LOGIC Similar tables can be set out for the other logical constants, such as ‘or’ and ‘if ’ ‘If p then q’ is written as ‘p ! q’ and is interpreted as a truthfunctional condition that is true in all cases except where ‘p’ is true and ‘q’ is false The simplest truth-table is the one for ‘not’: p T F $p F T This shows that a proposition is true when its negation is false, and vice versa Propositions of great length and complexity may be built up by repeated use of the logical constants, but however complex they are their truth-value can always be determined from the truth-values of the simple propositions that make them up (Wittgenstein, TLP 5.31) Consider the following proposition: If p and q, then not-p and q This is a truth-function of ‘p’ and ‘q’ as shown in the following table: p T F T F q T T F F p&q TTT FFT TFF FFF ! F T T T $p & q FTFT TFTT FTFF TFFF This table is constructed in the following manner First the columns under each occurrence of the single propositional variables are filled in by copying out the values given in the two left-hand columns, which represent a conventional arrangement to ensure that all possible combinations of truth-values are covered (TLP 4.31) Then in the fourth column from the right the truth-value of ‘not-p’ is filled in under the ‘$’ sign by reversing the truthvalue of ‘p’ Then the columns under the ‘&s’ are filled in by deriving the truth-value of the conjunct propositions via the table given earlier Finally the ‘!’ column is computed, the truth-values being derived from the truthfunctional definition of ‘if then’ This column shows the value of the whole complex formula for every possible combination of truth-values of its constituents It turns out to be false if ‘p & q’ is true, and to be true in all other cases 112