7.2: Parity and Counting Arguments
This section is concerned with two very powerful elements of the proofmaking arsenal: “ Parity ” is a way of referring to the result of an even/odd calculation; Counting arguments most often take the form of counting some collection in two different ways – and then comparing those results. These techniques have little to do with one another, but when they are applicable they tend to produce really elegant little arguments.
In (very) early computers and business machines, paper cards were used to store information. A so-called “punch card” or “Hollerith card” was used to store binary information by means of holes punched into it. Paper tape was also used in a similar fashion. A typical paper tape format would involve \(8\) positions in rows across the tape that might or might not be punched, often a column of smaller holes would appear as well which did not store information but were used to drive the tape through the reading mechanism on a sprocket. Tapes and cards could be “read” either by small sets of electrical contacts which would touch through a punched hole or be kept separate if the position wasn’t punched, or by using a photo-detector to sense whether light could pass through the hole or not. The mechanisms for reading and writing on these paper media were amazingly accurate, and allowed early data processing machines to use just a couple of large file cabinets to store what now fits in a jump drive one can wear on a necklace. (About \(10\) or \(12\) cabinets could hold a gigabyte of data).
Paper media was ideally suited to storing binary information, but of course most of the real data people needed to store and process would be alphanumeric 1 . There were several encoding schemes that served to translate between the character sets that people commonly used and the binary numerals that could be stored on paper. One of these schemes still survives today – ASCII. The American Standard Code for Information Interchange uses \(7\)-bit binary numerals to represent characters, so it contains \(128\) different symbols. This is more than enough to represent both upper- and lower-case letters, the \(10\) numerals, and the punctuation marks – many of the remaining spots in the ASCII code were used to contain so-called “control characters” that were associated with functionality that appeared on old-fashioned teletype equipment – things like “ring the bell,” “move the carriage backwards one space,” “move the carriage to the next line,” etc. These control characters are why modern keyboards still have a modifier key labeled “Ctrl” on them. The following listing gives the decimal and binary numerals from \(0\) to \(127\) and the ASCII characters associated with them – the non-printing and control characters have a \(2\) or \(3\) letter mnemonic designation.
\(\begin{array} &0\;\;\;\;\; &0000\; 0000\;\;\;\;\; &\text{NUL}\;\;\;\;\;\;\;\;\;\; &64\;\;\;\;\; &0100\; 0000\;\;\;\;\; &@ \\ 1\;\;\;\;\; &0000\; 0001 \;\;\;\;\;&\text{SOH}\;\;\;\;\;\;\;\;\;\; &65\;\;\;\;\; &0100\; 0001\;\;\;\;\; &\text{A}\\ 2\;\;\;\;\; &0000\; 0010 \;\;\;\;\;&\text{STX}\;\;\;\;\;\;\;\;\;\; &66\;\;\;\;\; &0100\; 0010\;\;\;\;\; &\text{B}\\ 3\;\;\;\;\; &0000\; 0011 \;\;\;\;\;&\text{ETX}\;\;\;\;\;\;\;\;\;\; &67\;\;\;\;\; &0100\; 0011\;\;\;\;\; &\text{C}\\ 4\;\;\;\;\; &0000\; 0100 \;\;\;\;\;&\text{EOT}\;\;\;\;\;\;\;\;\;\; &68\;\;\;\;\; &0100\; 0100\;\;\;\;\; &\text{D}\\ 5\;\;\;\;\; &0000\; 0101 \;\;\;\;\;&\text{ENQ}\;\;\;\;\;\;\;\;\;\; &69\;\;\;\;\; &0100\; 0101\;\;\;\;\; &\text{E}\\ 6\;\;\;\;\; &0000\; 0110 \;\;\;\;\;&\text{ACK}\;\;\;\;\;\;\;\;\;\; &70\;\;\;\;\; &0100 \;0110\;\;\;\;\; &\text{F}\\ 7\;\;\;\;\; &0000\; 0111 \;\;\;\;\;&\text{BEL}\;\;\;\;\;\;\;\;\;\; &71\;\;\;\;\; &0100 \;0111\;\;\;\;\; &\text{G}\\ 8\;\;\;\;\; &0000\; 1000 \;\;\;\;\;&\text{BS}\;\;\;\;\;\;\;\;\;\; &72\;\;\;\;\; &0100 \;1000\;\;\;\;\; &\text{H}\\ 9\;\;\;\;\; &0000\; 1001 \;\;\;\;\;&\text{TAB}\;\;\;\;\;\;\;\;\;\; &73\;\;\;\;\; &0100\; 1001\;\;\;\;\; &\text{I}\\ 10\;\;\;\;\; &0000\; 1010 \;\;\;\;\;&\text{LF}\;\;\;\;\;\;\;\;\;\; &74\;\;\;\;\; &0100 \;1010\;\;\;\;\; &\text{J}\\ 11\;\;\;\;\; &0000\; 1011 \;\;\;\;\;&\text{VT}\;\;\;\;\;\;\;\;\;\; &75\;\;\;\;\; &0100\; 1011\;\;\;\;\; &\text{K}\\ 12\;\;\;\;\; &0000\; 1100 \;\;\;\;\;&\text{FF}\;\;\;\;\;\;\;\;\;\; &76\;\;\;\;\; &0100 \;1100\;\;\;\;\; &\text{L}\\ 13\;\;\;\;\; &0000\; 1101 \;\;\;\;\;&\text{CR}\;\;\;\;\;\;\;\;\;\; &77\;\;\;\;\; &0100\; 1101\;\;\;\;\; &\text{M}\\ 14\;\;\;\;\; &0000\; 1110 \;\;\;\;\;&\text{SO}\;\;\;\;\;\;\;\;\;\; &78\;\;\;\;\; &0100 \;1110\;\;\;\;\; &\text{N}\\ 15\;\;\;\;\; &0000\; 1111 \;\;\;\;\;&\text{SI}\;\;\;\;\;\;\;\;\;\; &79\;\;\;\;\; &0100\; 1111 \;\;\;\;\;&\text{O}\\ 16\;\;\;\;\; &0001\; 0000 \;\;\;\;\;&\text{DLE}\;\;\;\;\;\;\;\;\;\; &80\;\;\;\;\; &0101\; 0000 \;\;\;\;\;&\text{P}\\ 17\;\;\;\;\; &0001\; 0001 ;\;\;\;\;&\text{DC1}\;\;\;\;\;\;\;\;\;\; &81\;\;\;\;\; &0101\; 0001 \;\;\;\;\;&\text{Q}\\ 18\;\;\;\;\; &0001\; 0010 \;\;\;\;\;&\text{DC2}\;\;\;\;\;\;\;\;\;\; &82\;\;\;\;\; &0101\; 0010 \;\;\;\;\;&\text{R}\\ 19\;\;\;\;\; &0001\; 0011 \;\;\;\;\;&\text{DC3}\;\;\;\;\;\;\;\;\;\; &83\;\;\;\;\; &0101\; 0011 \;\;\;\;\;&\text{S}\\ 20\;\;\;\;\; &0001\; 0100 \;\;\;\;\;&\text{DC4}\;\;\;\;\;\;\;\;\;\; &84\;\;\;\;\; &0101\; 0100 \;\;\;\;\;&\text{T}\\ 21\;\;\;\;\; &0001\; 0101 \;\;\;\;\;&\text{NAK}\;\;\;\;\;\;\;\;\;\; &85\;\;\;\;\; &0101\; 0101 \;\;\;\;\;&\text{U}\\ 22\;\;\;\;\; &0001\; 0110 \;\;\;\;\;&\text{SYN}\;\;\;\;\;\;\;\;\;\; &86\;\;\;\;\; &0101\; 0110 \;\;\;\;\;&\text{V}\\ 23\;\;\;\;\; &0001\; 0111 \;\;\;\;\;&\text{ETB}\;\;\;\;\;\;\;\;\;\; &87\;\;\;\;\; &0101\; 0111 \;\;\;\;\;&\text{W}\\ 24\;\;\;\;\; &0001\; 1000 \;\;\;\;\;&\text{CAN}\;\;\;\;\;\;\;\;\;\; &88\;\;\;\;\; &0101\; 1000 \;\;\;\;\;&\text{X}\\ 25\;\;\;\;\; &0001\; 1001 \;\;\;\;\;&\text{EM}\;\;\;\;\;\;\;\;\;\; &89\;\;\;\;\; &0101 \;1001 \;\;\;\;\;&\text{Y}\\ 26\;\;\;\;\; &0001\; 1010 \;\;\;\;\;&\text{SUB}\;\;\;\;\;\;\;\;\;\; &90\;\;\;\;\; &0101\; 1010 \;\;\;\;\;&\text{Z}\\ 27\;\;\;\;\; &0001 \;1011 \;\;\;\;\;&\text{ESC}\;\;\;\;\;\;\;\;\;\; &91\;\;\;\;\; &0101\; 1011 \;\;\;\;\;&[\\ 28\;\;\;\;\; &0001 \;1100 \;\;\;\;\;&\text{FS}\;\;\;\;\;\;\;\;\;\; &92\;\;\;\;\; &0101\; 1100 \;\;\;\;\;&\setminus \\ 29\;\;\;\;\; &0001 \;1101 \;\;\;\;\;&\text{GS}\;\;\;\;\;\;\;\;\;\; &93\;\;\;\;\; &0101\; 1101 \;\;\;\;\;&]\\ 30\;\;\;\;\; &0001 \;1110 \;\;\;\;\;&\text{RS}\;\;\;\;\;\;\;\;\;\; &94\;\;\;\;\; &0101\; 1110 \;\;\;\;\;&\text{^} \\ 31\;\;\;\;\; &0001 \;1111 \;\;\;\;\;&\text{US}\;\;\;\;\;\;\;\;\;\; &95\;\;\;\;\; &0101 \;1111 \;\;\;\;\;&\underline{\;}\\ 32\;\;\;\;\; &0010 \;0000 \;\;\;\;\;&\text{}\;\;\;\;\;\;\;\;\;\; &96\;\;\;\;\; &0110 \;0000 \;\;\;\;\;&‘\\ 33\;\;\;\;\; &0010 \;0001 \;\;\;\;\;&!\;\;\;\;\;\;\;\;\;\; &97\;\;\;\;\; &0110 \;0001 \;\;\;\;\;&\text{a}\\ 34\;\;\;\;\; &0010 \;0010 \;\;\;\;\;&" \;\;\;\;\;\;\;\;\;\;&98\;\;\;\;\; &0110 \;0010 \;\;\;\;\;&\text{b}\\ 35\;\;\;\;\; &0010 \;0011 \;\;\;\;\;&\#\;\;\;\;\;\;\;\;\;\; &99\;\;\;\;\; &0110\; 0011 \;\;\;\;\;&\text{c}\\ 36\;\;\;\;\; &0010 \;0100 \;\;\;\;\;&$ \;\;\;\;\;\;\;\;\;\;&100\;\;\;\;\; &0110\; 0100 \;\;\;\;\;&\text{d}\\ 37\;\;\;\;\; &0010 \;0101 \;\;\;\;\;&\% \;\;\;\;\;\;\;\;\;\;&101\;\;\;\;\; &0110\; 0101 \;\;\;\;\;&\text{e}\\38\;\;\;\;\; &0010 \;0110 \;\;\;\;\;&\& \;\;\;\;\;\;\;\;\;\;&102\;\;\;\;\; &0110\; 0110 \;\;\;\;\;&\text{f}\\ 39\;\;\;\;\; &0010 \;0111 \;\;\;\;\;&’ \;\;\;\;\;\;\;\;\;\;&103\;\;\;\;\; &0110\; 0111 \;\;\;\;\;&\text{g}\\ 40\;\;\;\;\; &0010 \;1000 \;\;\;\;\;&( \;\;\;\;\;\;\;\;\;\;&104\;\;\;\;\; &0110\; 1000 \;\;\;\;\;&\text{h}\\ 41\;\;\;\;\; &0010 \;1001 \;\;\;\;\;&) \;\;\;\;\;\;\;\;\;\;&105\;\;\;\;\; &0110 \;1001 \;\;\;\;\;&\text{i}\\ 42\;\;\;\;\; &0010 \;1010 \;\;\;\;\;&* \;\;\;\;\;\;\;\;\;\;&106\;\;\;\;\;&0110 \;1010 \;\;\;\;\;&\text{j}\\ 43\;\;\;\;\; &0010 \;1011 \;\;\;\;\;&+ \;\;\;\;\;\;\;\;\;\;&107\;\;\;\;\; &0110 \;1011 \;\;\;\;\;&\text{k}\\ 44\;\;\;\;\; &0010 \;1100 \;\;\;\;\;&, \;\;\;\;\;\;\;\;\;\;&108\;\;\;\;\; &0110 \;1100 \;\;\;\;\;&\text{l}\\ 45\;\;\;\;\; &0010 \;1101 \;\;\;\;\;&- \;\;\;\;\;\;\;\;\;\;&109\;\;\;\;\; &0110 \;1101 \;\;\;\;\;&\text{m}\\ 46\;\;\;\;\; &0010 \;1110 \;\;\;\;\;&. \;\;\;\;\;\;\;\;\;\;&110\;\;\;\;\; &0110 \;1110 \;\;\;\;\;&\text{n}\\ 47\;\;\;\;\; &0010 \;1111 \;\;\;\;\;&/ \;\;\;\;\;\;\;\;\;\;&111\;\;\;\;\; &0110 \;1111 \;\;\;\;\;&\text{o}\\ 48\;\;\;\;\; &0011 \;0000 \;\;\;\;\;&0 \;\;\;\;\;\;\;\;\;\;&112\;\;\;\;\; &0111 \;0000 \;\;\;\;\;&\text{p}\\ 49\;\;\;\;\; &0011 \;0001 \;\;\;\;\;&1 \;\;\;\;\;\;\;\;\;\;&113\;\;\;\;\; &0111 \;0001 \;\;\;\;\;&\text{q}\\ 