A Dictionary of Philosophy, Third Edition
. One among the decision procedures (see DECIDABLE) for the propositional CALCULUS. The TRUTH-VALUE, normally true or false, of a complex expression in the propositional calculus depends on the truth-values assigned to the VARIABLES in it. A truth table is a systematic exposition of all possible combinations of such assign-ments for the expression in question. Each combination occupies a row in the table, with the resulting truth-value for the whole expression placed, for example, at one end of the row.
These truth-values for the whole expression form a column, from which various properties of the expression can be read off, e.g. the expression is logically true if this column contains no truth-value but true.
The following example of a truth table for ‘Not (p and not q)’ (where p and q represent propositions, and ‘1’ means ‘true’ and ‘0’ means ‘false’) is written out at unnecessary length to show how it is constructed. Various alternative presentations are possible. In the present case each row should be read from left to right, and the final column shows that the whole expression is false when p is true and q is false, and true in all other cases.
p | q | not q | p and not q | Not (p and not q) |
1 | 1 | 0 | 0 | 1 |
1 | 0 | 1 | 1 | 0 |
0 | 1 | 0 | 0 | 1 |
0 | 0 | 1 | 0 | 1 |
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