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.

On the methodology of informal rigour: set theory, semantics, and intuitionism

“…44 Note in particular that the Completeness Theorem itself can be understood as providing a means of bridging the epistemological gap which Etchemendy suggests (cf. note 41) stands between our knowledge of the Tarskian Halbach The formalization of the validity argument we have given in §4.1.3 can also be compared to a recent treatment by Volker Halbach (2020a;2020b). Halbach suggests that Kreisel's policy of treating Val as a primitive predicate can be improved upon by developing an axiomatic theory of substitutional quantification within which it is possible to formalize the interpretation 'ϕ 1 is true in all structures' by quantifying over an appropriately broad class of substitution instances for the non-logical symbols in ϕ 1 .…”

“…by replacing his informal arguments for (1i,ii) with mathematical results (e.g. Lemma 4.3 and Theorem 4.4 in Halbach, 2020b) which must themselves be recognized as consequences of correct mathematical reasoning relative to a prior understanding of validity. Second, as we have discussed in §4.1.2, it is unclear that by simply providing a linguistic surrogate for class-sized interpretations, the envisioned approach does justice to the understanding of Val which Kreisel appears to have had in mind.…”

Classical Determinate Truth I

“…The axioms for truth can and should be added to a base theory that yields at least a comprehensive theory of syntax (in a direct or coded form); the base theory may also go far beyond a theory of syntax. In [22, 23] the second author formulated the truth theory over set theory as the base. Here, however, we start from Peano arithmetic, which is traditionally used as base theory for axiomatic theories of truth; but we consider it only as a simple model case.…”

“… 4 With modifications the theory can reformulated in a purely relational language without closed terms such as the language of set theory. For such a setting without function symbols, a satisfaction predicate might be a better fit than a unary truth predicate (see Halbach [22]). …”

Substitutional Validity for Modal Logic

“…There exists no model that matches the homophonic substitution, however, nor is there a model equivalent to any substitution which maps membership to some formula that fails to define a set, like x ̸ ∈ x. In contrast, for any model there exists a substitution where truth in that model is equivalent to (simple) truth under that substitution [12]. 3 A second advantage is the ability to model absolute generality.…”