50\;\;\;\;\; &0011 \;0010 \;\;\;\;\;&2 \;\;\;\;\;\;\;\;\;\;&114\;\;\;\;\; &0111 \;0010 \;\;\;\;\;&\text{r}\\ 51\;\;\;\;\; &0011 \;0011 \;\;\;\;\;&3 \;\;\;\;\;\;\;\;\;\;&115\;\;\;\;\; &0111 \;0011 \;\;\;\;\;&\text{s}\\ 52\;\;\;\;\; &0011 \;0100 \;\;\;\;\;&4 \;\;\;\;\;\;\;\;\;\;&116\;\;\;\;\; &0111 \;0100 \;\;\;\;\;&\text{t}\\ 53\;\;\;\;\; &0011 \;0101 \;\;\;\;\;&5 \;\;\;\;\;\;\;\;\;\;&117\;\;\;\;\; &0111\; 0101 \;\;\;\;\;&\text{u}\\ 54\;\;\;\;\; &0011 \;0110 \;\;\;\;\;&6 \;\;\;\;\;\;\;\;\;\;&118\;\;\;\;\; &0111 \;0110 \;\;\;\;\;&\text{v}\\ 55\;\;\;\;\; &0011 \;0111 \;\;\;\;\;&7 \;\;\;\;\;\;\;\;\;\;&119\;\;\;\;\; &0111 \;0111 \;\;\;\;\;&\text{w}\\ 56\;\;\;\;\; &0011 \;1000 \;\;\;\;\;&8 \;\;\;\;\;\;\;\;\;\;&120\;\;\;\;\; &0111 \;1000 \;\;\;\;\;&\text{x}\\ 57\;\;\;\;\; &0011 \;1001 \;\;\;\;\;&9 \;\;\;\;\;\;\;\;\;\;&121\;\;\;\;\; &0111 \;1001 \;\;\;\;\;&\text{y}\\ 58\;\;\;\;\; &0011 \;1010 \;\;\;\;\;&: \;\;\;\;\;\;\;\;\;\;&122\;\;\;\;\; &0111 \;1010 \;\;\;\;\;&\text{z}\\ 59\;\;\;\;\; &0011 \;1011 \;\;\;\;\;&; \;\;\;\;\;\;\;\;\;\;&123\;\;\;\;\; &0111 \;1011 \;\;\;\;\;&\{ \\ 60\;\;\;\;\; &0011 \;1100 \;\;\;\;\;&< \;\;\;\;\;\;\;\;\;\;&124\;\;\;\;\; &0111 \;1100 \;\;\;\;\;&| \\ 61\;\;\;\;\; &0011 \;1101 \;\;\;\;\;&= \;\;\;\;\;\;\;\;\;\;&125\;\;\;\;\; &0111 \;1101 \;\;\;\;\;&\} \\ 62\;\;\;\;\; &0011 \;1110 \;\;\;\;\;&> \;\;\;\;\;\;\;\;\;\;&126\;\;\;\;\; &0111 \;1110 \;\;\;\;\;& \sim \\ 63\;\;\;\;\; &0011 \;1111 \;\;\;\;\;&? \;\;\;\;\;\;\;\;\;\;&127\;\;\;\;\; &0111 \;1111 \;\;\;\;\;&\text{DEL} \end{array} \)
Now it only takes \(7\) bits to encode the \(128\) possible values in the ASCII system, which can easily be verified by noticing that the left-most bits in all of the binary representations above are \(0\). Most computers use \(8\) bit words or “bytes” as their basic units of information, and the fact that the ASCII code only requires \(7\) bits lead someone to think up a use for that additional bit. It became a “parity check bit.” If the seven bits of an ASCII encoding have an odd number of \(1\)’s, the parity check bit is set to \(1\) — otherwise, it is set to \(0\). The result of this is that, subsequently, all of the \(8\) bit words that encode ASCII data will have an even number of \(1\)’s. This is an example of a so-called error detecting code known as the “even code” or the “parity check code.” If data is sent over a noisy telecommunications channel, or is stored in fallible computer memory, there is some small but calculable probability that there will be a “bit error.” For instance, one computer might send \(10000111\) (which is the ASCII code that says “ring the bell”) but another machine across the network might receive \(10100111\) (the \(3^{\text{rd}}\) bit from the left has been received in error) now if we are only looking at the rightmost seven bits we will think that the ASCII code for a single quote has been received, but if we note that this piece of received data has an odd number of ones we’ll realize that something is amiss. There are other more advanced coding schemes that will let us not only detect an error, but (within limits) correct it as well! This rather amazing feat is what makes wireless telephony (not to mention communications with deep space probes — whoops! I mentioned it) work.
The concept of parity can be used in many settings to prove some fairly remarkable results.
In Section 6.3 , we introduced the idea of a graph. This notion was first used by Leonhard Euler to solve a recreational math problem posed by the citizens of Königsberg, Prussia (this is the city now known as Kaliningrad, Russia.) Königsberg was situated at a place where two branches of the Pregel river 2 come together – there is also a large island situated near this confluence. By Euler’s time, the city of Königsberg covered this island as well as the north and south banks of the river and also the promontory where the branches came together. A network of seven bridges had been constructed to connect all these land masses. The townsfolk are alleged to have become enthralled by the question of whether it was possible to leave one’s home and take a walk through town which crossed each of the bridges exactly once and, finally, return to one’s home.
Euler settled the question (it can’t be done) be converting the map of Königsberg into a graph and then making some simple observations about the parities of the nodes in this graph. The degree of a node in a graph is the number of edges that are incident with it (if a so-called “loop edge” is present it adds two to the node’s degree). The “parity of a node” is shorthand for the “parity of the degree of the node.”
The graph of Königsberg has \(4\) nodes: one of degree \(5\) and three of degree \(3\). All the nodes are odd. Would it be possible to either modify Königsberg or come up with an entirely new graph having some even nodes? Well, the answer to that is easy – just tear down one of the bridges, and two of the nodes will have their degree changed by one; they’ll both become even. Notice that, by removing one edge, we effected the parity of two nodes. Is it possible to create a graph with four nodes in which just one of them is even? More generally, given any short list of natural numbers, is it possible to draw a graph whose degrees are the listed numbers?
Try drawing graphs having the following lists of vertex degrees. (In some cases it will be impossible. . . )
- \(\{1, 1, 2, 3, 3\}\)
- \(\{1, 2, 3, 5\}\)
- \(\{1, 2, 3, 4\}\)
- \(\{4, 4, 4, 4, 5\}\)
- \(\{3, 3, 3, 3\}\)
- \(\{3, 3, 3, 3, 3\}\)
When it is possible to create a graph with a specified list of vertex degrees, it is usually easy to do. Of course, when it’s impossible you struggle a bit. . . To help get things rolling (just in case you haven’t really done the exercise) I’ll give a hint – for the first list it is possible to draw a graph, for the second it is not. Can you distinguish the pattern? What makes one list of vertex degrees reasonable and another not?
 BTW, if you did do it, good for you! You can either join with me now in sneering at all those people who are scurrying back to do the last one, or try the following:)
Figure out a way to distinguish a sequence of numbers that can be the degree sequence of some graph from the sequences that cannot be.
Okay, now if you’re reading this sentence you should know that every other list of vertex degrees above is impossible, you should have graphs drawn in the margin here for the \(1^{\text{st}}\), \(3^{\text{rd}}\) and \(5^{\text{th}}\) degree sequences, and you may have discovered some version of the following
In an undirected graph, the number of vertices having an odd degree is even.
A slightly pithier statement is: All graphs have an even number of odd nodes.
We’ll leave the proof of this theorem to the exercises but most of the work is done in proving the following equivalent result.
In an undirected graph, the sum of the degrees of the vertices is even.
- Proof
-
The sum of the degrees of all the vertices in a graph \(G\),
\[\sum_{v∈V (G)} \text{deg}(v),\]
counts every edge of \(G\) exactly twice.
Thus,
\[\sum_{v∈V (G)} \text{deg}(v) = 2 · |E(G)|.\]
In particular we see that this sum is even.
Q.E.D.
The question of whether a graph having a given list of vertex degrees can exist comes down to an elegant little argument using both of the techniques in the title of this section. We count the edge set of the graph in two ways – once in the usual fashion and once by summing the vertex degrees; we also note that since this latter count is actually a double count we can bring in the concept of parity.
Another perfectly lovely argument involving parity arises in questions concerning whether or not it is possible to tile a pruned chessboard with dominoes. We’ve seen dominoes before in Section 5.1 and we’re just hoping you’ve run across chessboards before. Usually, a chessboard is \(8 × 8\), but we would like to adopt a more liberal interpretation that a chessboard can be any rectangular grid of squares we might choose. 3 Suppose that we have a supply of dominoes that are of just the right size that if they are laid on a chessboard they perfectly cover two adjacent squares. Our first question is quite simple. Is it possible to perfectly tile an \(m × n\) chessboard with such dominoes?
First, let’s specify the question a bit more. By “perfectly tiling” a chessboard we mean that every domino lies (fully) on the board, covering precisely two squares, and that every square of the board is covered by a domino.
