Nothing else is a formula of . 
Repeated applications of these rules permits the construction of complex formulas. For example:
Example 1. Simple axiom system[edit]
Let , where , , , are defined as follows:
Then is defined as , and is defined as .
This system is used in Metamath set.mm formal proof database.
Example 2. Natural deduction system[edit]
Let , where , , , are defined as follows:
In the following example of a propositional calculus, the transformation rules are intended to be interpreted as the inference rules of a so-called  natural deduction system . The particular system presented here has no initial points, which means that its interpretation for logical applications derives its theorems from an empty axiom set.
Our propositional calculus has eleven inference rules. These rules allow us to derive other true formulas given a set of formulas that are assumed to be true. The first ten simply state that we can infer certain well-formed formulas from other well-formed formulas. The last rule however uses hypothetical reasoning in the sense that in the premise of the rule we temporarily assume an (unproven) hypothesis to be part of the set of inferred formulas to see if we can infer a certain other formula. Since the first ten rules don't do this they are usually described as non-hypothetical rules, and the last one as a hypothetical rule.
 In describing the transformation rules, we may introduce a metalanguage symbol . It is basically a convenient shorthand for saying "infer that". The format is , in which Γ is a (possibly empty) set of formulas called premises, and ψ is a formula called conclusion. The transformation rule means that if every proposition in Γ is a theorem (or has the same truth value as the axioms), then ψ is also a theorem. Note that considering the following rule Conjunction introduction, we will know whenever Γ has more than one formula, we can always safely reduce it into one formula using conjunction. So for short, from that time on we may represent Γ as one formula instead of a set. Another omission for convenience is when Γ is an empty set, in which case Γ may not appear.
  Negation introduction 
From and , infer .
That is, .
 Negation elimination 
From , infer .
That is, .
 Double negation elimination 
From , infer p.
That is, .
 Conjunction introduction 
From p and q, infer .
That is, .
 Conjunction elimination 
From , infer p.
From , infer q.
That is, and .
 Disjunction introduction 
From p, infer .
From q, infer .
That is, and .
 Disjunction elimination 
From and and , infer r.
That is, .
 Biconditional introduction 
From and , infer .
That is, .
 Biconditional elimination 
From , infer .
From , infer .
That is, and .
 Modus ponens (conditional elimination)
From p and , infer q.
That is, .
 Conditional proof (conditional introduction)
From [accepting p allows a proof of q], infer .
That is, . 
Basic and derived argument forms[edit]
 
Name
Sequent
Description
 
 Modus Ponens
If p then q; p; therefore q
 
 Modus Tollens
If p then q; not q; therefore not p
 
 Hypothetical Syllogism
If p then q; if q then r; therefore, if p then r
 
 Disjunctive Syllogism
Either p or q, or both; not p; therefore, q
 
 Constructive Dilemma
If p then q; and if r then s; but p or r; therefore q or s
 
 Destructive Dilemma
If p then q; and if r then s; but not q or not s; therefore not p or not r
 
Bidirectional Dilemma
If p then q; and if r then s; but p or not s; therefore q or not r
 
 Simplification
p and q are true; therefore p is true
 
 Conjunction
p and q are true separately; therefore they are true conjointly
 
 Addition
p is true; therefore the disjunction (p or q) is true
 
 Composition
If p then q; and if p then r; therefore if p is true then q and r are true
 
 De Morgan's Theorem (1)
The negation of (p and q) is equiv. to (not p or not q)
 
 De Morgan's Theorem (2)
The negation of (p or q) is equiv. to (not p and not q)
 
 Commutation (1)
(p or q) is equiv. to (q or p)
 
 Commutation (2)
(p and q) is equiv. to (q and p)
 
 Commutation (3)
(p is equiv. to q) is equiv. to (q is equiv. to p)
 
 Association (1)
p or (q or r) is equiv. to (p or q) or r
 
 Association (2)
p and (q and r) is equiv. to (p and q) and r
 
 Distribution (1)
p and (q or r) is equiv. to (p and q) or (p and r)
 
 Distribution (2)
p or (q and r) is equiv. to (p or q) and (p or r)
 
 Double Negation
 and 
p is equivalent to the negation of not p
 
 Transposition
If p then q is equiv. to if not q then not p
 
 Material Implication
If p then q is equiv. to not p or q
 
 Material Equivalence (1)
(p iff q) is equiv. to (if p is true then q is true) and (if q is true then p is true)
 
 Material Equivalence (2)
(p iff q) is equiv. to either (p and q are true) or (both p and q are false)
 
 Material Equivalence (3)
(p iff q) is equiv to., both (p or not q is true) and (not p or q is true)
 
 Exportation  [12] 
from (if p and q are true then r is true) we can prove (if q is true then r is true, if p is true)
 
 Importation
If p then (if q then r) is equivalent to if p and q then r
 
 Tautology (1)
p is true is equiv. to p is true or p is true
 
 Tautology (2)
p is true is equiv. to p is true and p is true
 
 Tertium non datur (Law of Excluded Middle)
p or not p is true
 
 Law of Non-Contradiction
p and not p is false, is a true statement
   
Proofs in propositional calculus[edit]
One of the main uses of a propositional calculus, when interpreted for logical applications, is to determine relations of logical equivalence between propositional formulas. These relationships are determined by means of the available transformation rules, sequences of which are called derivations or proofs.
 In the discussion to follow, a proof is presented as a sequence of numbered lines, with each line consisting of a single formula followed by a reason or justification for introducing that formula. Each premise of the argument, that is, an assumption introduced as an hypothesis of the argument, is listed at the beginning of the sequence and is marked as a "premise" in lieu of other justification. The conclusion is listed on the last line. A proof is complete if every line follows from the previous ones by the correct application of a transformation rule. (For a contrasting approach, see proof-trees).
Example of a proof in natural deduction system[edit]
 To be shown that A → A.
One possible proof of this (which, though valid, happens to contain more steps than are necessary) may be arranged as follows: 
Example of a proof
 
Number
Formula
Reason
 
1
 
premise
 
2
 
From ( 1 ) by disjunction introduction
 
3
 
From ( 1 ) and ( 2 ) by conjunction introduction
 
4
 
From ( 3 ) by conjunction elimination
 
5
 
Summary of ( 1 ) through ( 4 )
 
