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Full Idea
Using the definition of truth we are in a position to carry out the proof of consistency for deductive theories in which only (materially) true sentences are (formally) provable.
Gist of Idea
Using the definition of truth, we can prove theories consistent within sound logics
Source
Alfred Tarski (The Establishment of Scientific Semantics [1936], p.407)
Book Ref
Tarski,Alfred: 'Logic, Semantics, Meta-mathematics' [Hackett 1956], p.407
A Reaction
This is evidently what Tarski saw as the most important first fruit of his new semantic theory of truth.
| 10071 | Second Incompleteness: nice theories can't prove their own consistency [Gödel, by Smith,P] |
| 13341 | Using the definition of truth, we can prove theories consistent within sound logics [Tarski] |
| 13540 | A set of formulae is 'inconsistent' when there is no interpretation which can make them all true [Bostock] |
| 13541 | For 'negation-consistent', there is never |-(S)φ and |-(S)¬φ [Bostock] |
| 13542 | A proof-system is 'absolutely consistent' iff we don't have |-(S)φ for every formula [Bostock] |
| 12656 | P-and-Q gets its truth from the truth of P and truth of Q, but consistency isn't like that [Fodor] |
| 9552 | Sentences are consistent if they can all be true; for Frege it is that no contradiction can be deduced [Chihara] |
| 18785 | Consistency is semantic, but non-contradiction is syntactic [Mares] |
| 10119 | Consistency is a purely syntactic property, unlike the semantic property of soundness [George/Velleman] |
| 10126 | A 'consistent' theory cannot contain both a sentence and its negation [George/Velleman] |