ONT Model Theory
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I see that we are developing a bit of a communication breakdown
along both the "Inquiry Into Inquiry" and "Laws Of Logic" lines,
so I will attempt to bridge this burgeoning language barrier by
compiling a stock of standard options for nomenclature. Please
bear with me for a while, as this will take some time to anchor.
Jon Awbrey
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| 1.2 Model Theory for Sentential Logic
|
| 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
| that "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 (/) and the set $S$ itself.
|
| We now set up the sentential logic as a formal language.
| The symbols of our language are as follows:
|
| 1. connectives "&" (and), "~" (not).
| 2. parentheses "(", ")".
| 3. a nonempty set $S$ of sentence symbols.
|
| Intuitively, the sentence symbols stand for simple statements, and
| the connectives "&", "~" satnd for the words used to combine simple
| statements into compound statements. Formally, the 'sentences' of
| $S$ are defined as follows:
|
| 1.2.2 [Definition of 'sentence']
|
| 1. Every sentence symbol S is a sentence.
| 2. If q is a sentence then (~q) is a sentence.
| 3. If q, r are sentences, then (q & r) is a sentence.
| 4. A finite sequence of symbols is a sentence only if
| it can be shown to be a sentence by a finite number
| of applications of 1.2.2.1-3.
|
| C.C. Chang & H.J. Keisler, 'Model Theory',
| North-Holland, Amsterdam, Netherlands, 1973,
| pages 4-5.
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