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Contemporary philosophers generally conceive of consequence as necessary truth-preservation. They generally construe this necessity as logical, and operationalize it in substitutional, formal or model-theoretic terms as the absence of a counter-example. A minority tradition allows for grounding truth-preservation also on non-logical necessities, especially on the semantics of extra-logical constants. The present article reviews and updates the author's previous proposals to modify the received conception of consequence so as to require truth-preservation to be non-trivial (i.e. not a mere consequence of a necessarily true implicatum or a necessarily untrue implicans) and to allow variants of the substitutional, formal and model-theoretic realizations of the received conception where the condition underwriting truth-preservation is not purely formal. Indeed, the condition may be contingent rather than necessary. Allowing contingent non-trivial truth-preservation as a consequence relation fits our inferential practices, but turns out to be subject to counter-examples. We are left with an unhappy choice between an overly strict requirement that non-trivial truth-preservation be underwritten by a necessary truth and an overly loose recognition of non-trivial truth-preservation wherever some truth underwrites it. We need to look for a principled intermediate position between these alternatives.
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