6
 
From ( 5 ) by conditional proof
   
Interpret as "Assuming A, infer A". Read as "Assuming nothing, infer that A implies A", or "It is a tautology that A implies A", or "It is always true that A implies A".
Example of a proof in a classical propositional calculus system[edit]
We now prove the same theorem in the axiomatic system by Jan Łukasiewicz described above, which is an example of a Hilbert-style deductive system for the classical propositional calculus.
 The axioms are:
 (A1) 
(A2) 
(A3)  
And the proof is as follows:
        (instance of (A1))
       (instance of (A2))
       (from (1) and (2) by modus ponens)
       (instance of (A1))
       (from (4) and (3) by modus ponens) 
Soundness and completeness of the rules[edit]
The crucial properties of this set of rules are that they are  sound  and complete. Informally this means that the rules are correct and that no other rules are required. These claims can be made more formal as follows.
Note that the proofs for the soundness and completeness of the propositional logic are not themselves proofs in propositional logic ; these are theorems in ZFC used as a metatheory to prove properties of propositional logic.
 We define a truth assignment as a function that maps propositional variables to true or false. Informally such a truth assignment can be understood as the description of a possible state of affairs (or possible world) where certain statements are true and others are not. The semantics of formulas can then be formalized by defining for which "state of affairs" they are considered to be true, which is what is done by the following definition.
 We define when such a truth assignment A satisfies a certain well-formed formula with the following rules:
 A satisfies the propositional variable P if and only if A(P) = true
A satisfies ¬φ if and only if A does not satisfy φ
A satisfies (φ ∧ ψ) if and only if A satisfies both φ and ψ
A satisfies (φ ∨ ψ) if and only if A satisfies at least one of either φ or ψ
A satisfies (φ → ψ) if and only if it is not the case that A satisfies φ but not ψ
A satisfies (φ ↔ ψ) if and only if A satisfies both φ and ψ or satisfies neither one of them 
With this definition we can now formalize what it means for a formula φ to be implied by a certain set S of formulas. Informally this is true if in all worlds that are possible given the set of formulas S the formula φ also holds. This leads to the following formal definition: We say that a set S of well-formed formulas semantically entails (or implies) a certain well-formed formula φ if all truth assignments that satisfy all the formulas in S also satisfy φ.
 Finally we define syntactical entailment such that φ is syntactically entailed by S if and only if we can derive it with the inference rules that were presented above in a finite number of steps. This allows us to formulate exactly what it means for the set of inference rules to be sound and complete:
  Soundness: If the set of well-formed formulas S syntactically entails the well-formed formula φ then S semantically entails φ.
  Completeness: If the set of well-formed formulas S semantically entails the well-formed formula φ then S syntactically entails φ.
 For the above set of rules this is indeed the case.
Sketch of a soundness proof[edit]
(For most logical systems, this is the comparatively "simple" direction of proof)
 Notational conventions: Let G be a variable ranging over sets of sentences. Let A, B and C range over sentences. For "G syntactically entails A" we write "G proves A". For "G semantically entails A" we write "G implies A".
 We want to show: (A)(G) (if G proves A, then G implies A).
 We note that "G proves A" has an inductive definition, and that gives us the immediate resources for demonstrating claims of the form "If G proves A, then ...". So our proof proceeds by induction.
  Basis. Show: If A is a member of G, then G implies A. Basis. Show: If A is an axiom, then G implies A. Inductive step (induction on n, the length of the proof):
  Assume for arbitrary G and A that if G proves A in n or fewer steps, then G implies A. For each possible application of a rule of inference at step n + 1, leading to a new theorem B, show that G implies B.     
Notice that Basis Step II can be omitted for natural deduction systems because they have no axioms. When used, Step II involves showing that each of the axioms is a (semantic) logical truth.
 The Basis steps demonstrate that the simplest provable sentences from G are also implied by G, for any G. (The proof is simple, since the semantic fact that a set implies any of its members, is also trivial.) The Inductive step will systematically cover all the further sentences that might be provable—by considering each case where we might reach a logical conclusion using an inference rule—and shows that if a new sentence is provable, it is also logically implied. (For example, we might have a rule telling us that from "A" we can derive "A or B". In III.a We assume that if A is provable it is implied. We also know that if A is provable then "A or B" is provable. We have to show that then "A or B" too is implied. We do so by appeal to the semantic definition and the assumption we just made. A is provable from G, we assume. So it is also implied by G. So any semantic valuation making all of G true makes A true. But any valuation making A true makes "A or B" true, by the defined semantics for "or". So any valuation which makes all of G true makes "A or B" true. So "A or B" is implied.) Generally, the Inductive step will consist of a lengthy but simple case-by-case analysis of all the rules of inference, showing that each "preserves" semantic implication.
 By the definition of provability, there are no sentences provable other than by being a member of G, an axiom, or following by a rule; so if all of those are semantically implied, the deduction calculus is sound.
Sketch of completeness proof[edit]
(This is usually the much harder direction of proof.)
 We adopt the same notational conventions as above.
 We want to show: If G implies A, then G proves A. We proceed by contraposition: We show instead that if G does not prove A then G does not imply A. If we show that there is a model where A does not hold despite G being true, then obviously G does not imply A. The idea is to build such a model out of our very assumption that G does not prove A.
  G does not prove A. (Assumption) If G does not prove A, then we can construct an (infinite) Maximal Set, G ∗, which is a superset of G and which also does not prove A.
   Place an ordering (with order type ω) on all the sentences in the language (e.g., shortest first, and equally long ones in extended alphabetical ordering), and number them (E 1, E 2, ...) Define a series Gn of sets (G 0, G 1, ...) inductively:
  Define G ∗ as the union of all the Gn. (That is, G ∗ is the set of all the sentences that are in any Gn.) It can be easily shown that
    G ∗ contains (is a superset of) G (by (b.i));  G ∗ does not prove A (because the proof would contain only finitely many sentences and when the last of them is introduced in some Gn, that Gn would prove A contrary to the definition of Gn); and  G ∗ is a Maximal Set with respect to A: If any more sentences whatever were added to G ∗, it would prove A. (Because if it were possible to add any more sentences, they should have been added when they were encountered during the construction of the Gn, again by definition)       If G ∗ is a Maximal Set with respect to A, then it is truth-like. This means that it contains C if and only if it does not contain ¬C; If it contains C and contains "If C then B" then it also contains B; and so forth. In order to show this, one has to show the axiomatic system is strong enough for the following:
 For any formulas C and D, if it proves both C and ¬C, then it proves D. From this it follows, that a Maximal Set with respect to A cannot prove both C and ¬C, as otherwise it would prove A.
For any formulas C and D, if it proves both C→D and ¬C→D, then it proves D. This is used, together with the deduction theorem, to show that for any formula, either it or its negation is in G ∗: Let B be a formula not in G ∗; then G ∗ with the addition of B proves A. Thus from the deduction theorem it follows that G ∗ proves B→A. But suppose ¬B were also not in G ∗, then by the same logic G ∗ also proves ¬B→A; but then G ∗ proves A, which we have already shown to be false.
For any formulas C and D, if it proves C and D, then it proves C→D.
For any formulas C and D, if it proves C and ¬D, then it proves ¬(C→D).
For any formulas C and D, if it proves ¬C, then it proves C→D. 
If additional logical operation (such as conjunction and/or disjunction) are part of the vocabulary as well, then there are additional requirement on the axiomatic system (e.g. that if it proves C and D, it would also prove their conjunction). If G ∗ is truth-like there is a G ∗-Canonical valuation of the language: one that makes every sentence in G ∗ true and everything outside G ∗ false while still obeying the laws of semantic composition in the language. Note that the requirement that it is truth-like is needed to guarantee that the laws of semantic composition in the language will be satisfied by this truth assignment. A G ∗-canonical valuation will make our original set G all true, and make A false. If there is a valuation on which G are true and A is false, then G does not (semantically) imply A.  
Thus every system that has modus ponens as an inference rule, and proves the following theorems (including substitutions thereof) is complete:
The first five are used for the satisfaction of the five conditions in stage III above, and the last three for proving the deduction theorem.
Example[edit]
As an example, it can be shown that as any other tautology, the three axioms of the classical propositional calculus system described earlier can be proven in any system that satisfies the above, namely that has modus ponens as an inference rule, and proves the above eight theorems (including substitutions thereof). Out of the eight theorems, the last two are two of the three axioms; the third axiom, , can be proven as well, as we now show.
 For the proof we may use the hypothetical syllogism theorem (in the form relevant for this axiomatic system), since it only relies on the two axioms that are already in the above set of eight theorems.
The proof then is as follows:
        (instance of the 7th theorem)
       (instance of the 7th theorem)
       (from (1) and (2) by modus ponens)
       (instance of the hypothetical syllogism theorem)
       (instance of the 5th theorem)
       (from (5) and (4) by modus ponens)
       (instance of the 2nd theorem)
       (instance of the 7th theorem)
       (from (7) and (8) by modus ponens)
          (instance of the 8th theorem)    
       (from (9) and (10) by modus ponens)
       (from (3) and (11) by modus ponens)
       (instance of the 8th theorem)
       (from (12) and (13) by modus ponens)
       (from (6) and (14) by modus ponens) 
Verifying completeness for the classical propositional calculus system[edit]
We now verify that the classical propositional calculus system described earlier can indeed prove the required eight theorems mentioned above. We use several lemmas proven here:
 (DN1) - Double negation (one direction)
(DN2) - Double negation (another direction)
(HS1) - one form of Hypothetical syllogism 
(HS2) - another form of Hypothetical syllogism
(TR1) - Transposition 
(TR2) - another form of transposition.
(L1) 
(L3)  
We also use the method of the hypothetical syllogism metatheorem as a shorthand for several proof steps.
Another outline for a completeness proof[edit]
If a formula is a tautology, then there is a truth table for it which shows that each valuation yields the value true for the formula. Consider such a valuation. By mathematical induction on the length of the subformulas, show that the truth or falsity of the subformula follows from the truth or falsity (as appropriate for the valuation) of each propositional variable in the subformula. Then combine the lines of the truth table together two at a time by using "(P is true implies S) implies ((P is false implies S) implies S)". Keep repeating this until all dependencies on propositional variables have been eliminated. The result is that we have proved the given tautology. Since every tautology is provable, the logic is complete.
Interpretation of a truth-functional propositional calculus[edit]
An interpretation of a truth-functional propositional calculus is an assignment to each propositional symbol of of one or the other (but not both) of the truth values truth (T) and falsity (F), and an assignment to the connective symbols of of their usual truth-functional meanings. An interpretation of a truth-functional propositional calculus may also be expressed in terms of truth tables. [13] 
 For distinct propositional symbols there are distinct possible interpretations. For any particular symbol , for example, there are possible interpretations:
  is assigned T, or
 is assigned F. 
For the pair , there are possible interpretations:
 both are assigned T,
both are assigned F,
 is assigned T and is assigned F, or
 is assigned F and is assigned T. [13]   
Since has , that is, denumerably many propositional symbols, there are , and therefore uncountably many distinct possible interpretations of . [13] 
Interpretation of a sentence of truth-functional propositional logic[edit]
If φ and ψ are formulas of and is an interpretation of then the following definitions apply:
Some consequences of these definitions:
Alternative calculus[edit]
It is possible to define another version of propositional calculus, which defines most of the syntax of the logical operators by means of axioms, and which uses only one inference rule.
Axioms[edit]
Let φ, χ, and ψ stand for well-formed formulas. (The well-formed formulas themselves would not contain any Greek letters, but only capital Roman letters, connective operators, and parentheses.) Then the axioms are as follows:
Axioms
 
Name
Axiom Schema
Description
 
THEN-1
Add hypothesis χ, implication introduction
 
THEN-2
Distribute hypothesis over implication
 
AND-1
Eliminate conjunction
 
AND-2
 
 
AND-3
Introduce conjunction
 
OR-1
Introduce disjunction
 
OR-2
 
 
OR-3
Eliminate disjunction
 
NOT-1
Introduce negation
 
NOT-2
Eliminate negation
 
NOT-3
Excluded middle, classical logic
 
IFF-1
Eliminate equivalence
 
IFF-2
 
 
IFF-3
Introduce equivalence
   
 Axiom  THEN-2  may be considered to be a "distributive property of implication with respect to implication."
Axioms  AND-1  and  AND-2  correspond to "conjunction elimination". The relation between  AND-1  and  AND-2  reflects the commutativity of the conjunction operator.
Axiom  AND-3  corresponds to "conjunction introduction."
Axioms  OR-1  and  OR-2  correspond to "disjunction introduction." The relation between  OR-1  and  OR-2  reflects the commutativity of the disjunction operator.
Axiom  NOT-1  corresponds to "reductio ad absurdum."
Axiom  NOT-2  says that "anything can be deduced from a contradiction."
Axiom  NOT-3  is called "tertium non-datur" (Latin: "a third is not given") and reflects the semantic valuation of propositional formulas: a formula can have a truth-value of either true or false. There is no third truth-value, at least not in classical logic. Intuitionistic logicians do not accept the axiom  NOT-3 . 
Inference rule[edit]
The inference rule is modus ponens:
 . 
Meta-inference rule[edit]
Let a demonstration be represented by a sequence, with hypotheses to the left of the turnstile and the conclusion to the right of the turnstile. Then the deduction theorem can be stated as follows:
  If the sequence
    
 has been demonstrated, then it is also possible to demonstrate the sequence
 .   
This deduction theorem (DT) is not itself formulated with propositional calculus: it is not a theorem of propositional calculus, but a theorem about propositional calculus. In this sense, it is a meta-theorem, comparable to theorems about the soundness or completeness of propositional calculus.
 On the other hand, DT is so useful for simplifying the syntactical proof process that it can be considered and used as another inference rule, accompanying modus ponens. In this sense, DT corresponds to the natural conditional proof inference rule which is part of the first version of propositional calculus introduced in this article.
 The converse of DT is also valid:
  If the sequence
    
 has been demonstrated, then it is also possible to demonstrate the sequence
     
in fact, the validity of the converse of DT is almost trivial compared to that of DT:
  If
    
 then
 1: 
2:   
 and from (1) and (2) can be deduced
 3:   
 by means of modus ponens, Q.E.D.  
The converse of DT has powerful implications: it can be used to convert an axiom into an inference rule. For example, by axiom AND-1 we have,
   
which can be transformed by means of the converse of the deduction theorem into
   
which tells us that the inference rule
   
is admissible. This inference rule is conjunction elimination, one of the ten inference rules used in the first version (in this article) of the propositional calculus.
Example of a proof[edit]
The following is an example of a (syntactical) demonstration, involving only axioms  THEN-1  and  THEN-2 :
  Prove: (Reflexivity of implication).
  Proof:
 
