ONT Re: Model Theory
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| 1.6.2 Definition.
| A set G of propositions is 'consistent' if G |-/- _|_.
| In words: one cannot derive a contradiction from G.
| The consistency of G can be expressed in various other forms:
|
| 1.6.3 Lemma. The following three conditions are equivalent:
| 1. G is consistent.
| 2. For no q, G |- q and G |- ~q.
| 3. There is a least one q such that G |-/- q.
|
| Let us call G 'inconsistent' if G |- _|_,
| then we can just as well prove the equivalence of:
| 4. G is inconsistent.
| 5. There is a q such that G |- q and G |- ~q.
| 6. For all q, G |- q.
| ...
|
| Clause 6 shows us why inconsistent sets (theories) are devoid
| of mathematical interest. For, if everything is derivable,
| we cannot distinguish between "good" and "bad" propositions.
| Mathematics tries to find distinctions, not to blur them.
|
| In mathematical practice one tries to establish consistency
| by exhibiting a model (think of the consistency of the negation
| of Euclid's Fifth Postulate and the non-euclidean geometries).
| In the context of propositional logic this means looking for
| a suitable valuation.
|
| 1.6.4 Lemma. G is consistent if there is a valuation v
| such that v(r) = 1 for all r in G.
|
| DVD, pages 44-45.
|
| Dirk van Dalen,
|'Logic and Structure', Second Edition,
| Springer-Verlag, Berlin, Germany, 1983.
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