The answer is straightforward. If at least one of m or n is even it can be done. A necessary condition is that the number of squares be even (since every domino covers two squares) and so, if both \(m\) and \(n\) are odd we will be out of luck.
A “pruned board” is obtained by either literally removing some of the squares or perhaps by marking them as being off-limits in some way. When we ask questions about perfect tilings of pruned chessboards things get more interesting and the notion of parity can be used in several ways.
Here are two tiling problems regarding square chessboards:
- An even-sided square board (e.g. an ordinary \(8 × 8\) board) with diagonally opposite corners pruned.
- An odd-sided board with one square pruned.
Both of these situations satisfy the basic necessary condition that the number of squares on the board must be even. You may be able to determine another “parity” approach to these tiling problems by attempting the following
Below are two five-by-five chessboards each having a single square pruned. One can be tiled by dominoes and the other cannot. Which is which?
The pattern of black and white squares on a chessboard is an example of a sort of artificial parity, if we number the squares of the board appropriately then the odd squares will be white and the even squares will be black. We are used to chessboards having this alternating black/white pattern on them, but nothing about these tiling problems required that structure 4 . If we were used to monochromatic chessboards, we might never solve the previous two problems – unless of course we invented the coloring scheme in order to solve them. An odd-by-odd chessboard has more squares of one color than of the other. An odd-by-odd chessboard needs to have a square pruned in order for it to be possible for it to be tiled by dominoes – but if the wrong colored square is pruned it will still be impossible. Each domino covers two squares – one of each color! (So the pruned board must have the same number of white squares as black.)
We’ll close this section with another example of the technique of counting in two ways.
A magic square of order \(n\) is a square \(n×n\) array containing the numbers \(1, 2, 3, . . . , n^2\). The numbers must be arranged in such a way that every row and every column sum to the same number – this value is known as the magic sum.
For example, the following is an order \(3\) magic square.
| \(1\) | \(6\) | \(8\) |
| \(5\) | \(7\) | \(3\) |
| \(9\) | \(2\) | \(4\) |
The definition of a magic square requires that the rows and columns sum to the same number but says nothing about what that number must be. It is conceivable that we could produce magic squares (of the same order) having different magic sums. This is conceivable, but in fact the magic sum is determined completely by \(n\).
A magic square of order \(n\) has a magic sum equal to \(\dfrac{n^3 + n}{2}\).
- Proof
-
We count the total of the entries in the magic square in two ways. The sum of all the entries in the magic square is
\[S = 1 + 2 + 3 + . . . + n^2.\]
Using the formula for the sum of the first \(k\) naturals \(( \sum^{k}_{i=1} i = \dfrac{k^2+k}{2} )\) and evaluating at \(n^2\) gives
\[S = \dfrac{n^4 + n^2}{2}.\]
On the other hand, if the magic sum is \(M\), then each of the \(n\) rows has numbers in it which sum to \(M\) so
\[S = nM.\]
By equating these different expressions for \(S\) and solving for \(M\), we prove the desired result:
\[nM = \dfrac{n^4 + n^2}{2},\]
therefore
\[M = \dfrac{n^3 + n}{2} .\]
Q.E.D.
Exercises:
A walking tour of K¨onigsberg such as is described in this section, or more generally, a circuit through an arbitrary graph that crosses each edge precisely once and begins and ends at the same node is known as an Eulerian circuit . An Eulerian path also crosses every edge of a graph exactly once but it begins and ends at distinct nodes. For each of the following graphs determine whether an Eulerian circuit or path is possible, and if so, draw it.
Complete the proof of the fact that “Every graph has an even number of odd nodes.”
Provide an argument as to why an \(8 × 8\) chessboard with two squares pruned from diagonally opposite corners cannot be tiled with dominoes.
Prove that, if \(n\) is odd, any \(n × n\) chessboard with a square the same color as one of its corners pruned can be tiled by dominoes.
The five tetrominoes (familiar to players of the video game Tetris) are relatives of dominoes made up of four small squares.
All together these five tetrominoes contain \(20\) squares so it is conceivable that they could be used to tile a \(4 ×5\) chessboard. Prove that this is actually impossible.
State necessary and sufficient conditions for the existence of an Eulerian circuit in a graph.
State necessary and sufficient conditions for the existence of an Eulerian path in a graph.
Construct magic squares of order \(4\) and \(5\).
A magic hexagon of order \(2\) would consist of filling-in the numbers from \(1\) to \(7\) in the hexagonal array below. The magic condition means that each of the \(9\) “lines” of adjacent hexagons would have the same sum. Is this possible?
Is there a magic hexagon of order \(3\)?


7.6: The American Style
As we will see in a later chapter, jazz is a uniquely American style. American orchestral composers were becoming aware of jazz in the early twentieth century, and George Gershwin (1898-1937) was no exception. Gershwin was a brilliant talent who dropped out of school at the age of fifteen to begin a professional career playing piano in New York’s “Tin Pan Alley.” After several years of success as a performer and composer, he was asked by the famous band leader Paul Whiteman to compose a work that would help raise people’s perceptions of jazz as an art form. The resulting work, Rhapsody in Blue, combines the American blues style with the European symphonic tradition into a brilliant composition for piano and orchestra. Listen to how beautifully Gershwin combines these elements via the link below.
Gershwin, Rhapsody in Blue (1924)
www.youtube.com/watch?v=BxowOVIdnR0
In addition to Rhapsody in Blue , George Gershwin is also known for his opera, “Porgy and Bess.” Although not a true opera in the strict sense of the term (Gershwin dubbed it a “folk opera”), the piece is considered one of the great American operatic works of the century. The story is set in a tenement in Charleston, South Carolina. Based on DuBose Heyward’s novel Porgy , the opera incorporated classically trained black singers to depict the tragic love story between the two main title characters. Gershwin based the music for the opera on elements of folk music, referring to southern black musical style such as the blues and spirituals. Drawing on the nineteenth century opera tradition, Gershwin made use of leitmotifs to rep- resent people or places. Near the beginning of the opera, we hear the famous aria “Summertime,” which depicts the hot, hazy atmosphere in which the story is set.
George Gershwin – “summertime”
https://www.youtube.com/watch?v=O7-Qa92Rzbk
Like Gershwin, American born Aaron Copland (1900-1990) was instrumental in helping to define a distinct American sound by combining his European musical training with jazz and folk elements. As an early twentieth-century composer, Copland was active during the Great Depression, writing music for the new genre of radio, the phonograph, and motion pictures. El Salon Mexico (1935), Fanfare for the Common Man (1942) , and Appalachian Spring (1944) are three of Copland’s most famous works. He won a Pulitzer Prize for his music for the ballet Appalachian Spring and was also an Oscar-winning film composer. Appalachian Spring is a ballet depicting a pioneer wedding celebration in a newly-built farmhouse in Pennsylvania. It includes the now well-known Shaker song Simple Gifts .
Copland, Appalachian Spring (1944)
https://www.youtube.com/watch?v=ZNHWcHEMy-Q
Copland’s unique style evokes images of the landscape of the west, as we can hear in his score for the ballet Rodeo (1942) linked below.
Aaron Copland, Rodeo (1942)
https://www.youtube.com/watch?v=SXikDnYZYpM
One of the ways in which Copeland was able to capture the sense of vastness of the American landscape was through his use of certain harmonic intervals, that is, two notes played together, which sound “hollow” or “open.” These intervals, which are called “perfect 4ths” and “perfect 5ths,” have been used since medieval times, and were named so due to their simple harmonic ratios. The result is music that sounds vast and expansive. Perhaps the best example of this technique is found in Copland’s famous Fanfare for the Common Man .
While fanfares are typically associated with heralding the arrival of royalty, Copland wanted to create a fanfare that celebrated the lives of everyday people during a trying time in American history. The piece was premiered by the Cincinnati Symphony Orchestra on March 12, 1943 at the height of World War II. To this day, no other piece stirs up patriotic emotions like Fanfare for the Common Man . It has been used in countless movies, television shows, and even military recruitment ads. The piece came to define Copland’s uniquely American compositional style and remains one of the most popular patriotic pieces in the American repertoire.
|
LISTENING GUIDE For audio, go to: www.youtube.com/watch?v=FLMVB0B1_Ts |
|
Composer: Aaron Copland |
| Composition: Fanfare for the Common Man |
| Date: 1942 |
| Genre: Fanfare |
| Performing Forces: brass and percussion sections of symphony orchestra |
| Timing | Performing Forces, Melody, and Textures |
| 0:00 |
Opening crash heralds introduction by bass drum and timpani that slowly dies down.
|
| 0:31 |
Slow fanfare theme enters. The melody itself is comprised of many perfect 4ths and perfect 5th intervals which convey a sense of openness. Unison trumpets.
No harmonic accompaniment creates a sense of starkness. |
| 1:05 |
After brief notes from the percussion section, French horns enter, moving a perfect 5th below the trumpets.
|
| 1:46 |
Repeat of material from the introduction. Percussion. |
| 1:48 |
Clarinet states a contrasting melody. Melody over murmuring strings. |
| 1:59 |
Low brass enters with the main theme and is imitated by the horns and trumpets.
Full brass and percussion. |
| 3:01 |
Melody is restated at 1⁄2 speed (augmentation) and ends on climactic chord. Full brass and percussion. |


6.1: Ordinary Points
If \(x_0\) is an ordinary point of (6.1) , then it is possible to determine two power series (i.e., Taylor series) solutions for \(y = y(x)\) centered at \(x = x_0\). We illustrate the method of solution by solving two examples, with \(x_0 = 0\).
Find the general solution of \(y''+y=0\).
Solution
By now, you should know that the general solution is \(y(x) = a_0 \cos x + a_1 \sin x\), with \(a_0\) and \(a_1\) constants. To find a power series solution about the point \(x_0 = 0\), we write \[y(x)=\sum\limits_{n=0}^{\infty}a_nx^n;\nonumber\] and upon differentiating term-by-term \[y'(x)=\sum\limits_{n=1}^{\infty}na_nx^{n-1},\nonumber\] and \[y''(x)=\sum\limits_{n=2}^{\infty}n(n-1)a_nx^{n-2}.\nonumber\]
Substituting the power series for \(y\) and its derivatives into the differential equation to be solved, we obtain \[\label{eq:1}\sum\limits_{n=2}^{\infty}n(n-1)a_nx^{n-2}+\sum\limits_{n=0}^{\infty}a_nx^n=0.\]
The power-series solution method requires combining the two sums on the left-hand-side of \(\eqref{eq:1}\) into a single power series in \(x\). To shift the exponent of \(x^{n−2}\) in the first sum upward by two to obtain \(x^n\), we need to shift the summation index downward by two; that is, \[\sum\limits_{n=2}^{\infty}n(n-1)a_nx^{n-2}=\sum\limits_{n=0}^{\infty}(n+2)(n+1)a_{n+2}x^n.\nonumber\]
We can then combine the two sums in \(\eqref{eq:1}\) to obtain \[\label{eq:2}\sum\limits_{n=0}^{\infty}\left((n+2)(n+1)a_{n+2}+a_n\right)x^n=0.\]
For \(\eqref{eq:2}\) to be satisfied, the coefficient of each power of \(x\) must vanish separately. (This can be proved by setting \(x = 0\) after successive differentiation.) We therefore obtain the recurrence relation \[a_{n+2}=-\frac{a_n}{(n+2)(n+1)},\quad n=0,1,2,\ldots\nonumber\]
We observe that even and odd coefficients decouple. We thus obtain two independent sequences starting with first term \(a_0\) or \(a_1\). Developing these sequences, we have for the sequence beginning with \(a_0\):
\[\begin{aligned}a_0&, \\ a_2&=-\frac{1}{2}a_0, \\ a_4&=-\frac{1}{4\cdot 3}a_2=\frac{1}{4\cdot 3\cdot 2}a_0, \\ a_6&=-\frac{1}{6\cdot 5}a_4=-\frac{1}{6!}a_0;\end{aligned}\] and the general coefficient in this sequence for \(n = 0, 1, 2,\ldots\) is \[a_{2n}=\frac{(-1)^n}{(2n)!}a_0.\nonumber\]
Also, for the sequence beginning with \(a_1\):
\[\begin{aligned}a_1&, \\ a_3&=-\frac{1}{3\cdot 2}a_1, \\ a_5&=-\frac{1}{5\cdot 4}a_3=\frac{1}{5\cdot 4\cdot 3\cdot 2}a_1, \\ a_7&=-\frac{1}{7\cdot 6}a_5=-\frac{1}{7!}a_1;\end{aligned}\] and the general coefficient in this sequence for \(n = 0, 1, 2,\ldots\) is \[a_{2n+1}=\frac{(-1)^n}{(2n+1)!}a_1.\nonumber\]
Using the principle of superposition, the general solution is therefore \[\begin{aligned}y(x)&=a_0\sum\limits_{n=0}^\infty\frac{(-1)^n}{(2n)!}x^{2n}+a_1\sum\limits_{n=0}^\infty\frac{(-1)^n}{(2n+1)!}x^{2n+1} \\ &=a_0\left(1-\frac{x^2}{2!}+\frac{x^4}{4!}-\cdots\right)+a_1\left(x-\frac{x^3}{3!}+\frac{x^5}{5!}-\cdots\right) \\ &=a_0\cos x+a_1\sin x,\end{aligned}\] as expected.