 Axiom  THEN-2  with   
 Axiom  THEN-1  with   
 From (1) and (2) by modus ponens.  
 Axiom  THEN-1  with   
 From (3) and (4) by modus ponens.   
Equivalence to equational logics[edit]
The preceding alternative calculus is an example of a Hilbert-style deduction system. In the case of propositional systems the axioms are terms built with logical connectives and the only inference rule is modus ponens. Equational logic as standardly used informally in high school algebra is a different kind of calculus from Hilbert systems. Its theorems are equations and its inference rules express the properties of equality, namely that it is a congruence on terms that admits substitution.
 Classical propositional calculus as described above is equivalent to Boolean algebra, while intuitionistic propositional calculus is equivalent to Heyting algebra. The equivalence is shown by translation in each direction of the theorems of the respective systems. Theorems of classical or intuitionistic propositional calculus are translated as equations of Boolean or Heyting algebra respectively. Conversely theorems of Boolean or Heyting algebra are translated as theorems of classical or intuitionistic calculus respectively, for which is a standard abbreviation. In the case of Boolean algebra can also be translated as , but this translation is incorrect intuitionistically.
 In both Boolean and Heyting algebra, inequality can be used in place of equality. The equality is expressible as a pair of inequalities and . Conversely the inequality is expressible as the equality , or as . The significance of inequality for Hilbert-style systems is that it corresponds to the latter's deduction or entailment symbol . An entailment
       
is translated in the inequality version of the algebraic framework as
       
Conversely the algebraic inequality is translated as the entailment
   .   
The difference between implication and inequality or entailment or is that the former is internal to the logic while the latter is external. Internal implication between two terms is another term of the same kind. Entailment as external implication between two terms expresses a metatruth outside the language of the logic, and is considered part of the metalanguage. Even when the logic under study is intuitionistic, entailment is ordinarily understood classically as two-valued: either the left side entails, or is less-or-equal to, the right side, or it is not.
 Similar but more complex translations to and from algebraic logics are possible for natural deduction systems as described above and for the sequent calculus. The entailments of the latter can be interpreted as two-valued, but a more insightful interpretation is as a set, the elements of which can be understood as abstract proofs organized as the morphisms of a category. In this interpretation the cut rule of the sequent calculus corresponds to composition in the category. Boolean and Heyting algebras enter this picture as special categories having at most one morphism per homset, i.e., one proof per entailment, corresponding to the idea that existence of proofs is all that matters: any proof will do and there is no point in distinguishing them.
Graphical calculi[edit]
It is possible to generalize the definition of a formal language from a set of finite sequences over a finite basis to include many other sets of mathematical structures, so long as they are built up by finitary means from finite materials. What's more, many of these families of formal structures are especially well-suited for use in logic.
 For example, there are many families of graphs that are close enough analogues of formal languages that the concept of a calculus is quite easily and naturally extended to them. Many species of graphs arise as  parse graphs  in the syntactic analysis of the corresponding families of text structures. The exigencies of practical computation on formal languages frequently demand that text strings be converted into pointer structure renditions of parse graphs, simply as a matter of checking whether strings are well-formed formulas or not. Once this is done, there are many advantages to be gained from developing the graphical analogue of the calculus on strings. The mapping from strings to parse graphs is called  parsing  and the inverse mapping from parse graphs to strings is achieved by an operation that is called  traversing  the graph.
Other logical calculi[edit]
Propositional calculus is about the simplest kind of logical calculus in current use. It can be extended in several ways. (Aristotelian "syllogistic" calculus, which is largely supplanted in modern logic, is in some ways simpler – but in other ways more complex – than propositional calculus.) The most immediate way to develop a more complex logical calculus is to introduce rules that are sensitive to more fine-grained details of the sentences being used.
  First-order logic (a.k.a. first-order predicate logic) results when the "atomic sentences" of propositional logic are broken up into terms, variables, predicates, and quantifiers, all keeping the rules of propositional logic with some new ones introduced. (For example, from "All dogs are mammals" we may infer "If Rover is a dog then Rover is a mammal".) With the tools of first-order logic it is possible to formulate a number of theories, either with explicit axioms or by rules of inference, that can themselves be treated as logical calculi. Arithmetic is the best known of these; others include set theory and mereology. Second-order logic and other higher-order logics are formal extensions of first-order logic. Thus, it makes sense to refer to propositional logic as "zeroth-order logic", when comparing it with these logics.
  Modal logic also offers a variety of inferences that cannot be captured in propositional calculus. For example, from "Necessarily p" we may infer that p. From p we may infer "It is possible that p". The translation between modal logics and algebraic logics concerns classical and intuitionistic logics but with the introduction of a unary operator on Boolean or Heyting algebras, different from the Boolean operations, interpreting the possibility modality, and in the case of Heyting algebra a second operator interpreting necessity (for Boolean algebra this is redundant since necessity is the De Morgan dual of possibility). The first operator preserves 0 and disjunction while the second preserves 1 and conjunction.
  Many-valued logics are those allowing sentences to have values other than true and false. (For example, neither and both are standard "extra values"; "continuum logic" allows each sentence to have any of an infinite number of "degrees of truth" between true and false.) These logics often require calculational devices quite distinct from propositional calculus. When the values form a Boolean algebra (which may have more than two or even infinitely many values), many-valued logic reduces to classical logic; many-valued logics are therefore only of independent interest when the values form an algebra that is not Boolean.
Solvers[edit]
Deciding satisfiability of propositional logic formulas is an NP-complete problem. However, practical methods exist (e.g., DPLL algorithm, 1962; Chaff algorithm, 2001) that are very fast for many useful cases. Recent work has extended the SAT solver algorithms to work with propositions containing arithmetic expressions; these are the SMT solvers.
See also[edit]
Higher logical levels[edit]
  First-order logic 
 Second-order propositional logic 
 Second-order logic 
 Higher-order logic  
[edit]
  Boolean algebra (logic) 
 Boolean algebra (structure) 
 Boolean algebra topics 
 Boolean domain 
 Boolean function 
 Boolean-valued function 
 Categorical logic 
 Combinational logic 
 Combinatory logic 
 Conceptual graph 
 Disjunctive syllogism 
 Entitative graph 
 Equational logic 
 Existential graph 
 Frege's propositional calculus 
 Implicational propositional calculus 
 Intuitionistic propositional calculus 
 Jean Buridan 
  Laws of Form  
 List of logic symbols 
 Logical graph 
 Logical NOR 
 Logical value 
 Mathematical logic 
 Operation (mathematics) 
 Paul of Venice 
 Peirce's law 
 Peter of Spain (author) 
 Propositional formula 
 Symmetric difference 
 Tautology (rule of inference) 
 Truth function 
 Truth table 
 Walter Burley 
 William of Sherwood  
References[edit]
  ^   "Propositional Logic | Brilliant Math &amp; Science Wiki". brilliant.org. Retrieved 20 August 2020.
  ^  Bobzien, Susanne (1 January 2016). Zalta, Edward N. (ed.).  The Stanford Encyclopedia of Philosophy  – via Stanford Encyclopedia of Philosophy.
  ^   "Propositional Logic | Internet Encyclopedia of Philosophy". Retrieved 20 August 2020.
  ^  Marenbon, John (2007).  Medieval philosophy: an historical and philosophical introduction . Routledge. p. 137.
  ^  Peckhaus, Volker (1 January 2014). Zalta, Edward N. (ed.).  The Stanford Encyclopedia of Philosophy  – via Stanford Encyclopedia of Philosophy.
  ^  Hurley, Patrick (2007). A Concise Introduction to Logic 10th edition. Wadsworth Publishing. p. 392.
  ^  Beth, Evert W.; "Semantic entailment and formal derivability", series: Mededlingen van de Koninklijke Nederlandse Akademie van Wetenschappen, Afdeling Letterkunde, Nieuwe Reeks, vol. 18, no. 13, Noord-Hollandsche Uitg. Mij., Amsterdam, 1955, pp. 309–42. Reprinted in Jaakko Intikka (ed.) The Philosophy of Mathematics, Oxford University Press, 1969
^    a       b    Truth in Frege
^    a       b       c     "Russell: the Journal of Bertrand Russell Studies".
  ^   Anellis, Irving H. (2012). "Peirce's Truth-functional Analysis and the Origin of the Truth Table". History and Philosophy of Logic. 33: 87–97. doi:10.1080/01445340.2011.621702. S2CID 170654885.
  ^  Wernick, William (1942) "Complete Sets of Logical Functions," Transactions of the American Mathematical Society 51, pp. 117–132.
  ^  Toida, Shunichi (2 August 2009). "Proof of Implications". CS381 Discrete Structures/Discrete Mathematics Web Course Material. Department of Computer Science, Old Dominion University. Retrieved 10 March 2010.
^    a       b       c       d       e       f       g       h       i       j       k    Hunter, Geoffrey (1971). Metalogic: An Introduction to the Metatheory of Standard First-Order Logic. University of California Press. ISBN  0-520-02356-0 .
 
