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325 We should address how logic books belong in a list of mathematical works. It would be appropriate to briefly review the history of logic from a mathematical point of view. Aristotle is credited with developing the syllogism, a form of argument with two premises and a conclusion. An example would be the argument All men are animals. All animals are mortal. ⬖ All men are mortal. Aristotle cataloged all the various syllogism forms that constituted valid arguments. Necessity was introduced by Aristotle to supplement his theory of syllogisms. It then became possible for him to distinguish between propositions that are true and those that are necessarily true. Necessity and possibility are called “modalities,” and their logic is called “modal” logic. George Boolos says that, although Aristotle “developed the theory of the syllogism in almost perfect form,” Aristotle’s theory of modal syllogisms has been found “defective” even by “sympathetic commentators.”1 Some time later, symbolic logic was developed. Implication was denoted by 1 and negation by ¬. A statement of the form “all A are not B,” which might form one line of a syllogism, could be represented by a string of symbols, 5x (Ax 1 ¬ Bx). The era from the 1870s to the 1930s saw revolutionary work on the foundations of mathematics.2 The dual reductions that were achieved were (1) all mathematical objects came to be viewed as sets, and (2) all mathematical discourse was shown to be reducible (potentially at least) to formal logic. It became possible to answer the question, “What are we doing when we do mathematics?” by saying, “We use logic appendix c  Some Comments on the Place of Logic in Mathematics 1. George Boolos, The Logic of Provability, Cambridge, 1993: xv–xvi. 2. See the anthology by Jean van Heijenoort (From Frege to Gödel, a Source Book in Mathematical Logic, 1879–1931, Cambridge, 1967) of papers from this period. 326 appendix c to talk about sets.”3 Today, mathematics students at the graduate level have to become fluent in symbolic logic and basic set theory. In symbolic logic, the modality of necessity was denoted by n and np was read as “p is necessarily true.” The statement “p is possibly true” may be rendered as¬n¬p in this formalism.4 Various systems of modal logic have been developed, which might include as axioms or theorems statements like np 1 p, n(p 1 q) 1 (np 1 nq), or np 1 nn p. Many mathematicians, unlike philosophers, may not feel the need to distinguish between “p is true” and “p is necessarily true.” (If p is a strictly mathematical statement, many philosophers may not feel the need either.) A recent reinterpretation of the symbol n, however, makes modal logic more relevant to the mathematical enterprise. A 1931 paper by Kurt Gödel made the following argument. By reducing logical proof to a mechanistic process, Gödel showed that the provability of a statement in a formal system is equivalent to a statement in arithmetic. It follows that for a formal system that is sophisticated enough to contain arithmetic, the provability of a statement p in the system can be viewed as itself a statement in the system. Gödel ultimately used this to show that a formal system that encompassed at least arithmetic must allow propositions that are undecidable within the system, because they state, in so many words, their own unprovability. In today’s mathematics, now that more advanced techniques are known, hundreds of undecidable statements have been found and published in the literature of such mathematical subfields as topology , analysis, combinatorics, and set theory.5 If we interpret np as “p is provable” in the sense above, then the system of modal logic that arises is similar but distinct from the systems that went before. Called GL by Boolos, this system has n(p 1 q) 1 (np 1 nq) as axiomatic and np 1 nnp as a theorem, but np 1 p is not always a theorem. GL has the rather odd inference rule that whenever nA 1 A is a theorem, then one may derive A. Also, it has come to our attention that modal logic is being employed by theoretical computer scientists to cope with issues arising from parallel processing. Thus not only is formal logic at the very core of modern mathematical teaching and research, to the extent that it is required background for most graduate-level training in mathematics, but even modal logic...

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