ONT Re: Model Theory
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Note 6
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| 1. Introduction
|
| 1.2. Model Theory for Sentential Logic (cont.)
|
| Our definition of a sentence of $S$ may be
| restated as a recursive definition based on
| the length of a finite sequence of symbols:
|
| A single symbol is a sentence iff it is a sentential symbol;
|
| A sequence p of symbols of length n > 1 is a sentence
| iff there are sentences q and r of length less than n
| such that p is either (~q) or (q & r).
|
| Alternatively, our definition may be restated in set-theoretical terms:
|
| The set of all sentences of $S$ is the least set !S!
| of finite sequences of symbols of $S$ such that each
| sentence symbol S belongs to !S! and, whenever q, r
| are in !S!, then (~q), (q & r) belong to !S!.
|
| No matter how we may think of sentences, the important thing is that
| 'properties of sentences can only be established through an induction
| based on 1.2.2'. More precisely, to show that every sentence p has a
| given property P, we must establish three things:
|
| 1. Every sentence symbol S has the property P.
|
| 2. If p is (~q) and q has the property P,
| then p has the property P.
|
| 3. If p is (q & r) and q, r have the property P,
| then p has the property P.
|
| The reader may check his [or her] understanding
| of this point by proving through induction that
| every sentence p has the same number of right
| parentheses as it has left parentheses.
|
| How many sentences of $S$ are there? This depends on the number
| of sentence symbols S in $S$. Each sentence is a finite sequence
| of symbols. If the set $S$ is finite or countable, then there
| are countably many sentences of $S$. Of course, not every finite
| sequence of symbols is a sentence; for instance, (S_0 & (~S_5))
| is a sentence, but & & ) S_3 and S_0 & ~S_5 are not. If the set
| $S$ of sentence symbols has uncountable cardinal !a!, then the
| set of sentences of $S$ also has power !a!.
|
| Chang & Keisler, 'Model Theory', pages 5-6.
|
| C.C. Chang and H.J. Keisler, 'Model Theory',
| North-Holland, Amsterdam, Netherlands, 1973.
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