Further reading[edit]
 Brown, Frank Markham (2003), Boolean Reasoning: The Logic of Boolean Equations, 1st edition, Kluwer Academic Publishers, Norwell, MA. 2nd edition, Dover Publications, Mineola, NY.
 Chang, C.C. and Keisler, H.J. (1973), Model Theory, North-Holland, Amsterdam, Netherlands.
Kohavi, Zvi (1978), Switching and Finite Automata Theory, 1st edition, McGraw–Hill, 1970. 2nd edition, McGraw–Hill, 1978.
 Korfhage, Robert R. (1974), Discrete Computational Structures, Academic Press, New York, NY.
 Lambek, J. and Scott, P.J. (1986), Introduction to Higher Order Categorical Logic, Cambridge University Press, Cambridge, UK.
Mendelson, Elliot (1964), Introduction to Mathematical Logic, D. Van Nostrand Company. 
[edit]
   Hofstadter, Douglas (1979).  Gödel, Escher, Bach: An Eternal Golden Braid . Basic Books. ISBN  978-0-465-02656-2 .  
External links[edit]
  Klement, Kevin C. (2006), "Propositional Logic", in James Fieser and Bradley Dowden (eds.),  Internet Encyclopedia of Philosophy , Eprint.
 Formal Predicate Calculus, contains a systematic formal development along the lines of Alternative calculus 
  forall x: an introduction to formal logic , by P.D. Magnus, covers formal semantics and proof theory for sentential logic.
 Chapter 2 / Propositional Logic from Logic In Action 
 Propositional sequent calculus prover on Project Nayuki. (note: implication can be input in the form !X|Y, and a sequent can be a single formula prefixed with &gt; and having no commas)
 Propositional Logic - A Generative Grammar
Propositional calculus - Wikipedia
https://en.wikipedia.org/wiki/Propositional_calculus
https://academicguides.waldenu.edu/writingcenter/grammar/prepositions
Walden University Writing Center
2014-09-18
Beth Nastachowski
<div><div>
	<h3>Prepositions of Direction</h3>
<p>To refer to a direction, use the prepositions "to," "in," "into," "on," and "onto<em>."</em></p>
<ul>
	<li>She drove <strong><em>to</em></strong> the store.</li>
	<li>Don’t ring the doorbell. Come right <strong><em>in(to)</em></strong> the house.</li>
	<li>Drive <strong><em>on(to)</em></strong> the grass and park the car there.</li>
</ul>
<h3>Prepositions of Time</h3>
<p>To refer to one point in time, use the prepositions "in," "at," and "on."</p>
<blockquote>
<p>Use<em> </em>"in" with parts of the day (not specific times), months, years, and seasons.</p>
</blockquote>
<ul>
	<li>He reads <strong><em>in</em></strong> the evening.</li>
	<li>The weather is cold <strong><em>in</em></strong> December.</li>
	<li>She was born <strong><em>in</em></strong> 1996.</li>
	<li>We rake leaves <strong><em>in</em></strong> the fall.</li>
</ul>
<blockquote>
<p>Use "at"<strong><em> </em></strong>with the time of day. Also use "at" with noon, night, and midnight.</p>
</blockquote>
<ul>
	<li>I go to work <strong><em>at</em></strong> 8:00.</li>
	<li>He eats lunch <strong><em>at</em> </strong>noon.</li>
	<li>She often goes for a walk <strong><em>at</em></strong> night.</li>
	<li>They go to bed <strong><em>at</em></strong> midnight.</li>
</ul>
<blockquote>
<p>Use "on" with days.</p>
</blockquote>
<ul>
	<li>I work <strong><em>on</em></strong> Saturdays.</li>
	<li>He does laundry <strong><em>on</em></strong> Wednesdays.</li>
</ul>
<p>To refer to extended time, use the prepositions "since," "for," "by," "during," "from…to," "from…until," "with," and "within."</p>
<ul>
	<li>I have lived in Minneapolis <strong><em>since</em></strong> 2005. (I moved there in 2005 and still live there.)</li>
	<li>He will be in Toronto <strong><em>for</em></strong> 3 weeks. (He will spend 3 weeks in Toronto.)</li>
	<li>She will finish her homework <strong><em>by</em></strong> 6:00. (She will finish her homework sometime between now and 6:00.)</li>
	<li>He works part time <strong><em>during</em></strong> the summer. (For the period of time throughout the summer.)</li>
	<li>I will collect data <strong><em>from</em></strong> January <strong><em>to</em></strong> June. (Starting in January and ending in June.)</li>
	<li>They are in school <strong><em>from</em></strong> August <strong><em>until</em></strong> May. (Starting in August and ending in May.)</li>
	<li>She will graduate <strong><em>within</em></strong> 2 years. (Not longer than 2 years.)</li>
</ul>
<h3>Prepositions of Place</h3>
<p>To refer to a place, use the prepositions "in"<em> </em>(the point itself), "at"<em> </em>(the general vicinity), "on" (the surface), and "inside" (something contained).</p>
<ul>
	<li>They will meet<strong><em> in</em></strong> the lunchroom.</li>
	<li>She was waiting <strong><em>at</em></strong> the corner.</li>
	<li>He left his phone <strong><em>on</em></strong> the bed.</li>
	<li>Place the pen <strong><em>inside</em></strong> the drawer.</li>
</ul>
<p>To refer to an object higher than a point, use the prepositions "over" and "above." To refer to an object lower than a point, use the prepositions "below," "beneath," "under," and "underneath."</p>
<ul>
	<li>The bird flew <strong><em>over</em></strong> the house.</li>
	<li>The plates were on the shelf <strong><em>above</em></strong> the cups.</li>
	<li>Basements are dug <strong><em>below</em></strong> ground.</li>
	<li>There is hard wood <strong><em>beneath</em></strong> the carpet.</li>
	<li>The squirrel hid the nuts <strong><em>under</em></strong> a pile of leaves.</li>
	<li>The cat is hiding <strong><em>underneath</em></strong> the box.</li>
</ul>
<p> To refer to an object close to a point, use the prepositions "by," "near," "next to," "between," "among," and "opposite."</p>
<ul>
	<li>The gas station is <strong><em>by</em></strong> the grocery store.</li>
	<li>The park is <strong><em>near</em></strong> her house.</li>
	<li>Park your bike <strong><em>next to</em></strong> the garage.</li>
	<li>There is a deer <strong><em>between</em></strong> the two trees.</li>
	<li>There is a purple flower <strong><em>among</em></strong> the weeds.</li>
	<li>The garage is <strong><em>opposite</em></strong> the house.</li>
</ul>
<h3>Prepositions of Location</h3>
<p>To refer to a location, use the prepositions "in"<em> </em>(an area or volume), "at"<em> </em>(a point), and "on"<em> </em>(a surface).</p>
<ul>
	<li>They live <strong><em>in</em></strong> the country. (an area)</li>
	<li>She will find him <strong><em>at</em></strong> the library. (a point)</li>
	<li>There is a lot of dirt <strong><em>on</em></strong> the window. (a surface)</li>
</ul>
<h3>Prepositions of Spatial Relationships</h3>
<p>To refer to a spatial relationship, use the prepositions "above," "across," "against," "ahead of," "along," "among," "around," "behind," "below,"<br />
"beneath," "beside," "between," "from," "in front of," "inside," "near," "off," "out of," "through," "toward," "under," and "within."</p>
<ul>
	<li>The post office is <strong><em>across</em></strong> the street from the grocery store.</li>
	<li>We will stop at many attractions<strong><em> along</em></strong> the way.</li>
	<li>The kids are hiding <strong><em>behind</em></strong> the tree.</li>
	<li>His shirt is<strong><em> off</em></strong>.</li>
	<li>Walk <strong><em>toward</em></strong> the garage and then turn left.</li>
	<li>Place a check mark <strong><em>within</em></strong> the box.</li>
</ul>
	 </div>||||I||||Skip to Main Content

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 2. Academic Guides
 3. Writing Center
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 5. Prepositions

 Grammar: Prepositions

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 * Overview
 * Main Parts of Speech
 * Sentence Structure and Types of Sentences
 * Run-On Sentences and Sentence Fragments
 * Parallel Construction
 * Relative, Restrictive, and Nonrestrictive Clauses
 * Comparisons
 * Verb Tenses
 * Verb Forms: "-ing," Infinitives, and Past Participles
 * Subject-Verb Agreement
 * Transitive and Intransitive Verbs
 * Noun–Pronoun Agreement
 * Articles
 * Count and Noncount Nouns
 * Conjunctions
 * Modifiers
 * Prepositions
 + Preposition Basics
 + A Few Rules
 + Prepositions Following Verbs and Adjectives
 + Ending a Sentence With a Preposition
 + Prepositional Phrases and Wordiness
 + Unnecessary Prepositions
 + Related Resource
 + Knowledge Check: Prepositions
 + Webpage Feedback

 Preposition Basics

 A preposition is a word or group of words used before a noun, pronoun, or noun phrase to show direction, time, place, location, spatial relationships, or to introduce an object. Some examples of prepositions are words like "in," "at," "on," "of," and "to."

 Prepositions in English are highly idiomatic. Although there are some rules for usage, much preposition usage is dictated by fixed expressions. In these cases, it is best to memorize the phrase instead of the individual preposition.

 A Few Rules

 Prepositions of Direction

 To refer to a direction, use the prepositions "to," "in," "into," "on," and "onto."

 * She drove to the store.
 * Don’t ring the doorbell. Come right in(to) the house.
 * Drive on(to) the grass and park the car there.

 Prepositions of Time

 To refer to one point in time, use the prepositions "in," "at," and "on."

 Use "in" with parts of the day (not specific times), months, years, and seasons.

 * He reads in the evening.
 * The weather is cold in December.
 * She was born in 1996.
 * We rake leaves in the fall.

 Use "at" with the time of day. Also use "at" with noon, night, and midnight.

 * I go to work at 8:00.
 * He eats lunch at noon.
 * She often goes for a walk at night.
 * They go to bed at midnight.

 Use "on" with days.

 * I work on Saturdays.
 * He does laundry on Wednesdays.