In our next example, we will solve the Airy’s Equation . This differential equation arises in the study of optics, fluid mechanics, and quantum mechanics.
Find the general solution of \(y''-xy=0\).
Solution
With \[y(x)=\sum\limits_{n=0}^\infty a_nx^n,\nonumber\] the differential equation becomes \[\label{eq:3}\sum\limits_{n=2}^\infty n(n-1)a_nx^{n-2}-\sum\limits_{n=0}^\infty a_nx^{n+1}=0.\]
We shift the first sum to \(x^{n+1}\) by shifting the exponent up by three, i.e., \[\sum\limits_{n=2}^\infty n(n-1)a_nx^{n-2}=\sum\limits_{n=-1}^\infty (n+3)(n+2)a_{n+3}x^{n+1}.\nonumber\]
When combining the two sums in \(\eqref{eq:3}\), we separate out the extra \(n = −1\) term in the first sum given by \(2a_2\). Therefore, \(\eqref{eq:3}\) becomes \[\label{eq:4}2a_2+\sum\limits_{n=0}^\infty\left((n+3)(n+2)a_{n+3}-a_n\right)x^{n+1}=0.\]
Setting coefficients of powers of \(x\) to zero, we first find \(a_2 = 0\), and then obtain the recursion relation \[\label{eq:5}a_{n+3}=\frac{1}{(n+3)(n+2)}a_n.\]
Three sequences of coefficients—those starting with either \(a_0,\: a_1\) or \(a_2\)—decouple. In particular the three sequences are \[\begin{array}{l}a_0,a_3,a_6,a_9,\ldots ; \\ a_1,a_4,a_7,a_{10},\ldots ; \\ a_2,a_5,a_8,a_{11}\ldots\end{array}\nonumber\]
Since \(a_2 = 0\), we find immediately for the last sequence \[a_2=a_5=a_8=a_{11}=\cdots =0.\nonumber\]
We compute the first four nonzero terms in the power series with coefficients corresponding to the first two sequences. Starting with \(a_0\), we have \[\begin{aligned}a_0&, \\ a_3&=\frac{1}{3\cdot 2}a_0, \\ a_6&=\frac{1}{6\cdot 5\cdot 3\cdot 2}a_0, \\ a_9&=\frac{1}{9\cdot 8\cdot 6\cdot 5\cdot 3\cdot 2}a_0;\end{aligned}\] and starting with \(a_1\), \[\begin{aligned}a_1&, \\ a_4&=\frac{1}{4\cdot 3}a_1, \\ a_7&=\frac{1}{7\cdot 6\cdot 4\cdot 3}a_1, \\ a_{10}&=\frac{1}{10\cdot 9\cdot 7\cdot 6\cdot 4\cdot 3}a_1.\end{aligned}\]
The general solution for \(y = y(x)\), can therefore be written as \[\begin{aligned}y(x)&=a_0\left(1+\frac{x^3}{6}+\frac{x^6}{180}+\frac{x^9}{12960}+\cdot\right) +a_1\left(x+\frac{x^4}{12}+\frac{x^7}{504}+\frac{x^{10}}{45360}+\cdot\right) \\ &=a_0y_0(x)+a_1y_1(x).\end{aligned}\]
Suppose we would like to graph the solutions \(y = y_0(x)\) and \(y = y_1(x)\) versus \(x\) by solving the differential equation \(y'' − xy = 0\) numerically. What initial conditions should we use? Clearly, \(y = y_0(x)\) solves the ode with initial values \(y(0) = 1\) and \(y' (0) = 0\), while \(y = y_1(x)\) solves the ode with initial values \(y(0) = 0\) and \(y' (0) = 1\).
The numerical solutions, obtained using MATLAB, are shown in Fig. \(\PageIndex{1}\). Note that the solutions oscillate for negative \(x\) and grow exponentially for positive \(x\). This can be understood by recalling that \(y''+ y = 0\) has oscillatory sine and cosine solutions and \(y'' − y = 0\) has exponential hyperbolic sine and cosine solutions.


Skip to main content
-
ampule
-
a glass container that holds a single dose of liquid medication
-
aspiration
-
pulling back on the plunger to assess for blood return prior to injecting a medication
-
bevel
-
sharp, angled tip of the needle
-
buccal
-
placed between the gum and cheek
-
caplet
-
solid, compressed powder or granules
-
capsule
-
a powder or granules contained in a gelatin shell
-
compatibility
-
the capability of administering two or more medications together without the two interacting with one another
-
diluent
-
the liquid used to dilute a powdered medication into a liquid form
-
elixir
-
a liquid form of a medication that contains added sugars, flavors, and alcohol
-
emulsion
-
a solution that contains a medication, water, and oil mixture
-
enteric-coated tablet
-
a tablet covered in a substance that delays the medication from dissolving
-
gauge
-
the diameter of the hole in a needle
-
gelcap
-
a soft gelatin shell with a liquid medication inside
-
injector device
-
a spring-loaded syringe that delivers a single, preloaded dose of a medication when the device is firmly pressed against the body
-
intradermal (ID)
-
a medication route that injects the medication into the dermis
-
intramuscular (IM)
-
a medication route that injects the medication into a muscle
-
intrathecal
-
a medication route that injects the medication into the subarachnoid space around the spinal cord
-
intravenous (IV)
-
a medication route that injects the medication into a vein
-
lipoatrophy
-
the selective loss of subcutaneous fat without exudative reaction or fibrosis
-
lipohypertrophy
-
a lump of fatty tissue under the skin caused by repeated injections in the same place
-
lozenge
-
a medication with a candy base
-
Mantoux tuberculin skin test (TST)
-
a skin test used to diagnose latent TB
-
needle
-
a hollow metal tube with a sharp pointed tip, called a bevel, and a hub that attaches the needle to the syringe or a needleless system
-
needleless system
-
a system that allows parenteral medications to be administered without the use of a needle
-
parenteral
-
a medication route that bypasses the gastrointestinal (GI) system, eliminating the effects of first-pass metabolism when they are digested in the GI tract
-
powder
-
fine particles created from grinding a solid
-
prefilled syringe
-
a disposable syringe that is supplied with the necessary amount of the medication to be injected in a single dose
-
reconstitute
-
to add a liquid to a powdered medication to form a solution
-
subcutaneous (SQ)
-
a medication route that injects the medication into the adipose tissue
-
sublingual
-
placed under the tongue
-
suspension
-
undissolved medication particles that are mixed with a liquid agent
-
syringe
-
a sterile device that consists of a calibrated, cylinder-shaped barrel with a sliding plunger at one end and an open tip that attaches to a needle at the other end
-
syrup
-
a liquid form of a medication that contains added sugars and flavors
-
tablet
-
a compressed powder that has been formed into a solid, which disintegrates with body warmth and may be absorbed into the bloodstream
-
time-released
-
slow release of a medication for prolonged action
-
vial
-
a plastic or glass container with a rubber stopper on top that contains a liquid medication
-
wheal
-
a visible induration (elevation) of the skin
-
Z-track method
-
a method of IM medication administration that helps prevent the medication from leaking into the subcutaneous tissue


17.2: Creativity and Labor in Social Movement and Production Studies – A Snapshot
-
- Last updated
- Save as PDF
- Marwan M. Kraidy
- University of California Press
Social movement theorists have rarely discussed activism in terms of creativity or labor. Though creativity is sometimes mentioned in its prosaic meaning and the word occasionally appears in titles of books on social movements, rarely is it systematically theorized or critiqued as a conceptual category. 8 Jasper’s The Art of Moral Protest comes closest to a sustained conceptual treatment of creativity: the notion of artfulness is a cornerstone of the book’s “cultural” approach to protest, which intends “to increase [the focus on] explanatory factors . . . to concentrate on mechanisms, not grand theories . . . to give the voice back to the protestors we study.” 9 Jasper writes: “Protest movements work at the edge of a society’s understanding of itself and its surroundings. Like artists, they take inchoate intuitions and put flesh on them, formulating and elaborating them so that they can be debated. Without them, we would have only the inventions of corporations and state agencies, products and technologies created to enhance efficiency or profitability.” Jasper then concludes: “In order to understand these innovations, we need ‘moral innovators’ too: the artists, religious figures, and protestors who help us understand what we feel about new technologies.” 10 By comparing activists to artists, Jasper anchors artfulness in the socio-political realm of activism, valorizing innovation not in its potential for commodification but for its ability to generate political-rhetorical value. 11
For Jasper, artfulness refers to “experimental efforts to transmute existing traditions into new creations by problematizing elements that have been taken for granted.” 12 Artfulness articulates biography and culture: beginning as individual creativity, it becomes strategic once shaped by a group, and subsequently it is enacted in protest. Examples include deploying widely familiar and emotionally evocative symbols and grafting new meanings onto existing symbols. Language is a primary vehicle through which activists project, manipulate, and redefine symbols. Having elsewhere in the book compared activists to artists, Jasper writes that “at the most extreme, ideologists operate as poets; they define emerging structures of feeling with new terms and images.” 13 Invoking the “immense value we place on individual creativity,” 14 Jasper employs the notion of “tactical innovation,” a mainstay in the social movements literature, which emerges at “the interplay of protest groups and their opponents.” 15
Unlike studies of activism, research on cultural production does not focus on Political aspects of labor. 16 But the two are alike in rarely grappling directly with creativity as a central conceptual category. 17 One exception is Vicki Mayer’s study of workers in a television set factory in Manaus, Brazil, where the author endeavors to “deconstruct our received notions of creativity and to reconstruct a notion of creative action that is both social and individual in the practices of assembling.” 18 Following an argument made by Joas and others that social context is key to understanding creativity, Mayer develops notions of creativity that “conjoin the interiority of mental labor with the exteriority of a world that enables its articulation.” 19 In addition to emphasizing creativity’s social dimension, Mayer shows that as a discourse creativity is deployed with discrimination for purposes of social distinction and control. But it is Mayer’s discussion of creativity as a process of making do under structural constraints that is most relevant for my purposes, because it leads to two questions that are central to this chapter. What differences can we discern between deployments of “creativity” in media industries research and the trope of “creative resistance” used to describe some forms of dissent in the Arab uprisings? And how do these differences enable my elaboration of revolutionary creative labor?
