LOGIC sentences of ordinary language In order to express such sentences in Frege’s system one must graft his predicate calculus (the theory of quantifiers such as ‘some’ and ‘all’) on to a propositional calculus (the theory of connectives between sentences, such as ‘if’ and ‘and’) In Frege’s system of propositional logic the most important element is a sign for conditionality, roughly corresponding to ‘if’ in ordinary language The Stoic logician Philo, in ancient times, had defined ‘If p then q’ by saying that it was a proposition that was false in the case in which p was true and q false, and true in the three other possible cases.4 Frege defined his sign for conditionality (which we may render ‘!’) in a similar manner He warned that it did not altogether correspond to ‘if then’ in ordinary language If we take ‘p ! q’ as equivalent to ‘If p then q’ then propositions such as ‘If the sun is shining, Â ¼ 21’ and ‘If perpetual motion is possible, then pigs can fly’ turn out true—simply because the consequent of the first proposition is true, and the antecedent of the second proposition is false ‘If’ behaves differently in ordinary language; the use of it that is closest to ‘!’ is in sentences such as ‘If those curtains match that sofa, then I’m a Dutchman’ Frege’s sign can be looked on as a stripped-down version of the word ‘if’, designed to capture just that aspect of its meaning that is necessary for the formulation of rigorous proofs containing it In Frege’s terminology, ‘ ! ’ is a function that takes sentences as its arguments: its values, too, are sentences Whether the sentences that are its values (sentences of the form ‘p ! q’) are true or false will depend only on whether the sentences that are its arguments (‘p’ and ‘q’) are true or false We may call functions of this kind ‘truth-functions’ The conditional is not the only truth-function in Frege’s system So too is negation, represented by the sign ‘$’, since a negated sentence is true just in case the sentence negated is false, and vice versa With the aid of these two symbols Frege built up a complete system of propositional logic, deriving all the truths of that logic from a limited set of primitive truths or axioms, such as ‘(q ! p) ! ( $p ! $q)’ and ‘$$p ! p’ Connectives other than ‘if’, such as ‘and’ and ‘or’, are defined in terms of conditionality and negation Thus, ‘$q ! p’ rules out the case in which p is false and $q is true: it means that p and q are not both false, See vol I, p 138 105