 To refer to extended time, use the prepositions "since," "for," "by," "during," "from…to," "from…until," "with," and "within."

 * I have lived in Minneapolis since 2005. (I moved there in 2005 and still live there.)
 * He will be in Toronto for 3 weeks. (He will spend 3 weeks in Toronto.)
 * She will finish her homework by 6:00. (She will finish her homework sometime between now and 6:00.)
 * He works part time during the summer. (For the period of time throughout the summer.)
 * I will collect data from January to June. (Starting in January and ending in June.)
 * They are in school from August until May. (Starting in August and ending in May.)
 * She will graduate within 2 years. (Not longer than 2 years.)

 Prepositions of Place

 To refer to a place, use the prepositions "in" (the point itself), "at" (the general vicinity), "on" (the surface), and "inside" (something contained).

 * They will meet in the lunchroom.
 * She was waiting at the corner.
 * He left his phone on the bed.
 * Place the pen inside the drawer.

 To refer to an object higher than a point, use the prepositions "over" and "above." To refer to an object lower than a point, use the prepositions "below," "beneath," "under," and "underneath."

 * The bird flew over the house.
 * The plates were on the shelf above the cups.
 * Basements are dug below ground.
 * There is hard wood beneath the carpet.
 * The squirrel hid the nuts under a pile of leaves.
 * The cat is hiding underneath the box.

 To refer to an object close to a point, use the prepositions "by," "near," "next to," "between," "among," and "opposite."

 * The gas station is by the grocery store.
 * The park is near her house.
 * Park your bike next to the garage.
 * There is a deer between the two trees.
 * There is a purple flower among the weeds.
 * The garage is opposite the house.

 Prepositions of Location

 To refer to a location, use the prepositions "in" (an area or volume), "at" (a point), and "on" (a surface).

 * They live in the country. (an area)
 * She will find him at the library. (a point)
 * There is a lot of dirt on the window. (a surface)

 Prepositions of Spatial Relationships

 To refer to a spatial relationship, use the prepositions "above," "across," "against," "ahead of," "along," "among," "around," "behind," "below,"
 "beneath," "beside," "between," "from," "in front of," "inside," "near," "off," "out of," "through," "toward," "under," and "within."

 * The post office is across the street from the grocery store.
 * We will stop at many attractions along the way.
 * The kids are hiding behind the tree.
 * His shirt is off.
 * Walk toward the garage and then turn left.
 * Place a check mark within the box.

 Prepositions Following Verbs and Adjectives

 Some verbs and adjectives are followed by a certain preposition. Sometimes verbs and adjectives can be followed by different prepositions, giving the phrase different meanings. To find which prepositions follow the verb or an adjective, look up the verb or adjective in an online dictionary, such as Merriam Webster, or use a corpus, such as The Corpus of Contemporary American English. Memorizing these phrases instead of just the preposition alone is the most helpful.

 Some Common Verb + Preposition Combinations

 About: worry, complain, read

 * He worries about the future.
 * She complained about the homework.
 * I read about the flooding in the city.

 At: arrive (a building or event), smile, look

 * He arrived at the airport 2 hours early.
 * The children smiled at her.
 * She looked at him.

 From: differ, suffer

 * The results differ from my original idea.
 * She suffers from dementia.

 For: account, allow, search

 * Be sure to account for any discrepancies.
 * I returned the transcripts to the interviewees to allow for revisions to be made.
 * They are searching for the missing dog.

 In: occur, result, succeed

 * The same problem occurred in three out of four cases.
 * My recruitment strategies resulted in finding 10 participants.
 * She will succeed in completing her degree.

 Of: approve, consist, smell

 * I approve of the idea.
 * The recipe consists of three basic ingredients.
 * The basement smells of mildew.

 On: concentrate, depend, insist

 * He is concentrating on his work.
 * They depend on each other.
 * I must insist on following this rule.

 To: belong, contribute, lead, refer

 * Bears belong to the family of mammals.
 * I hope to contribute to the previous research.
 * My results will lead to future research on the topic.
 * Please refer to my previous explanation.

 With: (dis)agree, argue, deal

 * I (dis)agree with you.
 * She argued with him.
 * They will deal with the situation.

 Although verb + preposition combinations appear similar to phrasal verbs, the verb and the particle (in this case, the preposition) in these combinations cannot be separated like phrasal verbs. See more about this on our verb choice page.

 Some Common Adjective + Preposition Combinations

 About At By From For In Of To With
  
 Accustomed X 
  
 Aware X 
  
 Beneficial X 
  
 Capable X 
  
 Characteristic X 
  
 Composed X X 
  
 Different X 
  
 Disappointed X X 
  
 Employed X X 
  
 Essential X 
  
 About At By From For In Of To With
  
 Familiar X 
  
 Good X X 
  
 Grateful X X 
  
 Interested X 
  
 Happy X X X 
  
 Opposed X 
  
 Proud X 
  
 Responsible X 
  
 Similar X 
  
 Sorry X X 
  
 

 Ending a Sentence With a Preposition

 At one time, schools taught students that a sentence should never end with a preposition. This rule is associated with Latin grammar, and while many aspects of Latin have made their way into English, there are times when following this particular grammar rule creates unclear or awkward sentence structures. Since the purpose of writing is to clearly communicate your ideas, it is acceptable to end a sentence with a preposition if the alternative would create confusion or is too overly formal.

 Example: The car had not been paid for. (Ends with a preposition but is acceptable)

 Unclear Revision: Paid for the car had not been. (Unclear sentence.)

 Example: I would like to know where she comes from. (Ends with a preposition but is acceptable)

 Overly Grammatical Revision: I would like to know from where she comes. (Grammatical but overly formal. Nobody actually speaks like this.)

 However, in academic writing, you may decide that it is worth revising your sentences to avoid ending with a preposition in order to maintain a more formal scholarly voice.

 Example: My research will focus on the community the students lived in.

 Revision: My research will focus on the community in which the students lived.

 Example: I like the people I am working with.

 Revision: I like the people with whom I am working.

 Prepositional Phrases and Wordiness

 Like with pronouns, too many prepositional phrases can create wordiness in a sentence:

 Example: The author chose the mixed-method design to explain that the purpose of the study was to explore the leadership qualities of the principals in the schools as a means to gauge teacher satisfaction in the first year of teaching.

 This type of sentence could be shortened and condensed to minimize the prepositional phrases and bring clarity to the writer's intent:

 Revision: The author chose the mixed-method design to explore the principals' leadership qualities and their impact on first-year teachers' satisfaction.

 Unnecessary Prepositions

 If the preposition is unnecessary, leave it out. This creates more clear and concise writing.

 Example: Where are the plates at?

 Revision: Where are the plates?

 Example: She jumped off of the balance beam.

 Revision: She jumped off the balance beam.

 Related Resource

 * Mastering the Mechanics Part 1: Simple Sentences (webinar)

