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Lewis defined the concept of strict implication to resolve some problems of material implication. Basically, strict implication is defined as [](p --> q), where [] denotes the predicate 'is necessary"

Are there truth tables for strict implicaction?

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    No, the semantics of modal logic is not based on truth tables. See Kripke's semantics for Modal Logic. – Mauro ALLEGRANZA May 27 at 19:27
  • The motivation for strict implication was that truth tables for implication can not adequately express the natural language usage . So, no, strict implications can not be adequately represented by truth tables. – Graham Kemp May 28 at 3:25
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There are no truth tables that model strict implication, for a couple of reasons. As others have pointed out in the comments, Lewis devised strict implication precisely because he felt that material implication (the kind modelled by truth tables) was not a satisfactory account of the English conditional statement 'if, then...'. Further, the conditional statement you have provided is not a propositional statement but a modal one, meaning the semantics of propositional logic are inadequate in capturing this kind of sentence. Therefore, their logical form cannot be understood simply via truth tables, because they include logical operators (in this case the 'necessarily' operator) that are not present in classical logic. If you're interested, SEP has a pretty great article on truth tables and material implication.

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Let me add to the existing answer/comments:

The key term here is truth functional. A truth functional is roughly an operation on sentences such that the truth value of the output is completely determined by the truth value of the input. E.g. "AND" is a truth functional: "X AND Y" is true iff X is true and Y is true.

  • In my experience, although I don't know if this is formally entrenched, non-truth-functional operators - especially unary ones - are called modalities. Certainly modalities are non-truth-functional, so anything involving a modality - like strict implication, say - isn't going to be a truth functional unless somehow the modalities "cancel out" (e.g. "--- is necessary and --- is not-necessary" technically involves a modality but is clearly always false).

The first key point is that lots of sentence modifiers aren't truth-functional. E.g. "--- will be true tomorrow" is only a truth functional if we accept determinism. Similarly, Goldman/Vizzini have argued that the truth value of "--- is inconceivable" isn't a truth functional either since some true things are inconceivable.

The second key point is that truth tables only work for truth functionals. The whole point behind a truth table is that the truth of a compound proposition can be read off from the truth values of the components. So arguments against truth-functionality are inherently arguments against truth-tables.

That said, it is still true that the truth value of a proposition is (basically by definition) determined by all the "relevant facts." So we can develop generalizations of truth tables more appropriate to non-truth-functional situations. But the "naive" approach of the classical truth-table is invalid whenever we step away from truth functionals, and strict implication - indeed, anything involving a modality - does this.

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