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 Reference
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.