ONT Re: Model Theory
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Note 4
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| 1. Introduction
|
| 1.2. Model Theory for Sentential Logic
|
| In our introduction, Section 1.1, we gave a general idea of the
| flavor of model theory, but we were not yet ready to give many
| details. We shall now come down to earth and give a rigorous
| treatment of model theory for a very simple formal language,
| sentential logic (also known as propositional calculus).
| We shall quickly develop this "toy" model theory along
| lines parallel to the much deeper model theory for
| predicate logic. The basic ideas are the decision
| procedure via truth tables, due to Post (1921),
| and Lindenbaum's theorem with the compactness
| theorem which follows. This section will
| give a preview of what lies ahead in
| our book.
|
| We are assuming (see Preface) that the reader is already
| thoroughly familiar with sentential, and even predicate,
| logic. Thus we shall feel free to proceed at a fairly
| rapid pace. Nevertheless, we shall start from scratch,
| in order to show what sentential logic looks like when
| it is developed in the spirit of model theory.
|
| Classical sentential logic is designed to study a set $S$ of simple statements,
| and the compound statements built up from them. At the most intuitive level,
| an intended interpretation of these statements is a "possible world", in
| which each statement is either true or false. We wish to replace these
| intuitive interpretations by a collection of precise mathematical objects
| which we may use as our models. The first thing which comes to mind is
| a function F which associates with each simple statement S one of the
| truth values "true" or "false". Stripping away the inessentials,
| we shall instead take a model to be a subset A of $S$; the idea
| is that S in A indicates that the simple statement S is true,
| and S not in A indicates that the simple statement S is false.
|
| 1.2.1. By a 'model' A for $S$ we simply mean a subset A of $S$.
|
| Thus the set of all models has the power 2^|$S$|. Several relations and
| operations between models come to mind; for example, A c B, $S$ - A, and
| the intersection |^|_(i in I) A_i of a set {A_i : i in I} of models.
| Two distinguished models are the empty set Ø and the set $S$ itself.
|
| Chang & Keisler, 'Model Theory', page 4.
|
| C.C. Chang and H.J. Keisler, 'Model Theory',
| North-Holland, Amsterdam, Netherlands, 1973.
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