Does modal logic have truth tables?

Question

The title says it all. Does modal logic have truth tables.

I do not presuppose any system of modal logic.

Answer 1

NO, but there are decision procedures for modal logics.

See : James Garson Modal Logic for Philosophers (2nd ed 2013), page 187,

or : Brian Chellas, Modal Logic : An Introduction (1980), page 62.

Answer 2

Only for S5 - modeled by Lrst or universal access Kripke models - truth-tabular methods are available. The pioneers are H. Leonard and G. Massey who invented the partial truth table approach.

The partial-truth-tables method (Massey) works for S5 only. It might be relevant also to consider the Djugundi result, by which modal logics as such do not have characteristic matrices. If you are wondering what kind of species modal logic is -- look into First Order Logic (FOL) and take the fragment of FOL that remains invariant under a mathematical technique known as bisimulation.

Answer 3

Yes. From the primitive notions (t, ~, v, <>, p,q,r. etc.) we can establish truth tables for modal logic, eg. S5.

t = empirical truth. f = empirical falsity. T = logical truth. F = logical falsity.

1. t primitive
2. f =def ~t
3. T =def (t v f).
4. F =def ~T.
5. <>t =def T.
6. <>f =def T.
7. <>T =def T.
8. <>F =def F.
9. []p =def ~<>~p.
10. []T <-> T.
11. []t <-> F.
12. []f <-> F.

13. []F <-> F.

14. []p -> <>p.

proof:

[]T -> <>T, []t -> <>t, []f -> <>f, []F -> <>F.

T -> T, F -> T, F -> F, F -> F.

T, T, T, T.

[]p -> <>p, is logically true for all values of p, therefore it is tautologous.

All of the axiom of propositional logic and the axioms of modal logic (S5) are logically true!

Answer 4

It has been established that most of the Lewis systems of modal logic, which are by far the most prevalent, cannot be reduced to systems with a finite number of truth values. This does not necessarily prohibit some other kind of system of modal logic from employing three, four, or some other number of truth values.