ONT Re: Model Theory
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Note 7
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| 1. Introduction
|
| 1.2. Model Theory for Sentential Logic (cont.)
|
| We shall introduce abbreviations to our language in the usual way,
| in order to make sentences more readable. The symbols 'v' (or),
| '=>' (implies), and '<=>' (if and only if) are abbreviations
| defined as follows:
|
| (p v q) for (~((~p) & (~q))),
|
| (p => q) for ((~p) v q),
|
| (p <=> q) for ((p => q) & (q => p)).
|
| Of course, v, =>, and <=> could just as well have been
| included in our list of symbols as three more connectives.
| However, there are certain advantages to keeping our list of
| symbols short. For instance, 1.2.2 and proofs by induction
| based on it are shorter this way. At the other extreme,
| we could have managed with only a single connective,
| whose English translation is "neither ... nor ...".
| We did not do this because "neither ... nor ..."
| is a rather unnatural connective.
|
| Another abbreviation which we shall adopt is to
| leave out unnecessary parentheses. For instance,
| we shall never bother to write outer parentheses in
| a sentence -- thus ~S is our abbreviation for (~S).
| We shall follow the commonly accepted usage in dropping
| other parentheses. Thus ~ is considered more binding than
| & and v, which in turn are more binding than => and <=>.
| For instance, ~p v q => r & p means ((~p) v q) => (r & p).
|
| Hereafter we shall use the single symbol $S$ to denote both the
| set of sentence symbols and the language built on these symbols.
| There is no fear of confusion in this double usage since the
| language is determined uniquely, modulo the connectives,
| by the sentence symbols.
|
| Chang & Keisler, 'Model Theory', pages 6-7.
|
| C.C. Chang and H.J. Keisler, 'Model Theory',
| North-Holland, Amsterdam, Netherlands, 1973.
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