ONT Re: Model Theory
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| 4.7 Consistency
|
| The syntactic concept |- of derivability corresponds to
| the semantic concept |= of consequence. As a syntactic
| counterpart to satisfiability we define the concept of
| 'consistency'.
|
| 4.7.1 Definition.
|
| a. Q is 'consistent' (written: Con Q) if and only if
| there is no formula q such that Q |- q and Q |- ~q.
|
| b. Q is 'inconsistent' (written: Inc Q) if and only if
| Q is not consistent (that is, if there is a formula q
| such that Q |- q and Q |- ~q).
|
| First we show that from an inconsistent set one can derive any formula.
|
| 4.7.2 Lemma. For a set of formulas Q the following are equivalent:
|
| a. Inc Q.
| b. For all q, Q |- q.
|
| EFT, page 72.
|
| Ebbinghaus, H.-D., Flum, J., & Thomas, W.,
|'Mathematical Logic',
| Springer-Verlag, New York, NY, 1984 (145 A.P.)
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