“Creativity” is a strategic and discriminatory trope. It is strategic because its selective deployment reflects and perpetuates relations of politico-economic power. It is discriminatory because it is applied according to rules of exclusion and inclusion that serve criteria of social distinction. Considerations of power and distinction in creative labor differ between scholarship on media industries and research on Political forms of labor, such as activism and propaganda. In the television set factory Mayer studied, the discourse of creativity is reserved to operators in higher ranks of the industry, who exclude workers on the assembly line from creativity’s definitional scope. As Miller has shown, proponents of “creativity” have stretched the term to encompass most ways in which any activity that could remotely be described as cultural is monetized. 20 In contrast, the creative resistance trope operates primarily according to political and ideological imperatives. Creative resistance refers to propaganda by people we like—in this sense creative resistance is a more glamorous, bottom-up cousin of the great euphemism public diplomacy . During the war between Israel and Lebanon in 2006, Hezbollah launched a range of stylistically bold, visually compelling propaganda videos, some aimed at mobilizing supporters, others psyops clips, many in Hebrew, aimed at demoralizing Israeli soldiers. Though the notion of resistance is central to Hezbollah’s raison d’être, and though many of the videos were rhetorically sophisticated and aesthetically slick, to my knowledge no one called these “creative resistance.” Most mainstream media coverage in the West referred to them as “propaganda,” though in some aspects they resemble revolutionary videos of the Arab uprisings, and some of them even resemble U.S. Army recruitment commercials.


Section 1: Semantics for SL
This section provides a rigorous, formal characterization of truth in \(SL\) which builds on what we already know from doing truth tables. We were able to use truth tables to reliably test whether a sentence was a tautology in \(SL\), whether two sentences were equivalent, whether an argument was valid, and so on. For instance: \(\mathcal{A}\) is a tautology in \(SL\) if it is \(T\) on every line of a complete truth table.
This worked because each line of a truth table corresponds to a way the world might be. We considered all the possible combinations of 1 and 0 for the sentence letters that made a difference to the sentences we cared about. The truth table allowed us to determine what would happen given these different combinations.
Once we construct a truth table, the symbols ‘1’ and ‘0’ are divorced from their metalinguistic meaning of ‘true’ and ‘false’. We interpret ‘1’ as meaning ‘true’, but the formal properties of 1 are defined by the characteristic truth tables for the various connectives. The symbols in a truth table have a formal meaning that we can specific entirely in terms of how the connectives operate. For example, if \(A\) is value 1, then \(\neg A\) is value 0.
In short: Truth in \(SL\) just is the assignment of a 1 or a 0.
To formally define truth in \(SL\), then, we want a function that assigns a 1 or 0 to each of the \sentences of \(SL\). We can interpret this function as a definition of truth for \(SL\) if it assigns 1 to all of the true sentences of \(SL\) and 0 to all of the false sentences of \(SL\). Call this function ‘\(v\)’ (for ‘valuation’). We want \(v\) to be a function such that for any sentence \(\mathcal{A}\), \(v(\mathcal{A}) = 1\) if \(\mathcal{A}\) is true and \(v(\mathcal{A}) = 0\) if \(\mathcal{A}\) is false.
Recall that the recursive definition of a wff for \(SL\) had two stages: The first step said that atomic sentences (solitary sentence letters) are wffs. The second stage allowed for wffs to be constructed out of more basic wffs. There were clauses of the definition for all of the sentential connectives. For example, if \(\mathcal{A}\) is a wff, then \(\neg \mathcal{A}\) is a wff.
Our strategy for defining the truth function, \(v\), will also be in two steps. The first step will handle truth for atomic sentences; the second step will handle truth for compound sentences.
Truth in \(SL\)
How can we define truth for an atomic sentence of \(SL\)? Consider, for example, the sentence \(M\). Without an interpretation, we cannot say whether \(M\) is true or false. It might mean anything. If we use \(M\) to symbolize ‘The moon orbits the Earth’, then \(M\) is true. If we use \(M\) to symbolize ‘The moon is a giant turnip’, then \(M\) is false.
Moreover, the way you would discover whether or not \(M\) is true depends on what \(M\) means. If \(M\) means ‘It is Monday,’ then you would need to check a calendar. If \(M\) means ‘Jupiter’s moon Io has significant volcanic activity,’ then you would need to check an astronomy text— and astronomers know because they sent satellites to observe Io.
When we give a symbolization key for \(SL\), we provide an interpretation of the sentence letters that we use. The key gives an English language sentence for each sentence letter that we use. In this way, the interpretation specifies what each of the sentence letters means . However, this is not enough to determine whether or not that sentence is true. The sentences about the moon, for instance, require that you know some rudimentary astronomy.  She could understand what the sentence ‘The moon is a giant turnip’ means, but mistakenly think that it was true.
Consider another example: If \(M\) means ‘It is morning now’, then whether it is true or not depends on when you are reading this. I know what the sentence means, but— since I do not know when you will be reading this— I do not know whether it is true or false.
So an interpretation alone does not determine whether a sentence is true or false. Truth or falsity depends also on what the world is like. If \(M\) meant ‘The moon is a giant turnip’ and the real moon were a giant turnip, then \(M\) would be true. To put the point in a general way, truth or falsity is determined by an interpretation plus a way that the world is.
INTERPRETATION + STATE OF THE WORLD \(\implies\) TRUTH/FALSITY
In providing a logical definition of truth, we will not be able to give an account of how an atomic sentence is made true or false by the world. Instead, we will introduce a truth value assignment . Formally, this will be a function that tells us the truth value of all the atomic sentences. Call this function ‘\(a\)’ (for ‘assignment’). We define a for all sentence letters \(\mathcal{P}\), such that
\( a ( \mathcal { P } ) = \left\{ \begin{array} { l l } { 1 } & { \text { if } \mathcal { P } \text { is true } } \\ { 0 } & { \text { otherwise } } \end{array} \right. \)
This means that a takes any sentence of \(SL\) and assigns it either a one or a zero; one if the sentence is true, zero if the sentence is false. The details of the function \(a\) are determined by the meaning of the sentence letters together with the state of the world. If \(D\) means ‘It is dark outside’, then \(a\)(\(D\)) = 1 at night or during a heavy storm, while \(a\)(\(D\)) = 0 on a clear day.
You can think of \(a\) as being like a row of a truth table.  There are infinitely many sentence letters, and the truth value assignment gives a value to each of them. When constructing a truth table, we only care about sentence letters that affect the truth value of sentences that interest us. As such, we ignore the rest. Strictly speaking, every row of a truth table gives a partial truth value assignment.
It is important to note that the truth value assignment, \(a\), is not part of the language \(SL\). Rather, it is part of the mathematical machinery that we are using to describe \(SL\). It encodes which atomic sentences are true and which are false. We now define the truth function, v, using the same recursive structure that we used to define a wff of \(SL\).
1. If \(\mathcal{A}\) is a sentence letter, then \(v\)(\(\mathcal{A}\)) = \(a\)(\(\mathcal{A}\)).
2. If \(\mathcal{A}\) is ¬\(\mathcal{B}\) for some sentence \(\mathcal{A}\), then
\( v ( \mathcal { A } ) = \left\{ \begin{array} { l l } { 1 } & { \text { if } v ( \mathcal { B } ) = 0 } \\ { 0 } & { \text { otherwise } } \end{array} \right .\)
3. If \(\mathcal{A}\) is (\(\mathcal{B}\) & \(\mathcal{C}\) ) for some sentences \(\mathcal{B}\), \(\mathcal{C}\), then
\( v ( \mathcal { A } ) = \left\{ \begin{array} { l l } { 1 } & { \text { if } v ( \mathcal { B } ) = 1 \text { and } v ( C ) = 1 } \\ { 0 } & { \text { otherwise } } \end{array} \right .\)
It might seem as if this definition is circular, because it uses the word ‘and’ in trying to define ‘and.’ Notice, however, that this is not a definition of the English word ‘and’; it is a definition of truth for sentences of \(SL\) containing the logical symbol ‘&.’ We define truth for object language sentences containing the symbol ‘&’ using the metalanguage word ‘and.’ There is nothing circular about that.
4. If \(\mathcal{A}\) is (\(\mathcal{B}\)∨\(\mathcal{C}\)) for some sentences \(\mathcal{B}\),\(\mathcal{C}\), then
\( v ( \mathcal { A } ) = \left\{ \begin{array} { l l } { 0 } & { \text { if } v ( \mathcal { B } ) = 0 \text { and } v ( \mathcal { C } ) = 0 } \\ { 1 } & { \text { otherwise } } \end{array} \right .\)
5. If \(\mathcal{A}\) is (\(\mathcal{B}\) →\(\mathcal{C}\)) for some sentences \(\mathcal{B}\),\(\mathcal{C}\), then
\( v ( \mathcal { A } ) = \left\{ \begin{array} { l l } { 0 } & { \text { if } v ( \mathcal { B } ) = 1 \text { and } v ( C ) = 0 } \\ { 1 } & { \text { otherwise } } \end{array} \right .\)
6. If \(\mathcal{A}\) is (\(\mathcal{B}\)↔\(\mathcal{C}\)) for some sentences \(\mathcal{B}\),\(\mathcal{C}\), then
\( v ( \mathcal { A } ) = \left\{ \begin{array} { l l } { 1 } & { \text { if } v ( \mathcal { B } ) = v ( C ) } \\ { 0 } & { \text { otherwise } } \end{array} \right .\)
Since the definition of \(v\) has the same structure as the definition of a wff, we know that \(v\) assigns a value to every wff of \(SL\). Since the sentences of \(SL\) and the wffs of \(SL\) are the same, this means that \(v\) returns the truth value of every sentence of \(SL\).
Truth in \(SL\) is always truth relative to some truth value assignment, because the definition of truth for \(SL\) does not say whether a given sentence is true or false. Rather, it says how the truth of that sentence relates to a truth value assignment.
Other concepts in \(SL\)
Working with \(SL\) so far, we have done without a precise definition of ‘tautology’, ‘contradiction’, and so on. Truth tables provided a way to check i f a sentence was a tautology in \(SL\), but they did not define what it means to be a tautology in \(SL\). We will give definitions of these concepts for \(SL\) in terms of entailment.
The relation of semantic entailment, ‘\(\mathcal{A}\) entails \(\mathcal{B}\)’, means that there is no truth value assignment for which \(\mathcal{A}\) is true and \(\mathcal{B}\) is false. Put differently, it means that \(\mathcal{B}\) is true for any and all truth value assignments for which \(\mathcal{A}\) is true.
We abbreviate this with a symbol called the double turnstile : \(\mathcal{A}\) |= \(\mathcal{B}\) means ‘\(\mathcal{A}\) semantically entails \(\mathcal{B}\).’
We can talk about entailment between more than two sentences:
{\(\mathcal{A}\) 1 ,\(\mathcal{A}\) 2 ,\(\mathcal{A}\) 3 ,···}|= \(\mathcal{B}\)
means that there is no truth value assignment for which all of the sentences in the set {\(\mathcal{A}\) 1 ,\(\mathcal{A}\) 2 ,\(\mathcal{A}\) 3 ,···} are true and \(\mathcal{B}\) is false.
