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Suppose we have a logic with three or more truth values, how can we deal with semantic entailment then?

From what I understand a set of statements A, semantically entails B if B cannot be false if all in A are true. But this assumes we are dealing in a 2-valued logic.

I'm curious if there's a generalised concept of entailment, possibly even syntactic for an arbitrary logic.

Threnody
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In multi-valued logics there is still an ordering on the truth values. A statement $A$ entails a statement $B$ if $A$ is stronger than $B$ under this ordering (usually written $A \le B$). (Typically in such logics you can't view assumptions as a set, you have to treat them as a multiset, because $A \land A$ might be a stronger statement than $A$. Hence I'm replacing your set of assumption $A$ by a single assumption $A$ that is the conjunction of a multiset of assumptions.) Fuzzy logic, where the truth values are real numbers is a good example of such a logic. Multi-valued logics of this kind often don't admit the law of contraction: $A \to A \land A$ is not admissible.

Rob Arthan
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  • To be fair, it's worth noting that many multi-valued logics (most?) do admit contraction. – Noah Schweber Aug 09 '19 at 00:50
  • I don't know about "most", but I have corrected my unintended statement that contraction is not admissible in any multi-valued logic. Thanks for pointing that out. – Rob Arthan Aug 09 '19 at 15:18