section_path: Basic concepts/Premises, conclusions, and truth/Truth tables
section_name: Truth tables
paragraph_text: Truth tables
  
  Truth tables can be used to show how logical connectives work or how the truth values of complex propositions depends on their parts. They have a column for each input variable. Each row corresponds to one possible combination of the truth values these variables can take; for truth tables presented in the English literature, the symbols "T" and "F" or "1" and "0" are commonly used as abbreviations for the truth values "true" and "false". The first columns present all the possible truth-value combinations for the input variables. Entries in the other columns present the truth values of the corresponding expressions as determined by the input values. For example, the expression " p ∧ q {\displaystyle p\land q} " uses the logical connective ∧ {\displaystyle \land } (and). It could be used to express a sentence like "yesterday was Sunday and the weather was good". It is only true if both of its input variables, p {\displaystyle p} ("yesterday was Sunday") and q {\displaystyle q} ("the weather was good"), are true. In all other cases, the expression as a whole is false. Other important logical connectives are ¬ {\displaystyle \lnot } (not), ∨ {\displaystyle \lor } (or), → {\displaystyle \to } (if...then), and ↑ {\displaystyle \uparrow } (Sheffer stroke). Given the conditional proposition p → q {\displaystyle p\to q}, one can form truth tables of its converse q → p {\displaystyle q\to p}, its inverse (¬ p → ¬ q {\displaystyle \lnot p\to \lnot q}), and its contrapositive (¬ q → ¬ p {\displaystyle \lnot q\to \lnot p}). Truth tables can also be defined for more complex expressions that use several propositional connectives.