We can also use the symbol with just one sentence: |= \(\mathcal{C}\) means that \(\mathcal{C}\) is true for all truth value assignments. This is equivalent to saying that the sentence is entailed by anything.
The double turnstile symbol allows us to give concise definitions for various concepts of \(SL\):
A tautology in \(SL\) is a sentence \(\mathcal{A}\) such that |= \(\mathcal{A}\).
A contradiction in \(SL\) is a sentence \(\mathcal{A}\) such that |= ¬\(\mathcal{A}\).
A sentence is contingent in \(SL\) if and only if it is neither a tautology nor a contradiction.
An argument “ \(\mathcal{P}\) 1 ,\(\mathcal{P}\) 2 ,···, .˙. \(\mathcal{C}\) ” is valid in \(SL\) if and only if {\(\mathcal{P}\) 1 ,\(\mathcal{P}\) 2 ,···}|= \(\mathcal{C}\). Two sentences \(\mathcal{A}\) and \(\mathcal{B}\) are logically equivalent in \(SL\) if and only if both \(\mathcal{A}\) |= \(\mathcal{B}\) and \(\mathcal{B}\) |= \(\mathcal{A}\).
Logical consistency is somewhat harder to define in terms of semantic entailment. Instead, we will define it in this way:
The set {\(\mathcal{A}\) 1 ,\(\mathcal{A}\) 2 ,\(\mathcal{A}\) 3 ,···} is consistent in \(SL\) if and only if there is at least one truth value assignment for which all of the sentences are true. The set is inconsistent in \(SL\) if and if only there is no such assignment.


16.5: Substance Related and Addictive Disorders - A Special Case
Learning Objectives
- Recognize the goal of substance-related and addictive disorders treatment
- Discuss what makes for effective treatment
- Describe how comorbid disorders are treated
Addiction is often viewed as a chronic disease (Figure \(\PageIndex{1}\)). The choice to use a substance is initially voluntary; however, because chronic substance use can permanently alter the neural structure in the prefrontal cortex, an area of the brain associated with decision-making and judgment, a person becomes driven to use drugs and/or alcohol (Muñoz-Cuevas, Athilingam, Piscopo, & Wilbrecht, 2013). This helps explain why relapse rates tend to be high. About \(40\%-60\%\) of individuals relapse , which means they return to abusing drugs and/or alcohol after a period of improvement (National Institute on Drug Abuse [NIDA], 2008).
The goal of substance-related treatment is to help an addicted person stop compulsive drug-seeking behaviors (NIDA, 2012). This means an addicted person will need long-term treatment, similar to a person battling a chronic physical disease such as hypertension or diabetes. Treatment usually includes behavioral therapy and/or medication, depending on the individual (NIDA, 2012). Specialized therapies have also been developed for specific types of substance-related disorders, including alcohol, cocaine, and opioids (McGovern & Carroll, 2003). Substance-related treatment is considered much more cost-effective than incarceration or not treating those with addictions (NIDA, 2012). See figure below.
What makes Treatment Effective?
Specific factors make substance-related treatment much more effective. One factor is duration of treatment. Generally, the addict needs to be in treatment for at least three months to achieve a positive outcome (Simpson, 1981; Simpson, Joe, & Bracy, 1982; NIDA, 2012). This is due to the psychological, physiological, behavioral, and social aspects of abuse (Simpson, 1981; Simpson et al., 1982; NIDA, 2012). While in treatment, an addict might receive behavior therapy, which can help motivate the addict to participate in the treatment program and teach strategies for dealing with cravings and how to prevent relapse. Also, treatment needs to be holistic and address multiple needs, not just the drug addiction. This means that treatment will address factors such as communication, stress management, relationship issues, parenting, vocational concerns, and legal concerns (McGovern & Carroll, 2003; NIDA, 2012).
While individual therapy is used in the treatment of substance-related disorders, group therapy is the most widespread treatment modality (Weiss, Jaffee, de Menil, & Cogley, 2004). The rationale behind using group therapy for addiction treatment is that addicts are much more likely to maintain sobriety in a group format. It has been suggested that this is due to the rewarding and therapeutic benefits of the group, such as support, affiliation, identification, and even confrontation (Center for Substance Abuse Treatment, 2005). For teenagers, the whole family often needs to participate in treatment to address issues such as family dynamics, communication, and relapse prevention. Family involvement in teen drug addiction is vital. Research suggests that greater parental involvement is correlated with a greater reduction in use by teen substance abusers. Also, mothers who participated in treatment displayed better mental health and greater warmth toward their children (Bertrand et al., 2013). However, neither individual nor group therapy has been found to be more effective (Weiss et al., 2004). Regardless of the type of treatment service, the primary focus is on abstinence or at the very least a significant reduction in use (McGovern & Carroll, 2003).
Treatment also usually involves medications to detox the addict safely after an overdose, to prevent seizures and agitation that often occur in detox, to prevent reuse of the drug, and to manage withdrawal symptoms. Getting off drugs often involves the use of drugs—some of which can be just as addictive. Detox can be difficult and dangerous.
Comorbid Disorders
Frequently, a person who is addicted to drugs and/or alcohol has an additional psychological disorder. Saying a person has comorbid disorders means the individual has two or more diagnoses. This can often be a substance-related diagnosis and another psychiatric diagnosis, such as depression, bipolar disorder, or schizophrenia. These individuals fall into the category of mentally ill and chemically addicted (MICA)—their problems are often chronic and expensive to treat, with limited success. Compared with the overall population, substance abusers are twice as likely to have a mood or anxiety disorder. Drug abuse can cause symptoms of mood and anxiety disorders and the reverse is also true—people with debilitating symptoms of a psychiatric disorder may self-medicate and abuse substances.
In cases of comorbidity , the best treatment is thought to address both (or multiple) disorders simultaneously (NIDA, 2012). Behavior therapies are used to treat comorbid conditions, and in many cases, psychotropic medications are used along with psychotherapy. For example, evidence suggests that bupropion (trade names: Wellbutrin and Zyban), approved for treating depression and nicotine dependence, might also help reduce craving and use of the drug methamphetamine (NIDA, 2011). However, more research is needed to better understand how these medications work—particularly when combined in patients with comorbidities.
Summary
Addiction is often viewed as a chronic disease that rewires the brain. This helps explain why relapse rates tend to be high, around \(40\%-60\%\) (McLellan, Lewis, & O’Brien, & Kleber, 2000). The goal of treatment is to help an addict stop compulsive drug-seeking behaviors. Treatment usually includes behavioral therapy, which can take place individually or in a group setting. Treatment may also include medication. Sometimes a person has comorbid disorders, which usually means that they have a substance-related disorder diagnosis and another psychiatric diagnosis, such as depression, bipolar disorder, or schizophrenia. The best treatment would address both problems simultaneously.
Glossary
- comorbid disorder
- individual who has two or more diagnoses, which often includes a substance abuse diagnosis and another psychiatric diagnosis, such as depression, bipolar disorder, or schizophrenia
- relapse
- repeated drug use and/or alcohol use after a period of improvement from substance abuse
Contributors and Attributions
-
Rose M. Spielman with many significant contributors. The OpenStax name, OpenStax logo, OpenStax book covers, OpenStax CNX name, and OpenStax CNX logo are not subject to the creative commons license and may not be reproduced without the prior and express written consent of Rice University. For questions regarding this license, please contact<EMAIL_ADDRESS>.Textbook content produced by OpenStax College is licensed under a Creative Commons Attribution License 4.0 license. Download for free at<EMAIL_ADDRESS>.


12.3: Common Sense (Excerpt)
THOUGHTS ON THE PRESENT STATE OF AMERICAN AFFAIRS
In the following pages I offer nothing more than simple facts, plain arguments, and common sense; and have no other Preliminaries to settle with the reader, than that he will divest himself of prejudice and prepossession, and suffer his reason and his feelings to determine for themselves; that he will put ON, or rather that he will not put OFF the true character of a man, and generously enlarge his views beyond the present day.
Volumes have been written on the subject of the struggle between England and America. Men of all ranks have embarked in the controversy, from different motives, and with various designs; but all have been ineffectual, and the period of debate is closed. Arms, as the last resource, decide this contest; the appeal was the choice of the king, and the continent hath accepted the challenge.
It hath been reported of the late Mr. Pelham (who tho’ an able minister was not without his faults) that on his being attacked in the house of commons, on the score, that his measures were only of a temporary kind, replied “THEY WILL LAST MY TIME.” Should a thought so fatal and unmanly possess the colonies in the present contest, the name of ancestors will be remembered by future generations with detestation.
The sun never shined on a cause of greater worth. ‘Tis not the affair of a city, a county, a province, or a kingdom, but of a continent—of at least one eighth part of the habitable globe. ‘Tis not the concern of a day, a year, or an age; posterity are virtually involved in the contest, and will be more or less affected, even to the end of time, by the proceedings now. Now is the seed-time of continental union, faith and honour. The least fracture now will be like a name engraved with the point of a pin on the tender rind of a young oak; the wound will enlarge with the tree, and posterity read it in full grown characters.
By referring the matter from argument to arms, a new aera for politics is struck; a new method of thinking hath arisen. All plans, proposals, &c. prior to the nineteenth of April, i. e. to the commencement of hostilities, are like the almanacs of the last year; which, though proper then are superseded and useless now. Whatever was advanced by the advocates on either side of the question then, terminated in one and the same point. viz. a union with Great-Britain: the only difference between the parties was the method of effecting it; the one proposing force, the other friendship; but it hath so far happened that the first hath failed, and the second hath withdrawn her influence.
As much hath been said of the advantages of reconciliation which, like an agreeable dream, hath passed away and left us as we were, it is but right, that we should examine the contrary side of the argument, and inquire into some of the many material injuries which these colonies sustain, and always will sustain, by being connected with, and dependent on Great Britain: To examine that connection and dependence, on the principles of nature and common sense, to see what we have to trust to, if separated, and what we are to expect, if dependant.
I have heard it asserted by some, that as America hath flourished under her former connection with Great Britain that the same connection is necessary towards her future happiness, and will always have the same effect. Nothing can be more fallacious than this kind of argument. We may as well assert that because a child has thrived upon milk that it is never to have meat, or that the first twenty years of our lives is to become a precedent for the next twenty. But even this is admitting more than is true, for I answer roundly, that America would have flourished as much, and probably much more, had no European power had any thing to do with her. The commerce, by which she hath enriched herself, are the necessaries of life, and will always have a market while eating is the custom of Europe.
But she has protected us, say some. That she has engrossed us is true, and defended the continent at our expense as well as her own is admitted, and she would have defended Turkey from the same motive, viz. the sake of trade and dominion.
Alas, we have been long led away by ancient prejudices, and made large sacrifices to superstition. We have boasted the protection of Great Britain, without considering, that her motive was INTEREST not ATTACHMENT; that she did not protect us from OUR ENEMIES on OUR ACCOUNT, but from HER ENEMIES on HER OWN ACCOUNT, from those who had no quarrel with us on any OTHER ACCOUNT, and who will always be our enemies on the SAME ACCOUNT. Let Britain wave her pretensions to the continent, or the continent throw off the dependence, and we should be at peace with France and Spain were they at war with Britain. The miseries of Hanover last war ought to warn us against connections.
It has lately been asserted in parliament, that the colonies have no relation to each other but through the parent country, i. e. that Pennsylvania and the Jerseys, and so on for the rest, are sister colonies by the way of England; this is certainly a very round-about way of proving relationship, but it is the nearest and only true way of proving enemyship, if I may so call it. France and Spain never were, nor perhaps ever will be our enemies as AMERICANS, but as our being the subjects of GREAT BRITAIN.
