LOGIC When we construct truth-tables for complex propositions in this manner, we sometimes find that they take the same truth-value for every possible truth-value of the elementary propositions Thus, the proposition ‘p or not p’ is true whether ‘p’ is true or false, as we see thus: p T F p V $p T T FT F T TF On the other hand, the proposition ‘p and not-p’ is false whatever ‘p’ may be: p T F p & $p T F FT F F TF A proposition that is true for all truth-possibilities of its elementary propositions is called a tautology; a proposition that is false for all truth-possibilities is called a contradiction (TLP 4.46) The tautology set out above corresponds to the law of excluded middle The tautology that is the negation of the contradiction set out above corresponds to the law of non-contradiction These two laws were two of the three traditional laws of thought In this way the study of tautologies links with old-fashioned logics, but it also marks an advance on Frege’s handling of propositional logic It can be shown that all formulae that are tautologous by Wittgenstein’s test are either axioms or theorems of Frege’s system, and conversely that anything that can be proved from Frege’s axioms will be a tautology The truth-table method and the axiomatic system thus turn out to be two devices for handling the same material, namely the logical truisms of the propositional calculus But the truth-table method has several advantages over the axiomatic method First, it represents all logical truths as on a level with each other, whereas Frege’s system and the system of Principia privilege an arbitrarily chosen set of them as axioms Second, there is no need to appeal to any self-evidence in logic: the truth-table method is entirely mechanical, in the sense that it can be carried out by a machine Finally, given a formula of the propositional calculus we can always settle, by the use of a truth-table, whether or not it is a tautology An axiomatic system offers nothing comparable To be sure, if we discover a proof we know the formula is a 113