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Full Idea
A plausible account is that the syntactic notion of consequence is for positive results, that some form of argument is valid; the semantic notion is required for negative results, that some argument is invalid, because a counterexample can be found.
Gist of Idea
Syntactic consequence is positive, for validity; semantic version is negative, with counterexamples
Source
Michael Dummett (The Justification of Deduction [1973], p.292)
Book Ref
Dummett,Michael: 'Truth and Other Enigmas' [Duckworth 1978], p.292
A Reaction
This rings true for the two strategies of demonstration, the first by following the rules in steps, the second by using your imagination (or a tableau) to think up problems.
19058 | Syntactic consequence is positive, for validity; semantic version is negative, with counterexamples [Dummett] |
9718 | Validity is either semantic (what preserves truth), or proof-theoretic (following procedures) [Enderton] |
10259 | The two standard explanations of consequence are semantic (in models) and deductive [Shapiro] |
10691 | Logical consequence needs either proofs, or absence of counterexamples [Beall/Restall] |
13253 | There are several different consequence relations [Beall/Restall] |
10753 | Logical consequence is intuitively semantic, and captured by model theory [Rossberg] |