Logical Consequence: From Logical Terms to Semantic Constraints

In the paper I propose a new framework for extensional logic, where the explication of the notion of logical consequence is the primary aim. The framework may also be applicable in the study of natural language, especially in illuminating various semantic relations between expressions. In the paper I discuss a prevailing view by which logical terms determine the forms of sentences and therefore the logical validity of arguments. This view is common to those who hold that there is a principled distinction between logical and nonlogical terms and those holding relativistic accounts. I adopt the Tarskian tradition by which logical validity is determined by form, but reject the centrality of logical terms. I propose an alternative framework for logic where logical terms no longer play a distinctive role. This account employs a new notion of semantic constraints. The paper includes some preliminary definitions and results in the new framework. A semantic constraint for L is a sentence in the metalanguage that somehow constrains or limits the admissible models for L (and can be viewed as a meaning rule). Logical terms (or more precisely, rules defining logical terms) are merely a special case of semantic constraints, while all the semantic constraints in a system are involved in determining logical consequence. Take, for instance, the terms allRed and allGreen.1 These are paradigmatic cases of nonlogical terms in mainstream logic. There are good reasons for not fixing the extensions of color-terms completely. But we could fix their mutual dependencies, and have a rule in our system that says that the intersection of their extensions is empty in all models. A rule like this, I contend, is not essentially different from a rule fixing the interpretation of a logical term. In both cases, there is a rule that consists in restricting admissible models. We may say that whereas the interpretations for logical terms are completely fixed, other terms may be partly fixed. Let L be a language including a set of terms (the primitive expressions of the language) and phrases (strings of terms, possibly including auxiliary devices such as parentheses). Let a model for L be a pair 〈D,I〉 where D is a nonempty set (the domain), and I an interpretation function, assigning extensions to phrases in L. As we deal here with models set up on a classical set-theoretic foundation, the metalanguage used in all the examples includes, but is not confined to, the language of set theory. Semantic constraints include implicit universal quantification over models (domains and interpretation functions).