Theory theory

Let us explore the theory of theories, the theory of models, the theory of models of theories, and the theory of theories of models.

Another installment from A Panorama of Logic, an introduction to topics in logic for philosophers, mathematicians, and computer scientists.

A theory is simply any set of sentences in a given language. A model of such a theory T is a structure M in which every sentence of the theory is true, that is, M ⊨ σ for every sentence σ in T, and in this case we write M ⊨ T.

Logical consequence

We say that a theory T entails a sentence σ, written

\( T ⊨ \sigma,\)

if every model of T is a model of σ. In this situation we also say that the theory T logically implies the sentence σ, that σ is a logical consequence of T, or that σ is valid in the theory T—all synonymous terminology for this central concept. One theory entails another, written T ⊨ S, if T entails every sentence σ in S. Two theories are equivalent if each entails the other—each theory has all sentences in the other theory as logical consequences.

I find it insightful to notice that this notion of entailment or validity has nothing to do with proof or argument or one's reasoning process—it is not about our knowledge of the entailment or about any epistemological concern, but rather a notion entirely of the semantic realm, about the nature and range of logical possibility, about what kinds of models there are, including perhaps infinite models of vast uncountable size, and whether they all satisfy σ. Namely, if it happens to be the case that every model of T is also a model of σ, then T entails σ and otherwise not. In this sense, logical consequence belongs not to epistemology, but rather perhaps to ontology or metaphysics.

Let us explore several natural logical operators on theories and their models.