 Knowledge Check: Prepositions

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Prepositions of Direction
To refer to a direction, use the prepositions "to," "in," "into," "on," and "onto." 
	She drove  to  the store.
	Don’t ring the doorbell. Come right  in(to)  the house.
	Drive  on(to)  the grass and park the car there.
Prepositions of Time
To refer to one point in time, use the prepositions "in," "at," and "on."
Use "in" with parts of the day (not specific times), months, years, and seasons.
	He reads  in  the evening.
	The weather is cold  in  December.
	She was born  in  1996.
	We rake leaves  in  the fall.
Use "at"   with the time of day. Also use "at" with noon, night, and midnight.
	I go to work  at  8:00.
	He eats lunch  at noon.
	She often goes for a walk  at  night.
	They go to bed  at  midnight.
Use "on" with days.
	I work  on  Saturdays.
	He does laundry  on  Wednesdays.
To refer to extended time, use the prepositions "since," "for," "by," "during," "from…to," "from…until," "with," and "within."
	I have lived in Minneapolis  since  2005. (I moved there in 2005 and still live there.)
	He will be in Toronto  for  3 weeks. (He will spend 3 weeks in Toronto.)
	She will finish her homework  by  6:00. (She will finish her homework sometime between now and 6:00.)
	He works part time  during  the summer. (For the period of time throughout the summer.)
	I will collect data  from  January  to  June. (Starting in January and ending in June.)
	They are in school  from  August  until  May. (Starting in August and ending in May.)
	She will graduate  within  2 years. (Not longer than 2 years.)
Prepositions of Place
To refer to a place, use the prepositions "in" (the point itself), "at" (the general vicinity), "on" (the surface), and "inside" (something contained).
	They will meet  in  the lunchroom.
	She was waiting  at  the corner.
	He left his phone  on  the bed.
	Place the pen  inside  the drawer.
To refer to an object higher than a point, use the prepositions "over" and "above." To refer to an object lower than a point, use the prepositions "below," "beneath," "under," and "underneath."
	The bird flew  over  the house.
	The plates were on the shelf  above  the cups.
	Basements are dug  below  ground.
	There is hard wood  beneath  the carpet.
	The squirrel hid the nuts  under  a pile of leaves.
	The cat is hiding  underneath  the box.
 To refer to an object close to a point, use the prepositions "by," "near," "next to," "between," "among," and "opposite."
	The gas station is  by  the grocery store.
	The park is  near  her house.
	Park your bike  next to  the garage.
	There is a deer  between  the two trees.
	There is a purple flower  among  the weeds.
	The garage is  opposite  the house.
Prepositions of Location
To refer to a location, use the prepositions "in" (an area or volume), "at" (a point), and "on" (a surface).
	They live  in  the country. (an area)
	She will find him  at  the library. (a point)
	There is a lot of dirt  on  the window. (a surface)
Prepositions of Spatial Relationships
To refer to a spatial relationship, use the prepositions "above," "across," "against," "ahead of," "along," "among," "around," "behind," "below,"
"beneath," "beside," "between," "from," "in front of," "inside," "near," "off," "out of," "through," "toward," "under," and "within."
	The post office is  across  the street from the grocery store.
	We will stop at many attractions  along  the way.
	The kids are hiding  behind  the tree.
	His shirt is  off .
	Walk  toward  the garage and then turn left.
	Place a check mark  within  the box.
Walden University Writing Center
https://academicguides.waldenu.edu/writingcenter/grammar/prepositions
https://academicguides.waldenu.edu/writingcenter/grammar/prepositions
Walden University Writing Center
2014-09-18
Beth Nastachowski
<div><div>
	<h3>Prepositions of Direction</h3>
<p>To refer to a direction, use the prepositions "to," "in," "into," "on," and "onto<em>."</em></p>
<ul>
	<li>She drove <strong><em>to</em></strong> the store.</li>
	<li>Don’t ring the doorbell. Come right <strong><em>in(to)</em></strong> the house.</li>
	<li>Drive <strong><em>on(to)</em></strong> the grass and park the car there.</li>
</ul>
<h3>Prepositions of Time</h3>
<p>To refer to one point in time, use the prepositions "in," "at," and "on."</p>
<blockquote>
<p>Use<em> </em>"in" with parts of the day (not specific times), months, years, and seasons.</p>
</blockquote>
<ul>
	<li>He reads <strong><em>in</em></strong> the evening.</li>
	<li>The weather is cold <strong><em>in</em></strong> December.</li>
	<li>She was born <strong><em>in</em></strong> 1996.</li>
	<li>We rake leaves <strong><em>in</em></strong> the fall.</li>
</ul>
<blockquote>
<p>Use "at"<strong><em> </em></strong>with the time of day. Also use "at" with noon, night, and midnight.</p>
</blockquote>
<ul>
	<li>I go to work <strong><em>at</em></strong> 8:00.</li>
	<li>He eats lunch <strong><em>at</em> </strong>noon.</li>
	<li>She often goes for a walk <strong><em>at</em></strong> night.</li>
	<li>They go to bed <strong><em>at</em></strong> midnight.</li>
</ul>
<blockquote>
<p>Use "on" with days.</p>
</blockquote>
<ul>
	<li>I work <strong><em>on</em></strong> Saturdays.</li>
	<li>He does laundry <strong><em>on</em></strong> Wednesdays.</li>
</ul>
<p>To refer to extended time, use the prepositions "since," "for," "by," "during," "from…to," "from…until," "with," and "within."</p>
<ul>
	<li>I have lived in Minneapolis <strong><em>since</em></strong> 2005. (I moved there in 2005 and still live there.)</li>
	<li>He will be in Toronto <strong><em>for</em></strong> 3 weeks. (He will spend 3 weeks in Toronto.)</li>
	<li>She will finish her homework <strong><em>by</em></strong> 6:00. (She will finish her homework sometime between now and 6:00.)</li>
	<li>He works part time <strong><em>during</em></strong> the summer. (For the period of time throughout the summer.)</li>
	<li>I will collect data <strong><em>from</em></strong> January <strong><em>to</em></strong> June. (Starting in January and ending in June.)</li>
	<li>They are in school <strong><em>from</em></strong> August <strong><em>until</em></strong> May. (Starting in August and ending in May.)</li>
	<li>She will graduate <strong><em>within</em></strong> 2 years. (Not longer than 2 years.)</li>
</ul>
<h3>Prepositions of Place</h3>
<p>To refer to a place, use the prepositions "in"<em> </em>(the point itself), "at"<em> </em>(the general vicinity), "on" (the surface), and "inside" (something contained).</p>
<ul>
	<li>They will meet<strong><em> in</em></strong> the lunchroom.</li>
	<li>She was waiting <strong><em>at</em></strong> the corner.</li>
	<li>He left his phone <strong><em>on</em></strong> the bed.</li>
	<li>Place the pen <strong><em>inside</em></strong> the drawer.</li>
</ul>
<p>To refer to an object higher than a point, use the prepositions "over" and "above." To refer to an object lower than a point, use the prepositions "below," "beneath," "under," and "underneath."</p>
<ul>
	<li>The bird flew <strong><em>over</em></strong> the house.</li>
	<li>The plates were on the shelf <strong><em>above</em></strong> the cups.</li>
	<li>Basements are dug <strong><em>below</em></strong> ground.</li>
	<li>There is hard wood <strong><em>beneath</em></strong> the carpet.</li>
	<li>The squirrel hid the nuts <strong><em>under</em></strong> a pile of leaves.</li>
	<li>The cat is hiding <strong><em>underneath</em></strong> the box.</li>
</ul>
<p> To refer to an object close to a point, use the prepositions "by," "near," "next to," "between," "among," and "opposite."</p>
<ul>
	<li>The gas station is <strong><em>by</em></strong> the grocery store.</li>
	<li>The park is <strong><em>near</em></strong> her house.</li>
	<li>Park your bike <strong><em>next to</em></strong> the garage.</li>
	<li>There is a deer <strong><em>between</em></strong> the two trees.</li>
	<li>There is a purple flower <strong><em>among</em></strong> the weeds.</li>
	<li>The garage is <strong><em>opposite</em></strong> the house.</li>
</ul>
<h3>Prepositions of Location</h3>
<p>To refer to a location, use the prepositions "in"<em> </em>(an area or volume), "at"<em> </em>(a point), and "on"<em> </em>(a surface).</p>
<ul>
	<li>They live <strong><em>in</em></strong> the country. (an area)</li>
	<li>She will find him <strong><em>at</em></strong> the library. (a point)</li>
	<li>There is a lot of dirt <strong><em>on</em></strong> the window. (a surface)</li>
</ul>
<h3>Prepositions of Spatial Relationships</h3>
<p>To refer to a spatial relationship, use the prepositions "above," "across," "against," "ahead of," "along," "among," "around," "behind," "below,"<br />
"beneath," "beside," "between," "from," "in front of," "inside," "near," "off," "out of," "through," "toward," "under," and "within."</p>
<ul>
	<li>The post office is <strong><em>across</em></strong> the street from the grocery store.</li>
	<li>We will stop at many attractions<strong><em> along</em></strong> the way.</li>
	<li>The kids are hiding <strong><em>behind</em></strong> the tree.</li>
	<li>His shirt is<strong><em> off</em></strong>.</li>
	<li>Walk <strong><em>toward</em></strong> the garage and then turn left.</li>
	<li>Place a check mark <strong><em>within</em></strong> the box.</li>
</ul>
	 </div>||||I||||Skip to Main Content

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 2. Academic Guides
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 5. Prepositions

 Grammar: Prepositions

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 * Overview
 * Main Parts of Speech
 * Sentence Structure and Types of Sentences
 * Run-On Sentences and Sentence Fragments
 * Parallel Construction
 * Relative, Restrictive, and Nonrestrictive Clauses
 * Comparisons
 * Verb Tenses
 * Verb Forms: "-ing," Infinitives, and Past Participles
 * Subject-Verb Agreement
 * Transitive and Intransitive Verbs
 * Noun–Pronoun Agreement
 * Articles
 * Count and Noncount Nouns
 * Conjunctions
 * Modifiers
 * Prepositions
 + Preposition Basics
 + A Few Rules
 + Prepositions Following Verbs and Adjectives
 + Ending a Sentence With a Preposition
 + Prepositional Phrases and Wordiness
 + Unnecessary Prepositions
 + Related Resource
 + Knowledge Check: Prepositions
 + Webpage Feedback

 Preposition Basics

 A preposition is a word or group of words used before a noun, pronoun, or noun phrase to show direction, time, place, location, spatial relationships, or to introduce an object. Some examples of prepositions are words like "in," "at," "on," "of," and "to."

 Prepositions in English are highly idiomatic. Although there are some rules for usage, much preposition usage is dictated by fixed expressions. In these cases, it is best to memorize the phrase instead of the individual preposition.

 A Few Rules

 Prepositions of Direction

 To refer to a direction, use the prepositions "to," "in," "into," "on," and "onto."

 * She drove to the store.
 * Don’t ring the doorbell. Come right in(to) the house.
 * Drive on(to) the grass and park the car there.

 Prepositions of Time

 To refer to one point in time, use the prepositions "in," "at," and "on."

 Use "in" with parts of the day (not specific times), months, years, and seasons.

 * He reads in the evening.
 * The weather is cold in December.
 * She was born in 1996.
 * We rake leaves in the fall.

 Use "at" with the time of day. Also use "at" with noon, night, and midnight.

 * I go to work at 8:00.
 * He eats lunch at noon.
 * She often goes for a walk at night.
 * They go to bed at midnight.

 Use "on" with days.

 * I work on Saturdays.
 * He does laundry on Wednesdays.

 To refer to extended time, use the prepositions "since," "for," "by," "during," "from…to," "from…until," "with," and "within."

 * I have lived in Minneapolis since 2005. (I moved there in 2005 and still live there.)
 * He will be in Toronto for 3 weeks. (He will spend 3 weeks in Toronto.)
 * She will finish her homework by 6:00. (She will finish her homework sometime between now and 6:00.)
 * He works part time during the summer. (For the period of time throughout the summer.)
 * I will collect data from January to June. (Starting in January and ending in June.)
 * They are in school from August until May. (Starting in August and ending in May.)
 * She will graduate within 2 years. (Not longer than 2 years.)

 Prepositions of Place

 To refer to a place, use the prepositions "in" (the point itself), "at" (the general vicinity), "on" (the surface), and "inside" (something contained).

 * They will meet in the lunchroom.
 * She was waiting at the corner.
 * He left his phone on the bed.
 * Place the pen inside the drawer.

 To refer to an object higher than a point, use the prepositions "over" and "above." To refer to an object lower than a point, use the prepositions "below," "beneath," "under," and "underneath."

 * The bird flew over the house.
 * The plates were on the shelf above the cups.
 * Basements are dug below ground.
 * There is hard wood beneath the carpet.
 * The squirrel hid the nuts under a pile of leaves.
 * The cat is hiding underneath the box.

 To refer to an object close to a point, use the prepositions "by," "near," "next to," "between," "among," and "opposite."

 * The gas station is by the grocery store.
 * The park is near her house.
 * Park your bike next to the garage.
 * There is a deer between the two trees.
 * There is a purple flower among the weeds.
 * The garage is opposite the house.

