nLab
abstract model theory

Contents

Idea

Abstract model theory is the study of the general properties of the model theory of extensions of (classical untyped) first-order logic.

Originally motivated by Lindström's theorem that characterizes first-order logic, the field has subsequently been extended to provide alternative characterizations and include different logics within its range.

The basic concept of abstract model theory is that of an abstract logic which is a triple =(S,Φ,)\mathcal{L}=(S,\Phi ,\models) where \models is a binary relation between the class of \mathcal{L}-‘structures’ SS and the class of \mathcal{L}-‘sentences’ Φ\Phi to be thought of as minimalistic version of the satisfaction relation.

Remark

As the logical relations studied by abstract model theory are of a functorial nature, some category theory entered the picture already in Barwise (1974). The theory of institutions, aka institution-independent model theory (Diaconescu 2008), constitutes abstract categorical model theory proper. In a similar abstract categorical vein is the functorial approach to geometric theories described in Johnstone (2002, sec. B4.2).

References