Models and Logical Consequence

Abstract

This paper deals with the adequacy of the model-theoretic definition of logical consequence. Logical consequence is commonly described as a necessary relation that can be determined by the form of the sentences involved. In this paper, necessity is assumed to be a metaphysical notion, and formality is viewed as a means to avoid dealing with complex metaphysical questions in logical investigations. Logical terms are an essential part of the form of sentences and thus have a crucial role in determining logical consequence. Gila Sher and Stewart Shapiro each propose a formal criterion for logical terms within a model-theoretic framework, based on the idea of invariance under isomorphism. The two criteria are formally equivalent, and thus we have a common ground for evaluating and comparing Sher and Shapiro philosophical justification of their criteria. It is argued that Shapiro's blended approach, by which models represent possible worlds under interpretations of the language, is preferable to Sher’s formal-structural view, according to which models represent formal structures. The advantages and disadvantages of both views’ reliance on isomorphism are discussed.

Appendix

1.1 A.1 Equivalence Result

Let L be a first order language with logical terms of level 1 or 2 (truth-functional connectives, predicates and functional expressions), and countably many nonlogical predicates from each arity. We assume also that the model theory is a standard classical one. Then the isomorphism property holds for L, with the class of models for L, if and only if the predicates and functional expressions which are logical terms in L are invariant under isomorphisms.

Proof Outline

The left to right direction (from the isomorphism property to Sher’s criterion) is proved by constructing, for each logical term C, a formula φ where C is the only logical term, and then using the isomorphism property on φ. We show this only for the simple case of first-level predicates.

Let C be an n-place first-level predicate for some n ≥ 1, and assume that C is a logical term in L. We need to show that it satisfies Sher’s criterion, i.e. that it is invariant under isomorphic structures. Sher assigns to each logical term C a function f _C such that for each model M, f _C (M) is C’s extension in M. Let M and M′ be models with domains A and A′, and 〈b ₁, … , b _n 〉 ∈ A ⁿ, 〈b′₁, … , b′ _n 〉 ∈ A′ⁿ such that the structures 〈A, 〈b ₁, … , b _n 〉〉 and 〈A′, 〈b′₁, … , b′ _n 〉〉 are isomorphic. We need to show that 〈b ₁, … , b _n 〉 ∈ f _C (M) iff 〈b′₁, … , b′ _n 〉 ∈ f _C (M′). We now define assignments s and s′ for M and M′ respectively, thus:

$$s(x_i) = \left\{\begin{array}{l} b_{i}\quad \text{if}\; 1\leq i\leq n\\ b_{n}\quad \text{if}\; i>n \end{array}\right\} \qquad s^{\prime}(x_i) = \left\{\begin{array}{l} b^{\prime}_{i}\quad\text{if}\;1\leq i\leq n\\ b^{\prime}_{n}\quad\text{if}\; i>n \end{array}\right\} $$

Note that the structures 〈A, 〈b ₁, … , b _n 〉, s〉 and 〈A′, 〈b′₁, … , b′ _n 〉, s′〉 are isomorphic.

Now consider the formula φ = C(x ₁, … , x _n ). From the previous claim we get that 〈M, s〉 and 〈M′, s′〉 are isomorphic with respect to the nonlogical terms in φ (the nonlogical terms in φ include only variables²³). So by the assumption that the isomorphism property holds, 〈M, s〉 ⊧ φ iff 〈M′, s′〉 ⊧ φ. Therefore, by the definitions of s and s′, M ⊧ C[b ₁, … , b _n ] iff M′ ⊧ C[b′₁, … , b′ _n ], thus 〈b ₁, … , b _n 〉 ∈ f _C (M) iff 〈b′₁, … , b′ _n 〉 ∈ f _C (M′), as required.

The right to left direction is proved by induction, the details of which we leave to the reader.²⁴