 Prepositions of Location

 To refer to a location, use the prepositions "in" (an area or volume), "at" (a point), and "on" (a surface).

 * They live in the country. (an area)
 * She will find him at the library. (a point)
 * There is a lot of dirt on the window. (a surface)

 Prepositions of Spatial Relationships

 To refer to a spatial relationship, use the prepositions "above," "across," "against," "ahead of," "along," "among," "around," "behind," "below,"
 "beneath," "beside," "between," "from," "in front of," "inside," "near," "off," "out of," "through," "toward," "under," and "within."

 * The post office is across the street from the grocery store.
 * We will stop at many attractions along the way.
 * The kids are hiding behind the tree.
 * His shirt is off.
 * Walk toward the garage and then turn left.
 * Place a check mark within the box.

 Prepositions Following Verbs and Adjectives

 Some verbs and adjectives are followed by a certain preposition. Sometimes verbs and adjectives can be followed by different prepositions, giving the phrase different meanings. To find which prepositions follow the verb or an adjective, look up the verb or adjective in an online dictionary, such as Merriam Webster, or use a corpus, such as The Corpus of Contemporary American English. Memorizing these phrases instead of just the preposition alone is the most helpful.

 Some Common Verb + Preposition Combinations

 About: worry, complain, read

 * He worries about the future.
 * She complained about the homework.
 * I read about the flooding in the city.

 At: arrive (a building or event), smile, look

 * He arrived at the airport 2 hours early.
 * The children smiled at her.
 * She looked at him.

 From: differ, suffer

 * The results differ from my original idea.
 * She suffers from dementia.

 For: account, allow, search

 * Be sure to account for any discrepancies.
 * I returned the transcripts to the interviewees to allow for revisions to be made.
 * They are searching for the missing dog.

 In: occur, result, succeed

 * The same problem occurred in three out of four cases.
 * My recruitment strategies resulted in finding 10 participants.
 * She will succeed in completing her degree.

 Of: approve, consist, smell

 * I approve of the idea.
 * The recipe consists of three basic ingredients.
 * The basement smells of mildew.

 On: concentrate, depend, insist

 * He is concentrating on his work.
 * They depend on each other.
 * I must insist on following this rule.

 To: belong, contribute, lead, refer

 * Bears belong to the family of mammals.
 * I hope to contribute to the previous research.
 * My results will lead to future research on the topic.
 * Please refer to my previous explanation.

 With: (dis)agree, argue, deal

 * I (dis)agree with you.
 * She argued with him.
 * They will deal with the situation.

 Although verb + preposition combinations appear similar to phrasal verbs, the verb and the particle (in this case, the preposition) in these combinations cannot be separated like phrasal verbs. See more about this on our verb choice page.

 Some Common Adjective + Preposition Combinations

 About At By From For In Of To With
  
 Accustomed X 
  
 Aware X 
  
 Beneficial X 
  
 Capable X 
  
 Characteristic X 
  
 Composed X X 
  
 Different X 
  
 Disappointed X X 
  
 Employed X X 
  
 Essential X 
  
 About At By From For In Of To With
  
 Familiar X 
  
 Good X X 
  
 Grateful X X 
  
 Interested X 
  
 Happy X X X 
  
 Opposed X 
  
 Proud X 
  
 Responsible X 
  
 Similar X 
  
 Sorry X X 
  
 

 Ending a Sentence With a Preposition

 At one time, schools taught students that a sentence should never end with a preposition. This rule is associated with Latin grammar, and while many aspects of Latin have made their way into English, there are times when following this particular grammar rule creates unclear or awkward sentence structures. Since the purpose of writing is to clearly communicate your ideas, it is acceptable to end a sentence with a preposition if the alternative would create confusion or is too overly formal.

 Example: The car had not been paid for. (Ends with a preposition but is acceptable)

 Unclear Revision: Paid for the car had not been. (Unclear sentence.)

 Example: I would like to know where she comes from. (Ends with a preposition but is acceptable)

 Overly Grammatical Revision: I would like to know from where she comes. (Grammatical but overly formal. Nobody actually speaks like this.)

 However, in academic writing, you may decide that it is worth revising your sentences to avoid ending with a preposition in order to maintain a more formal scholarly voice.

 Example: My research will focus on the community the students lived in.

 Revision: My research will focus on the community in which the students lived.

 Example: I like the people I am working with.

 Revision: I like the people with whom I am working.

 Prepositional Phrases and Wordiness

 Like with pronouns, too many prepositional phrases can create wordiness in a sentence:

 Example: The author chose the mixed-method design to explain that the purpose of the study was to explore the leadership qualities of the principals in the schools as a means to gauge teacher satisfaction in the first year of teaching.

 This type of sentence could be shortened and condensed to minimize the prepositional phrases and bring clarity to the writer's intent:

 Revision: The author chose the mixed-method design to explore the principals' leadership qualities and their impact on first-year teachers' satisfaction.

 Unnecessary Prepositions

 If the preposition is unnecessary, leave it out. This creates more clear and concise writing.

 Example: Where are the plates at?

 Revision: Where are the plates?

 Example: She jumped off of the balance beam.

 Revision: She jumped off the balance beam.

 Related Resource

 * Mastering the Mechanics Part 1: Simple Sentences (webinar)

 Knowledge Check: Prepositions

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Prepositions of Direction
To refer to a direction, use the prepositions "to," "in," "into," "on," and "onto." 
	She drove  to  the store.
	Don’t ring the doorbell. Come right  in(to)  the house.
	Drive  on(to)  the grass and park the car there.
Prepositions of Time
To refer to one point in time, use the prepositions "in," "at," and "on."
Use "in" with parts of the day (not specific times), months, years, and seasons.
	He reads  in  the evening.
	The weather is cold  in  December.
	She was born  in  1996.
	We rake leaves  in  the fall.
Use "at"   with the time of day. Also use "at" with noon, night, and midnight.
	I go to work  at  8:00.
	He eats lunch  at noon.
	She often goes for a walk  at  night.
	They go to bed  at  midnight.
Use "on" with days.
	I work  on  Saturdays.
	He does laundry  on  Wednesdays.
To refer to extended time, use the prepositions "since," "for," "by," "during," "from…to," "from…until," "with," and "within."
	I have lived in Minneapolis  since  2005. (I moved there in 2005 and still live there.)
	He will be in Toronto  for  3 weeks. (He will spend 3 weeks in Toronto.)
	She will finish her homework  by  6:00. (She will finish her homework sometime between now and 6:00.)
	He works part time  during  the summer. (For the period of time throughout the summer.)
	I will collect data  from  January  to  June. (Starting in January and ending in June.)
	They are in school  from  August  until  May. (Starting in August and ending in May.)
	She will graduate  within  2 years. (Not longer than 2 years.)
Prepositions of Place
To refer to a place, use the prepositions "in" (the point itself), "at" (the general vicinity), "on" (the surface), and "inside" (something contained).
	They will meet  in  the lunchroom.
	She was waiting  at  the corner.
	He left his phone  on  the bed.
	Place the pen  inside  the drawer.
To refer to an object higher than a point, use the prepositions "over" and "above." To refer to an object lower than a point, use the prepositions "below," "beneath," "under," and "underneath."
	The bird flew  over  the house.
	The plates were on the shelf  above  the cups.
	Basements are dug  below  ground.
	There is hard wood  beneath  the carpet.
	The squirrel hid the nuts  under  a pile of leaves.
	The cat is hiding  underneath  the box.
 To refer to an object close to a point, use the prepositions "by," "near," "next to," "between," "among," and "opposite."
	The gas station is  by  the grocery store.
	The park is  near  her house.
	Park your bike  next to  the garage.
	There is a deer  between  the two trees.
	There is a purple flower  among  the weeds.
	The garage is  opposite  the house.
Prepositions of Location
To refer to a location, use the prepositions "in" (an area or volume), "at" (a point), and "on" (a surface).
	They live  in  the country. (an area)
	She will find him  at  the library. (a point)
	There is a lot of dirt  on  the window. (a surface)
Prepositions of Spatial Relationships
To refer to a spatial relationship, use the prepositions "above," "across," "against," "ahead of," "along," "among," "around," "behind," "below,"
"beneath," "beside," "between," "from," "in front of," "inside," "near," "off," "out of," "through," "toward," "under," and "within."
	The post office is  across  the street from the grocery store.
	We will stop at many attractions  along  the way.
	The kids are hiding  behind  the tree.
	His shirt is  off .
	Walk  toward  the garage and then turn left.
	Place a check mark  within  the box.
Walden University Writing Center
https://academicguides.waldenu.edu/writingcenter/grammar/prepositions
https://www.grammarly.com/blog/prepositions/
Prepositions: Definition and Examples
2016-11-30
Brittney Ross
<div><article><div><ul>
<li>Prepositions indicate relationships between other words in a sentence.</li>
<li>Many prepositions tell you where something is or when something happened.</li>
<li>Most prepositions have several definitions, so the meaning changes quite a bit in different contexts.</li>
<li>Ending a sentence with a preposition is not a grammatical error.</li>
</ul>
<h2>What is a preposition?</h2>
<p><i>“Vampires! Zombies! Werewolves!”</i></p>
<p><i>“Where?!”</i></p>
<p><i>“Behind you!”</i></p>
<p>Thank goodness for prepositions. Imagine not knowing where the danger lay….</p>
<p>Prepositions tell us where or when something is in relation to something else. When monsters are approaching, it’s good to have these special words to tell us where those monsters are. Are they <b>behind</b> us or <b>in front of</b> us? Will they be arriving <b>in</b> three seconds or <b>at</b> midnight?</p>
<p>Prepositions often tell us where one<a href="https://www.grammarly.com/blog/common-noun/"> noun</a> is in relation to another (e.g., <i>The coffee is </i><b><i>on</i></b><i> the table </i><b><i>beside</i></b><i> you</i>). But they can also indicate more abstract ideas, such as purpose or contrast (e.g., <i>We went </i><b><i>for</i></b><i> a walk </i><b><i>despite</i></b><i> the rain</i>).</p>
<p><b>Here’s a tip:</b> Want to make sure your <a href="https://www.grammarly.com/writing">writing</a> shines? Grammarly can <a href="https://www.grammarly.com/spell-checker">check your spelling</a> and save you from <a href="https://www.grammarly.com/grammar">grammar</a> and <a href="https://www.grammarly.com/punctuation">punctuation</a> mistakes. It even <a href="https://www.grammarly.com/proofreading">proofreads</a> your text, so your work is extra polished wherever you write. </p>
<h2>Types of prepositions</h2>
<p>Prepositions indicate direction, time, location, and spatial relationships, as well as other abstract types of relationships.</p>
<p><b>Direction:</b> Look <i>to</i> the left and you’ll see our destination.</p>
<p><b>Time:</b> We’ve been working <i>since</i> this morning.</p>
<p><b>Location:</b> We saw a movie <i>at</i> the theater.</p>
<p><b>Space:</b> The dog hid <i>under</i> the table.</p>
<h2>Preposition examples</h2>
<p>Unfortunately, there’s no reliable formula for determining which preposition to use with a particular combination of words. The best way to learn which prepositions go with which words is to read as much<a href="https://www.grammarly.com/writing"> high-quality writing</a> as you can and pay attention to which combinations sound right. Here are a few examples of the most common prepositions used in sentences.</p>
<p>You can also use tools like Google Ngrams to see which prepositions most commonly occur with particular words—but remember, this tool can’t explain the difference in meaning between different<a href="https://www.grammarly.com/blog/prepositional-phrase/"> prepositional phrases</a> like “pay for” (to purchase) and “pay off” (to bribe). For that, you may want to refer to a list of prepositions that includes the meanings of common combinations.</p>
<h2>Ending a sentence with a preposition</h2>
<p>The old claim that<a href="https://www.grammarly.com/blog/youve-been-lied-to-heres-why-you-absolutely-can-end-a-sentence-with-a-preposition/"> it’s wrong to end a sentence with a preposition</a> has been utterly debunked. It’s not true and it never was true. Writers who always insist that a preposition can’t end a sentence often end up with stilted and unnatural sentences:</p>
<p>That said, it is sometimes more elegant to move a preposition to an earlier spot in a sentence, especially in very serious and<a href="https://www.grammarly.com/blog/formal-vs-informal-writing/"> formal writing</a>. But if you do move the preposition, remember to delete it from the end.</p>
<h2>Unnecessary prepositions</h2>
<p>One of the most common preposition mistakes is adding an unnecessary <i>at</i> to the end of a question.</p>
<p>Although this is common in<a href="https://www.grammarly.com/blog/the-primary-differences-among-major-international-english-dialects/"> some English dialects</a>, it’s considered an error in writing. You can fix the problem by simply deleting the <i>at</i>.</p>
<p>On the bright side, if you’re not sure which preposition to use, sometimes you can just get rid of it altogether. In fact, you should always get rid of unnecessary prepositional phrases. Too many prepositions can be a sign of flabby writing. Look at how many prepositions appear in the sentence below:</p>
<p>Getting rid of the prepositions forces you to tighten up the sentence. The result is shorter, more direct, and easier to understand:</p>
<p>Here’s another example:</p>
<p>Get rid of the <i>up</i>. You don’t need it:</p>
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 Prepositions: Definition and Examples

