A Profile of Mathematical Logic (Dover Books on Mathematics) by Howard DeLong

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By Howard DeLong

Publish 12 months note: First released in 1971
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This textual content explores the ancient purposes for the formation of Aristotelian common sense, the increase of mathematical good judgment, the character of the formal axiomatic strategy and its use, and the most result of metatheory and their import.

From 1971 edition

Includes 22 figures and 19 tables. Appendixes. Bibliography. Indexes.

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12 Metaphysical Conclusions Whether the eclecticist chooses the traditional route of Carnapian tolerance or the modernized version of Beall and Restall, the following seems hard to evade: In logic, there will be no morals, and very little metaphysics. Presumably, the eclecticist who endorses both classical and intuitionistic logic as intelligible will simply be committed to the same metaphysical upshots31 as the monogamous classicist. I am more concerned with the first of the questions Dummett raised at the beginning of the last section than the second one.

Intuitionists have wavered between these two ways of motivating ECQ, and I will come back to the distinction when I discuss negation in empirical contexts. As they will come up again, let us find two names for these lines: The first might be called a truly constructive conversion, the second an empty promise11 conversion. Kolmogorov and others after him thought that there is something fishy about both ways this argument could be played out; for example, worries about this argument have led to the development of minimal logic, where a contradiction does not entail everything.

That is, we could come to recognize a proof of either of them if we were presented with one. What, now, would a proof of the conjunction look like? Well, it would simply be a proof of the first conjunct, followed by a proof of the second conjunct. The case of disjunction is equally trivial, but what about negations? The intuitionist tries to express everything in terms of constructions and proofs that are obtainable by us. A negation, in contrast, seems to be telling us something about the impossibility of proof and construction.

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