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FIRST‐ORDER LOGICAL VALIDITY AND THE HILBERT‐BERNAYS THEOREM
Author(s) -
Ebbs Gary,
Goldfarb Warren
Publication year - 2018
Publication title -
philosophical issues
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.638
H-Index - 18
eISSN - 1758-2237
pISSN - 1533-6077
DOI - 10.1111/phis.12120
Subject(s) - schema (genetic algorithms) , predicate (mathematical logic) , sentence , first order logic , logical equivalence , quine , mathematics , computer science , philosophy , discrete mathematics , linguistics , epistemology , machine learning , equivalence (formal languages) , programming language
What we call the Hilbert‐Bernays (HB) Theorem establishes that for any satisfiable first‐order quantificational schema S , there are expressions of elementary arithmetic that yield a true sentence of arithmetic when they are substituted for the predicate letters in S . Our goals here are, first, to explain and defend W. V. Quine's claim that the HB theorem licenses us to define the first‐order logical validity of a schema in terms of predicate substitution; second, to clarify the theorem by sketching an accessible and illuminating new proof of it; and, third, to explain how Quine's substitutional definition of logical notions can be modified and extended in ways that make it more attractive to contemporary logicians.