Formal Notes on the Substitutional Analysis of Logical Consequence

Abstract

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 iff there is no substitution instance with true premisses 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 thus model-theoretic consequence. Every modeltheoretic 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, second-order logic, and theories and languages other than set theory are sketched. 1 The Substitutional Analysis of Logical Consequence In what could be called semantic theories of logical consequence – as opposed to proof-theoretic analyses –, logical consequence is defined as truth preservation under all interpretations. In model-theoretic semantics, interpretations are conceived as formal set-theoretic models. However, this is only a very recent understanding of interpretation. Traditionally, in order to refute the formal validity of an argument, logicians showed that there is a substitution instance with true premisses but a false conclusion. Such an interpretation is a substitutional counterexample to the argument in question. In the present paper I attempt to revive the substitutional understanding of interpretation and make the informal substitutional account precise in a mathematical setting for first-order predicate logic. The substitutional account of logical consequence advanced in this paper contrasts with earlier substitutional definitions of logical truth and consequence by Quine [20] 2010 Mathematics Subject Classification: Primary 03A05, 03B10