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I'm new to modal logic.

We have truth tables for operators in propositional logic. For example, for $\neg P$, we have:

$$\begin{array}{|c|c|} \hline P & \neg P \\ \hline \text{T} & \text{F} \\ \hline \text{F} & \text{T} \\ \hline \end{array}$$

And the same for other operators. Now I'm curious to know if we have truth tables for modal operators in modal logic as well? For example, is it possible to draw a truth table for $\Box P$ or $\Diamond P$ or $\Box(P\to Q)$?

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2 Answers 2

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No. Modal operators are not truth-functional, that is, it is not possible to compute the truth value of a formula with a modal operator from the truth values of the component formulas alone. So it is not possible to write the semantics down in a table format.

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Not with any of the accepted systems of modal logic (The Lewis systems). It may be possible to develop a non-Lewis system of modal logic based on the 3-valued logic of Lukasiewicz, but this approach has generally been abandoned by logicians.

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