LOGIC and therefore is equivalent to ‘p or q’ (in modern symbols, ‘p V q’) ‘p and q’ (‘p & q’), on the other hand, is rendered by Frege as ‘$ ðq ! $pÞ’ As Frege realized, a different system would be possible in which conjunction was primitive, and conditionality was defined in terms of conjunction and negation But in logic, he maintained, deduction is more important than conjunction, and that is why ‘if’ and not ‘and’ is taken as primitive Earlier logicians had drawn up a number of rules of inference, rules for passing from one proposition to another One of the best known was called modus ponens: ‘From ‘‘p’’ and ‘‘If p then q’’ infer ‘‘q’’ ’ In his system Frege claims to prove all the laws of logic using this as a single rule of inference The other rules are either axioms of his system or theorems proved from them Thus the rule traditionally called contraposition, which allows the inference from If John is snoring, John is asleep to If John is not asleep, John is not snoring, is justified by the first of the axioms quoted above When we put together Frege’s propositional calculus and his predicate calculus we can symbolize the universal sentences of ordinary language, making use of both the sign of generality and the sign of conditionality The expression (x)(Fx ! Gx) can be read For all x, if Fx then Gx, which means that whatever x may be, if ‘Fx’ is true then ‘Gx’ is true If we substitute ‘is a man’ for ‘F’ and ‘is mortal’ for ‘G’ then we obtain ‘For all x, if x is a man, x is mortal’, which is what Frege offers as the translation of ‘All men are mortal’ The contradictory of this, ‘Some men are not mortal’, comes out as ‘$(x)(x is a man ! x is mortal)’ and its contrary, ‘No man is mortal’, comes out as ‘(x)(x is a man !$ x is mortal)’ By the use of these translations, Frege is able to prove as part of his system theorems corresponding to the entire corpus of Aristotelian syllogistic 106