Logical Constants and Unrestricted Quantification

From Firenze University Press Journal: Journal for the Philosophy of Mathematics

Volker Halbach, University of Oxford

An argument is logically valid if, and only if, there is no interpretation of the non-logical vocabulary underwhich the premisses are all true and the conclusion is false. On this semantic definition, logical consequencedepends on the distinction between the logical and non-logical vocabulary. The logical vocabulary is widelytaken to be the vocabulary that is independent of the subject matter and behaves on all objects in the sameway. Logicians have tried to cast this characterization into a more formal criterion: An expression is a logicalconstant if, and only if, the behaviour of the expression cannot be changed by replacing every object with aproxy in such a way that every object is a proxy of something. In other words, the behaviour of logical constantsis invariant under arbitrary permutations.Logicians have discussed various versions of this invariance criterion. Usually, the behaviour of a term orthe operations expressed by the term is said to be logical if, and only if, the behaviour of the term or operation,when interpreted in a model with a certain domain, is not affected by permuting the model’s domain. Thisyields a criterion relative to a model; but one can usually choose the ‘intended’ interpretation over the domain.This model-theoretic version of the invariance criterion is formulated in set theory, even if it is applied totype-theoretic languages.The model-theoretic criterion of logicality deviates from the initial informal characterization as follows:

According to the formal criterion, a term or operation is defined as logical if, and only if, it behaves on all objects of any domain in the same way, while the original informal characterization takes a term or operation to be logical if, and only if it behaves on all objects in the same way; no mention is made of domains.

In this paper I discuss a formal version of the invariance criterion without domains. It is closer to the original conception of independence from the subject matter than the model-theoretic. Not only is my criterion closer to the initialinformal characterization of logical constants; it also avoids certain well-known problems that are generated bythe use of domains.Of course, domains are essential to the understanding of interpretations as (set-theoretic) models and,consequently, to the model-theoretic definition of logical consequence. However, there are also definitions oflogical consequence that do not rely on domains. For such definitions my invariance criterion is much bettersuited than the model-theoretic. Before delving into the discussion of invariance criteria, I explain why one might want to eliminate domains from the definition of logical validity.

DOI: https://doi.org/10.36253/jpm-2935

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