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PEIRCE TO STRAWSON sitional calculus in an axiomatic manner in which all the laws of propositional logic are derived, by a specified method of inference, from a number of primitive propositions The actual symbolism that Frege invented for this purpose is difficult to print, and has long been superseded in the presentation of the calculus; but the operations that it expressed continue to be fundamental in mathematical logic It was not, however, the propositional calculus, but the predicate calculus, that was Frege’s greatest contribution to logic This is the branch of logic that deals with the internal structure of propositions rather than with propositions considered as atomic units Frege invented a novel notation for quantification, that is to say, a method of symbolizing and rigorously displaying those inferences that depend for their validity on expressions such as ‘all’ or ‘some’, ‘no’ or ‘none’ With this notation he presented a predicate calculus that greatly improved upon the Aristotelian syllogistic that had hitherto been looked upon as the be-all and end-all of logic Frege’s calculus allowed formal logic, for the first time, to cope with sentences containing multiple quantification, such as ‘Nobody knows everything’ and ‘Every boy loves some girl’.2 Though Begriffsschrift is a classical text in the history of logic, Frege’s purpose in writing it was concerned more with mathematics than with logic He wanted to put forward a formal system of arithmetic as well as a formal system of logic, and most importantly, he wanted to show that the two systems were intimately linked All the truths of arithmetic, he claimed, could be shown to follow from truths of logic without the need of any extra support How this thesis (which came to be known as ‘logicism’) was to be demonstrated was sketched in Begriffsschrift, and set out more fully in two later works, Grundlagen der Arithmetik (‘Foundations of Arithmetic’) of 1884 and Die Grundgesetze der Arithmetik (‘The Fundamental Laws of Arithmetic’) of 1893 and 1903 The most important step in Frege’s logicist programme was to define arithmetical notions, such as that of number, in terms of purely logical notions, such as that of class Frege achieves this by treating the cardinal numbers as classes of equivalent classes, that is to say, of classes with the same number of members Thus the number two is the class of pairs, and the number three the class of trios Such a definition at first sight appears See Ch below 39
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