Chapter

Higher-order Logics

Hale Bob

in Necessary Beings

Published in print September 2013 | ISBN: 9780199669578
Published online January 2014 | e-ISBN: 9780191765186 | DOI: https://dx.doi.org/10.1093/acprof:oso/9780199669578.003.0009
Higher-order Logics

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The conception of properties advocated in early chapters has an obvious bearing upon the interpretation of higher-order logics. In particular, interpreting second-order variables as ranging over properties so understood puts us in a position to answer Quine’s charge that second-order logic is just ‘set theory in sheep’s clothing’ and related charges that it must involve existential assumptions comparable to those of standard set theory. From the standpoint of the conception of properties advocated here, the standard semantics of second-order logic is inadmissible, and does indeed involve something like an assimilation of second-order logic to a form of set theory. Although, properly understood, properties should be individuated intensionally, there is a non-standard model-theoretic semantics which accords much more closely with the spirit of our interpretation. With respect to this semantics, second-order logic is compact, complete and the Löwenheim-Skolem Theorems hold—in contrast with the standard semantics. The consequent failure of categoricity results for second-order arithmetic, analysis and set theory might be thought a decisive objection, but it is argued that it is not so. Finally it is argued that when second-order logic is interpreted in accordance with the alternative semantics proposed, there is no reason to proscribe impredicative comprehension axioms. An appendix sketches proofs of some meta-theorems.

Keywords: properties; sets; second-order logic; standard semantics; existence-assumptions; comprehension; Quine’s objections; impredicativity

Chapter.  14418 words. 

Subjects: Metaphysics ; Philosophy of Mathematics and Logic

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