LOGIC Frege’s logical calculus is not just more systematic than Aristotle’s; it is also more comprehensive His symbolism is able, for instance, to mark the difference between Every boy loves some girl ¼ (x)(x is a boy ! Ey(y is a girl & x loves y) ) and the apparently similar (but much less plausible) passive version of the sentence Some girl is loved by every boy ¼ (Ey(y is a girl & (x)(x is a boy ! x loves y) ) Aristotelian logicians in earlier ages had sought in vain to find a simple and conspicuous way of bringing out such differences of meaning in ambiguous sentences of ordinary language A final subtlety of Frege’s system must be mentioned The sentence ‘Socrates is mortal’, as we have seen, can be analysed as having ‘Socrates’ for argument, and ‘ is mortal’ as function But the function ‘ is mortal’ can itself be regarded as an argument of a different function, a function operating at a higher level This is what happens when we complete the function ‘ is mortal’ not with a determinate argument, but with a quantifier, as in ‘(x)(x is mortal)’ The quantifier ‘(x)(x )’ can then be regarded as a second-level function of the first-level function ‘ is mortal’ The initial function, Frege always emphasizes, is incomplete; but it may be completed in two ways, either by having an argument inserted in its argument place, or by itself becoming the argument of a second-level function This is what happens when the ellipsis in ‘ is mortal’ is filled with a quantifier such as ‘Everything’ Induction and Abduction in Peirce A number of Frege’s innovations in logic occurred, quite independently, to C S Peirce; but Peirce was never able to incorporate his results into a rigorous system, much less to publish them in a definitive form Peirce’s importance in the history of logic derives rather from his investigations into the structure of scientific inquiry Deductive logic assists us in organizing our knowledge; but the kind of reasoning that extends our knowledge (‘ampliative inference’ as Peirce calls it) is of three kinds: induction, hypothesis, and analogy All of these inferences, Peirce claimed, 107