Reconstructing the conditional's truth table from natural deduction Can the conditional's truth table be reconstructed using the rules from natural deduction?
 A: Three elementary reminders might help to clarify the situation:


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*The standard natural deduction rules for the conditional (modus ponens and conditional proof) are constant across many logics -- including not only classical logic but e.g. intuitionist logic. Intuitionist logic is not truth-functional. So that's enough to remind us that the standard natural deduction rules for the conditional by themselves do not determine how to interpret the conditional semantically: it can all depend on what else is going on in the natural deduction system.

*If we add the standard natural deduction rules for the conditional into the context of classical rules for  the other connectives, in particular the classical negation rules, then of course the usual soundness and completeness result with respect to two-valued semantics determines that the connectives can be interpreted truth-functionally -- and if the logic is interpreted truth-functionally, the conditional governed by those natural deduction rules must have the usual two truth-table.

*But note that there is also a sense in which even the full classical rules still don't fix the semantics -- for they can interpreted in Boolean Algebras other than the usual two-valued one. That's why I said "if interpreted truth-functionally [with the usual to truth-values!],  the conditional must have the usual [four line] truth-table".

