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May 2020 Formal Notes on the Substitutional Analysis of Logical Consequence
Volker Halbach
Notre Dame J. Formal Logic 61(2): 317-339 (May 2020). DOI: 10.1215/00294527-2020-0009


Logical consequence in first-order predicate logic is defined substitutionally in set theory augmented with a primitive satisfaction predicate: an argument is defined to be logically valid if and only if there is no substitution instance with true premises and a false conclusion. Substitution instances are permitted to contain parameters. Variants of this definition of logical consequence are given: logical validity can be defined with or without identity as a logical constant, and quantifiers can be relativized in substitution instances or not. It is shown that the resulting notions of logical consequence are extensionally equivalent to versions of first-order provability and model-theoretic consequence. Every model-theoretic interpretation has a substitutional counterpart, but not vice versa. In particular, in contrast to the model-theoretic account, there is a trivial intended interpretation on the substitutional account, namely, the homophonic interpretation that does not substitute anything. Applications to free logic, and theories and languages other than set theory are sketched.


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Volker Halbach. "Formal Notes on the Substitutional Analysis of Logical Consequence." Notre Dame J. Formal Logic 61 (2) 317 - 339, May 2020.


Received: 5 November 2019; Accepted: 8 December 2019; Published: May 2020
First available in Project Euclid: 7 April 2020

zbMATH: 07222694
MathSciNet: MR4092538
Digital Object Identifier: 10.1215/00294527-2020-0009

Primary: 03A05
Secondary: 03B10 , Z003

Keywords: logical consequence , logical validity

Rights: Copyright © 2020 University of Notre Dame


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Vol.61 • No. 2 • May 2020
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