But Britain is the parent country, say some. Then the more shame upon her conduct. Even brutes do not devour their young, nor savages make war upon their families; wherefore the assertion, if true, turns to her reproach; but it happens not to be true, or only partly so and the phrase PARENT or MOTHER COUNTRY hath been jesuitically adopted by the king and his parasites, with a low papistical design of gaining an unfair bias on the credulous weakness of our minds. Europe, and not England, is the parent country of America. This new world hath been the asylum for the persecuted lovers of civil and religious liberty from EVERY PART of Europe. Hither have they fled, not from the tender embraces of the mother, but from the cruelty of the monster; and it is so far true of England, that the same tyranny which drove the first emigrants from home, pursues their descendants still.
In this extensive quarter of the globe, we forget the narrow limits of three hundred and sixty miles (the extent of England) and carry our friendship on a larger scale; we claim brotherhood with every European Christian, and triumph in the generosity of the sentiment.
It is pleasant to observe by what regular gradations we surmount the force of local prejudice, as we enlarge our acquaintance with the world. A man born in any town in England divided into parishes, will naturally associate most with his fellow-parishioners (because their interests in many cases will be common) and distinguish him by the name of NEIGHBOUR; if he meet him but a few miles from home, he drops the narrow idea of a street, and salutes him by the name of TOWNSMAN; if he travel out of the county, and meet him in any other, he forgets the minor divisions of street and town, and calls him COUNTRYMAN, i. e. COUNTRYMAN; but if in their foreign excursions they should associate in France or any other part of EUROPE, their local remembrance would be enlarged into that of ENGLISHMEN. And by a just parity of reasoning, all Europeans meeting in America, or any other quarter of the globe, are COUNTRYMEN; for England, Holland, Germany, or Sweden, when compared with the whole, stand in the same places on the larger scale, which the divisions of street, town, and county do on the smaller ones; distinctions too limited for continental minds. Not one third of the inhabitants, even of this province, are of English descent. Wherefore I reprobate the phrase of parent or mother country applied to England only, as being false, selfish, narrow and ungenerous.
But admitting, that we were all of English descent, what does it amount to? Nothing. Britain, being now an open enemy, extinguishes every other name and title: And to say that reconciliation is our duty, is truly farcical. The first king of England, of the present line (William the Conqueror) was a Frenchman, and half the Peers of England are descendants from the same country; therefore, by the same method of reasoning, England ought to be governed by France.
Much hath been said of the united strength of Britain and the colonies, that in conjunction they might bid defiance to the world. But this is mere presumption; the fate of war is uncertain, neither do the expressions mean any thing; for this continent would never suffer itself to be drained of inhabitants, to support the British arms in either Asia, Africa, or Europe.
Besides what have we to do with setting the world at defiance? Our plan is commerce, and that, well attended to, will secure us the peace and friendship of all Europe; because, it is the interest of all Europe to have America a FREE PORT. Her trade will always be a protection, and her barrenness of gold and silver secure her from invaders.
I challenge the warmest advocate for reconciliation, to shew, a single advantage that this continent can reap, by being connected with Great Britain. I repeat the challenge, not a single advantage is derived. Our corn will fetch its price in any market in Europe, and our imported goods must be paid for, buy them where we will.
But the injuries and disadvantages we sustain by that connection, are without number; and our duty to mankind at large, as well as to ourselves, instruct us to renounce the alliance: Because, any submission to, or dependence on Great Britain, tends directly to involve this continent in European wars and quarrels; and sets us at variance with nations, who would otherwise seek our friendship, and against whom, we have neither anger nor complaint. As Europe is our market for trade, we ought to form no partial connection with any part of it. It is the true interest of America to steer clear of European contentions, which she never can do, while by her dependence on Britain, she is made the make-weight in the scale of British politics.
Europe is too thickly planted with kingdoms to be long at peace, and whenever a war breaks out between England and any foreign power, the trade of America goes to ruin, BECAUSE OF HER CONNECTION WITH ENGLAND. The next war may not turn out like the last, and should it not, the advocates for reconciliation now, will be wishing for separation then, because, neutrality in that case, would be a safer convoy than a man of war. Every thing that is right or natural pleads for separation. The blood of the slain, the weeping voice of nature cries, ‘TIS TIME TO PART. Even the distance at which the Almighty hath placed England and America, is a strong and natural proof, that the authority of the one, over the other, was never the design of Heaven. The time likewise at which the continent was discovered, adds weight to the argument, and the manner in which it was peopled increases the force of it. The reformation was preceded by the discovery of America, as if the Almighty graciously meant to open a sanctuary to the Persecuted in future years, when home should afford neither friendship nor safety.
The authority of Great Britain over this continent, is a form of government, which sooner or later must have an end: And a serious mind can draw no true pleasure by looking forward under the painful and positive conviction, that what he calls “the present constitution” is merely temporary. As parents, we can have no joy, knowing that THIS GOVERNMENT is not sufficiently lasting to ensure any thing which we may bequeath to posterity: And by a plain method of argument, as we are running the next generation into debt, we ought to do the work of it, otherwise we use them meanly and pitifully. In order to discover the line of our duty rightly, we should take our children in our hand, and fix our station a few years farther into life; that eminence will present a prospect, which a few present fears and prejudices conceal from our sight.
Though I would carefully avoid giving unnecessary offense, yet I am inclined to believe, that all those who espouse the doctrine of reconciliation, may be included within the following descriptions. Interested men, who are not to be trusted; weak men, who CANNOT see; prejudiced men, who WILL NOT see; and a certain set of moderate men, who think better of the European world than it deserves; and this last class, by an ill-judged deliberation, will be the cause of more calamities to this continent, than all the other three.
It is the good fortune of many to live distant from the scene of sorrow; the evil is not sufficient brought to their doors to make THEM feel the precariousness with which all American property is possessed. But let our imaginations transport us for a few moments to Boston, that seat of wretchedness will teach us wisdom, and instruct us for ever to renounce a power in whom we can have no trust. The inhabitants of that unfortunate city, who but a few months ago were in ease and affluence, have now, no other alternative than to stay and starve, or turn and beg. Endangered by the fire of their friends if they continue within the city, and plundered by the soldiery if they leave it. In their present condition they are prisoners without the hope of redemption, and in a general attack for their relief, they would be exposed to the fury of both armies.
Men of passive tempers look somewhat lightly over the offenses of Britain, and, still hoping for the best, are apt to call out, “COME, COME, WE SHALL BE FRIENDS AGAIN, FOR ALL THIS.” But examine the passions and feelings of mankind, Bring the doctrine of reconciliation to the touchstone of nature, and then tell me, whether you can hereafter love, honor, and faithfully serve the power that hath carried fire and sword into your land? If you cannot do all these, then are you only deceiving yourselves, and by your delay bringing ruin upon posterity. Your future connection with Britain, whom you can neither love nor honor will be forced and unnatural, and being formed only on the plan of present convenience, will in a little time fall into a relapse more wretched than the first. But if you say, you can still pass the violations over, then I ask, Hath your house been burnt? Hath your property been destroyed before your face! Are your wife and children destitute of a bed to lie on, or bread to live on?  If you have not, then are you not a judge of those who have. But if you have, and still can shake hands with the murderers, then are you unworthy of the name of husband, father, friend, or lover, and whatever may be your rank or title in life, you have the heart of a coward, and the spirit of a sycophant.
This is not inflaming or exaggerating matters, but trying them by those feelings and affections which nature justifies, and without which, we should be incapable of discharging the social duties of life, or enjoying the felicities of it. I mean not to exhibit horror for the purpose of provoking revenge, but to awaken us from fatal and unmanly slumbers, that we may pursue determinately some fixed object. It is not in the power of Britain or of Europe to conquer America, if she do not conquer herself by DELAY and TIMIDITY. The present winter is worth an age if rightly employed, but if lost or neglected, the whole continent will partake of the misfortune; and there is no punishment which that man will not deserve, be he who, or what, or where he will, that may be the means of sacrificing a season so precious and useful.
It is repugnant to reason, to the universal order of things, to all examples from former ages, to suppose, that this continent can longer remain subject to any external power. The most sanguine in Britain does not think so. The utmost stretch of human wisdom cannot, at this time, compass a plan short of separation, which can promise the continent even a year’s security. Reconciliation is NOW a fallacious dream. Nature hath deserted the connection, and Art cannot supply her place. For, as Milton wisely expresses, “never can true reconcilement grow, where wounds of deadly hate have pierced so deep.”
Every quiet method for peace hath been ineffectual. Our prayers have been rejected with disdain; and only tended to convince us, that nothing flatters vanity, or confirms obstinacy in Kings more than repeated petitioning—and nothing hath contributed more than that very measure to make the Kings of Europe absolute: Witness Denmark and Sweden. Wherefore, since nothing but blows will do, for God’s sake, let us come to a final separation, and not leave the next generation to be cutting throats, under the violated unmeaning names of parent and child.
To say, they will never attempt it again is idle and visionary, we thought so at the repeal of the stamp-act, yet a year or two undeceived us; as well may we suppose that nations, which have been once defeated, will never renew the quarrel.
As to government matters, it is not in the power of Britain to do this continent justice: The business of it will soon be too weighty, and intricate, to be managed with any tolerable degree of convenience, by a power so distant from us, and so very ignorant of us; for if they cannot conquer us, they cannot govern us. To be always running three or four thousand miles with a tale or a petition, waiting four or five months for an answer, which when obtained requires five or six more to explain it in, will in a few years be looked upon as folly and childishness—There was a time when it was proper, and there is a proper time for it to cease.
Small islands not capable of protecting themselves, are the proper objects for kingdoms to take under their care; but there is something very absurd, in supposing a continent to be perpetually governed by an island. In no instance hath nature made the satellite larger than its primary planet, and as England and America, with respect to each other, reverses the common order of nature, it is evident they belong to different systems; England to Europe, America to itself.
I am not induced by motives of pride, party, or resentment to espouse the doctrine of separation and independance; I am clearly, positively, and conscientiously persuaded that it is the true interest of this continent to be so; that every thing short of THAT is mere patchwork, that it can afford no lasting felicity, —that it is leaving the sword to our children, and shrinking back at a time, when, a little more, a little farther, would have rendered this continent the glory of the earth.
As Britain hath not manifested the least inclination towards a compromise, we may be assured that no terms can be obtained worthy the acceptance of the continent, or any ways equal to the expense of blood and treasure we have been already put to.
The object, contended for, ought always to bear some just proportion to the expense. The removal of North, or the whole detestable junto, is a matter unworthy the millions we have expended. A temporary stoppage of trade, was an inconvenience, which would have sufficiently balanced the repeal of all the acts complained of, had such repeals been obtained; but if the whole continent must take up arms, if every man must be a soldier, it is scarcely worth our while to fight against a contemptible ministry only. Dearly, dearly, do we pay for the repeal of the acts, if that is all we fight for; for in a just estimation, it is as great a folly to pay a Bunker-hill price for law, as for land. As I have always considered the independancy of this continent, as an event, which sooner or later must arrive, so from the late rapid progress of the continent to maturity, the event could not be far off. Wherefore, on the breaking out of hostilities, it was not worth while to have disputed a matter, which time would have finally redressed, unless we meant to be in earnest; otherwise, it is like wasting an estate on a suit at law, to regulate the trespasses of a tenant, whose lease is just expiring. No man was a warmer wisher for reconciliation than myself, before the fatal nineteenth of April 1775, but the moment the event of that day was made known, I rejected the hardened, sullen tempered Pharaoh of England for ever; and disdain the wretch, that with the pretended title of FATHER OF HIS PEOPLE can unfeelingly hear of their slaughter, and composedly sleep with their blood upon his soul.
