Kripke Models

In Alan Berger (ed.), Saul Kripke. Cambridge University Press (2011)
Abstract
Saul Kripke has made fundamental contributions to a variety of areas of logic, and his name is attached to a corresponding variety of objects and results. 1 For philosophers, by far the most important examples are ‘Kripke models’, which have been adopted as the standard type of models for modal and related non-classical logics. What follows is an elementary introduction to Kripke’s contributions in this area, intended to prepare the reader to tackle more formal treatments elsewhere.2 2. WHAT IS A MODEL THEORY? Traditionally, a statement is regarded as logically valid if it is an instance of a logically valid form, where a form is regarded as logically valid if every instance is true. In modern logic, forms are represented by formulas involving letters and special symbols, and logicians seek therefore to define a notion of model and a notion of a formula’s truth in a model in such a way that every instance of a form will be true if and only if a formula representing that form is true in every model. Thus the unsurveyably vast range of instances can be replaced for purposes of logical evaluation by the range of models, which may be more tractable theoretically and perhaps practically. Consideration of the familiar case of classical sentential logic should make these ideas clear. Here a formula, say (p & q) ∨ ¬p ∨ ¬q, will be valid if for all statements P..
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