 Brittney Ross
 Updated on January 14, 2021Grammar
 * Prepositions indicate relationships between other words in a sentence.
 * Many prepositions tell you where something is or when something happened.
 * Most prepositions have several definitions, so the meaning changes quite a bit in different contexts.
 * Ending a sentence with a preposition is not a grammatical error.

 Confused about prepositions?
 Grammarly provides suggestions, wherever you write.
 Write with Grammarly

 What is a preposition?

 “Vampires! Zombies! Werewolves!”

 “Where?!”

 “Behind you!”

 Thank goodness for prepositions. Imagine not knowing where the danger lay….

 Prepositions tell us where or when something is in relation to something else. When monsters are approaching, it’s good to have these special words to tell us where those monsters are. Are they behind us or in front of us? Will they be arriving in three seconds or at midnight?

 Prepositions often tell us where one noun is in relation to another (e.g., The coffee is on the table beside you ). But they can also indicate more abstract ideas, such as purpose or contrast (e.g., We went for a walk despite the rain ).

 Here’s a tip: Want to make sure your writing shines? Grammarly can check your spelling and save you from grammar and punctuation mistakes. It even proofreads your text, so your work is extra polished wherever you write.

 Types of prepositions

 Prepositions indicate direction, time, location, and spatial relationships, as well as other abstract types of relationships.

 Direction: Look to the left and you’ll see our destination.

 Time: We’ve been working since this morning.

 Location: We saw a movie at the theater.

 Space: The dog hid under the table.

 Preposition examples

 Unfortunately, there’s no reliable formula for determining which preposition to use with a particular combination of words. The best way to learn which prepositions go with which words is to read as much high-quality writing as you can and pay attention to which combinations sound right. Here are a few examples of the most common prepositions used in sentences.

 I should rewrite the introduction of my essay.

 Sam left his jacket in the car.

 Did you send that letter to your mother?

 We’re cooking for ten guests tonight.

 Dan ate lunch with his boss.

 You can also use tools like Google Ngrams to see which prepositions most commonly occur with particular words—but remember, this tool can’t explain the difference in meaning between different prepositional phrases like “pay for” (to purchase) and “pay off” (to bribe). For that, you may want to refer to a list of prepositions that includes the meanings of common combinations.

 Ending a sentence with a preposition

 The old claim that it’s wrong to end a sentence with a preposition has been utterly debunked. It’s not true and it never was true. Writers who always insist that a preposition can’t end a sentence often end up with stilted and unnatural sentences:

 There’s no one else to hide behind . (Grammatically correct and natural)

 There’s no one else behind whom to hide. (Grammatically correct, but unnatural)

 Where did you come from ? (Grammatically correct and natural)

 From where did you come? (Grammatically correct, but unnatural)

 That said, it is sometimes more elegant to move a preposition to an earlier spot in a sentence, especially in very serious and formal writing . But if you do move the preposition, remember to delete it from the end.

 This is something we must meditate on .

 This is something on which we must meditate.

 This is something on which we must meditate on .

 Unnecessary prepositions

 One of the most common preposition mistakes is adding an unnecessary at to the end of a question.

 Where is your brother at ?

 Although this is common in some English dialects , it’s considered an error in writing. You can fix the problem by simply deleting the at .

 Where is your brother?

 On the bright side, if you’re not sure which preposition to use, sometimes you can just get rid of it altogether. In fact, you should always get rid of unnecessary prepositional phrases. Too many prepositions can be a sign of flabby writing. Look at how many prepositions appear in the sentence below:

 For many people, the reality of an entry into a new area of employment is cause for a host of anxieties.

 Getting rid of the prepositions forces you to tighten up the sentence. The result is shorter, more direct, and easier to understand:

 Changing careers makes many people anxious.

 Here’s another example:

 Alex hit the baseball up over the fence.

 Get rid of the up . You don’t need it:

 Alex hit the baseball over the fence.

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Prepositions indicate relationships between other words in a sentence.
Many prepositions tell you where something is or when something happened.
Most prepositions have several definitions, so the meaning changes quite a bit in different contexts.
Ending a sentence with a preposition is not a grammatical error.
What is a preposition?
 “Vampires! Zombies! Werewolves!” 
 “Where?!” 
 “Behind you!” 
Thank goodness for prepositions. Imagine not knowing where the danger lay….
Prepositions tell us where or when something is in relation to something else. When monsters are approaching, it’s good to have these special words to tell us where those monsters are. Are they behind us or in front of us? Will they be arriving in three seconds or at midnight?
Prepositions often tell us where one noun is in relation to another (e.g., The coffee is   on   the table   beside   you). But they can also indicate more abstract ideas, such as purpose or contrast (e.g., We went   for   a walk   despite   the rain).
 Here’s a tip: Want to make sure your writing shines? Grammarly can check your spelling and save you from grammar and punctuation mistakes. It even proofreads your text, so your work is extra polished wherever you write. 
Types of prepositions
Prepositions indicate direction, time, location, and spatial relationships, as well as other abstract types of relationships.
 Direction: Look to the left and you’ll see our destination.
 Time: We’ve been working since this morning.
 Location: We saw a movie at the theater.
 Space: The dog hid under the table.
Preposition examples
Unfortunately, there’s no reliable formula for determining which preposition to use with a particular combination of words. The best way to learn which prepositions go with which words is to read as much high-quality writing as you can and pay attention to which combinations sound right. Here are a few examples of the most common prepositions used in sentences.
You can also use tools like Google Ngrams to see which prepositions most commonly occur with particular words—but remember, this tool can’t explain the difference in meaning between different prepositional phrases like “pay for” (to purchase) and “pay off” (to bribe). For that, you may want to refer to a list of prepositions that includes the meanings of common combinations.
Ending a sentence with a preposition
The old claim that it’s wrong to end a sentence with a preposition has been utterly debunked. It’s not true and it never was true. Writers who always insist that a preposition can’t end a sentence often end up with stilted and unnatural sentences:
That said, it is sometimes more elegant to move a preposition to an earlier spot in a sentence, especially in very serious and formal writing. But if you do move the preposition, remember to delete it from the end.
Unnecessary prepositions
One of the most common preposition mistakes is adding an unnecessary at to the end of a question.
Although this is common in some English dialects, it’s considered an error in writing. You can fix the problem by simply deleting the at.
On the bright side, if you’re not sure which preposition to use, sometimes you can just get rid of it altogether. In fact, you should always get rid of unnecessary prepositional phrases. Too many prepositions can be a sign of flabby writing. Look at how many prepositions appear in the sentence below:
Getting rid of the prepositions forces you to tighten up the sentence. The result is shorter, more direct, and easier to understand:
Here’s another example:
Get rid of the up. You don’t need it:
Prepositions: Definition and Examples
https://www.grammarly.com/blog/prepositions/