But admitting that matters were now made up, what would be the event? I answer, the ruin of the continent. And that for several reasons.
FIRST. The powers of governing still remaining in the hands of the king, he will have a negative over the whole legislation of this continent.  And is there any inhabitant in America so ignorant as not to know, that according to what is called the PRESENT CONSTITUTION, that this continent can make no laws but what the king gives leave to; and is there any man so unwise, as not to see, that (considering what has happened) he will suffer no law to be made here, but such as suit HIS purpose. We may be as effectually enslaved by the want of laws in America, as by submitting to laws made for us in England. After matters are made up (as it is called) can there be any doubt, but the whole power of the crown will be exerted, to keep this continent as low and humble as possible? Instead of going forward we shall go backward, or be perpetually quarrelling or ridiculously petitioning. —WE are already greater than the king wishes us to be, and will he not hereafter endeavour to make us less? To bring the matter to one point. Is the power who is jealous of our prosperity, a proper power to govern us? Whoever says No to this question, is an INDEPENDANT, for independancy means no more, than, whether we shall make our own laws, or whether the king, the greatest enemy this continent hath, or can have, shall tell us “THERE SHALL BE NO LAWS BUT SUCH AS I LIKE.”
But the king you will say has a negative in England; the people there can make no laws without his consent. In point of right and good order, there is something very ridiculous, that a youth of twenty-one (which hath often happened) shall say to several millions of people, older and wiser than himself, I forbid this or that act of yours to be law. But in this place I decline this sort of reply, though I will never cease to expose the absurdity of it, and only answer, that England being the King’s residence, and America not so, makes quite another case. The king’s negative HERE is ten times more dangerous and fatal than it can be in England, for THERE he will scarcely refuse his consent to a bill for putting England into as strong a state of defense as possible, and in America he would never suffer such a bill to be passed.
America is only a secondary object in the system of British politics, England consults the good of THIS country, no farther than it answers her OWN purpose. Wherefore, her own interest leads her to suppress the growth of OURS in every case which doth not promote her advantage, or in the least interferes with it. A pretty state we should soon be in under such a secondhand government, considering what has happened! Men do not change from enemies to friends by the alteration of a name: And in order to shew that reconciliation now is a dangerous doctrine, I affirm, THAT IT WOULD BE POLICY IN THE KING AT THIS TIME, TO REPEAL THE ACTS FOR THE SAKE OF REINSTATING HIMSELF IN THE GOVERNMENT OF THE PROVINCES; in order, that HE MAY ACCOMPLISH BY CRAFT AND SUBTLETY, IN THE LONG RUN, WHAT HE CANNOT DO BY FORCE AND VIOLENCE IN THE SHORT ONE. Reconciliation and ruin are nearly related.
SECONDLY. That as even the best terms, which we can expect to obtain, can amount to no more than a temporary expedient, or a kind of government by guardianship, which can last no longer than till the colonies come of age, so the general face and state of things, in the interim, will be unsettled and unpromising. Emigrants of property will not choose to come to a country whose form of government hangs but by a thread, and who is every day tottering on the brink of commotion and disturbance; and numbers of the present inhabitants would lay hold of the interval, to dispense of their effects, and quit the continent.
But the most powerful of all arguments, is, that nothing but independence, i.e. a continental form of government, can keep the peace of the continent and preserve it inviolate from civil wars. I dread the event of a reconciliation with Britain now, as it is more than probable, that it will be followed by a revolt somewhere or other, the consequences of which may be far more fatal than all the malice of Britain.
Thousands are already ruined by British barbarity; (thousands more will probably suffer the same fate.) Those men have other feelings than us who have nothing suffered. All they NOW possess is liberty, what they before enjoyed is sacrificed to its service, and having nothing more to lose, they disdain submission. Besides, the general temper of the colonies, towards a British government, will be like that of a youth, who is nearly out of his time; they will care very little about her. And a government which cannot preserve the peace, is no government at all, and in that case we pay our money for nothing; and pray what is it that Britain can do, whose power will be wholly on paper, should a civil tumult break out the very day after reconciliation! I have heard some men say, many of whom I believe spoke without thinking, that they dreaded an independence, fearing that it would produce civil wars. It is but seldom that our first thoughts are truly correct, and that is the case here; for there are ten times more to dread from a patched up connection than from independence. I make the sufferers case my own, and I protest, that were I driven from house and home, my property destroyed, and my circumstances ruined, that as man, sensible of injuries, I could never relish the doctrine of reconciliation, or consider myself bound thereby.
The colonies have manifested such a spirit of good order and obedience to continental government, as is sufficient to make every reasonable person easy and happy on that head. No man can assign the least pretence for his fears, on any other grounds, than such as are truly childish and ridiculous, viz. that one colony will be striving for superiority over another.
Where there are no distinctions there can be no superiority, perfect equality affords no temptation. The republics of Europe are all (and we may say always) in peace. Holland and Switzerland are without wars, foreign or domestic: Monarchical governments, it is true, are never long at rest; the crown itself is a temptation to enterprising ruffians at HOME; and that degree of pride and insolence ever attendant on regal authority, swells into a rupture with foreign powers, in instances, where a republican government, by being formed on more natural principles, would negotiate the mistake.
If there is any true cause of fear respecting independence, it is because no plan is yet laid down. Men do not see their way out— Wherefore, as an opening into that business, I offer the following hints; at the same time modestly affirming, that I have no other opinion of them myself, than that they may be the means of giving rise to something better. Could the straggling thoughts of individuals be collected, they would frequently form materials for wise and able men to improve into useful matter.
LET the assemblies be annual, with a President only. The representation more equal. Their business wholly domestic, and subject to the authority of a Continental Congress.
Let each colony be divided into six, eight, or ten, convenient districts, each district to send a proper number of delegates to Congress, so that each colony send at least thirty. The whole number in Congress will be at least 390. Each Congress to sit and to choose a president by the following method. When the delegates are met, let a colony be taken from the whole thirteen colonies by lot, after which, let the whole Congress choose (by ballot) a president from out of the delegates of that province. In the next Congress, let a colony be taken by lot from twelve only, omitting that colony from which the president was taken in the former Congress, and so proceeding on till the whole thirteen shall have had their proper rotation. And in order that nothing may pass into a law but what is satisfactorily just not less than three fifths of the Congress to be called a majority— He that will promote discord, under a government so equally formed as this, would have joined Lucifer in his revolt.
But as there is a peculiar delicacy, from whom, or in what manner, this business must first arise, and as it seems most agreeable and consistent, that it should come from some intermediate body between the governed and the governors, that is, between the Congress and the people. Let a CONTINENTAL CONFERENCE be held, in the following manner, and for the following purpose.
A committee of twenty-six members of Congress, viz. two for each colony. Two Members from each House of Assembly, or Provincial Convention; and five representatives of the people at large, to be chosen in the capital city or town of each province, for and in behalf of the whole province, by as many qualified voters as shall think proper to attend from all parts of the province for that purpose; or, if more convenient, the representatives may be chosen in two or three of the most populous parts thereof. In this conference, thus assembled, will be united, the two grand principles of business KNOWLEDGE and POWER. The members of Congress, Assemblies, or Conventions, by having had experience in national concerns, will be able and useful counsellors, and the whole, being empowered by the people, will have a truly legal authority.
The conferring members being met, let their business be to frame a CONTINENTAL CHARTER, or Charter of the United Colonies; (answering to what is called the Magna Carta of England) fixing the number and manner of choosing members of Congress, members of Assembly, with their date of sitting, and drawing the line of business and jurisdiction between them: (Always remembering, that our strength is continental, not provincial:) Securing freedom and property to all men, and above all things, the free exercise of religion, according to the dictates of conscience; with such other matter as is necessary for a charter to contain. Immediately after which, the said Conference to dissolve, and the bodies which shall be chosen comformable to the said charter, to be the legislators and governors of this continent for the time being: Whose peace and happiness may God preserve, Amen.
Should any body of men be hereafter delegated for this or some similar purpose, I offer them the following extracts from that wise observer on governments DRAGONETTI. “The science” says he “of the politician consists in fixing the true point of happiness and freedom. Those men would deserve the gratitude of ages, who should discover a mode of government that contained the greatest sum of individual happiness, with the least national expense.”[1]
But where, says some, is the King of America? I’ll tell you. Friend, he reigns above, and doth not make havoc of mankind like the Royal Brute of Britain. Yet that we may not appear to be defective even in earthly honors, let a day be solemnly set apart for proclaiming the charter; let it be brought forth placed on the divine law, the word of God; let a crown be placed thereon, by which the world may know, that so far as we approve of monarchy, that in America THE LAW IS KING. For as in absolute governments the King is law, so in free countries the law OUGHT to be King; and there ought to be no other. But lest any ill use should afterwards arise, let the crown at the conclusion of the ceremony, be demolished, and scattered among the people whose right it is.
A government of our own is our natural right: And when a man seriously reflects on the precariousness of human affairs, he will become convinced, that it is infinitely wiser and safer, to form a constitution of our own in a cool deliberate manner, while we have it in our power, than to trust such an interesting event to time and chance. If we omit it now, some[2] Massanello may hereafter arise, who laying hold of popular disquietudes, may collect together the desperate and the discontented, and by assuming to themselves the powers of government, may sweep away the liberties of the continent like a deluge. Should the government of America return again into the hands of Britain, the tottering situation of things will be a temptation for some desperate adventurer to try his fortune; and in such a case, that relief can Britain give? Ere she could hear the news, the fatal business might be done; and ourselves suffering like the wretched Britons under the oppression of the Conqueror. Ye that oppose independence now, ye know not what ye do; ye are opening a door to eternal tyranny, by keeping vacant the seat of government. There are thousands, and tens of thousands, who would think it glorious to expel from the continent that barbarous and hellish power, which hath stirred up the Indians and Negroes to destroy us; the cruelty hath a double guilt, it is dealing brutally by us, and treacherously by them.
To talk of friendship with those in whom our reason forbids us to have faith, and our affections wounded through a thousand pores instruct us to detest, is madness and folly. Every day wears out the little remains of kindred between us and them, and can there be any reason to hope, that as the relationship expires, the affection will increase, or that we shall agree better, when we have ten times more and greater concerns to quarrel over than ever?
Ye that tell us of harmony and reconciliation, can ye restore to us the time that is past? Can ye give to prostitution its former innocence? Neither can ye reconcile Britain and America. The last cord now is broken, the people of England are presenting addresses against us. There are injuries which nature cannot forgive; she would cease to be nature if she did. As well can the lover forgive the ravisher of his mistress, as the continent forgive the murders of Britain. The Almighty hath implanted in us these unextinguishable feelings for good and wise purposes. They are the guardians of his image in our hearts. They distinguish us from the herd of common animals. The social compact would dissolve, and justice be extirpated the earth, or have only a casual existence were we callous to the touches of affection. The robber, and the murderer, would often escape unpunished, did not the injuries which our tempers sustain, provoke us into justice.
O ye that love mankind! Ye that dare oppose, not only the tyranny, but the tyrant, stand forth! Every spot of the old world is overrun with oppression. Freedom hath been hunted round the globe. Asia, and Africa, have long expelled her—Europe regards her like a stranger, and England hath given her warning to depart. O! receive the fugitive, and prepare in time an asylum for mankind.