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(a) $P\rightarrow \diamondsuit Q \wedge \square P \wedge \neg Q$

(b) $\neg \square \neg Q, \diamondsuit \neg P \wedge \square (\square \neg P \rightarrow \diamondsuit Q)$

I want to state for each of these formulae, a and b, whether it should be an intuitively satisfiable proposition of alethic modal logic. Satisfiable means that it is true at some possible world of some models and is often proved by Kripke models.

Since I do not have a Kripke model M for this I thought I would solve it by using truth values, ie replace p and q with either true or false, as I know this works in formulas such as $P \rightarrow Q\wedge P$. But now I think about whether it is really possible to do this in terms of modal logic where we use the modal operators Necessarily and Possibly?

Does anyone know if I can use truth values to solve a and b, which are expressions within alethic modal logic? If so, how should the modal operators be handled?

Troposphere
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kabin
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  • If $P$ is not true, then the first statement is satisfied. Now, can you come up with a model for $(a)$? – Ethan Aug 10 '21 at 22:12

1 Answers1

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You definitely cannot stick to two truth values for modal formulas.

The entire reason why Kripke models were invented in the first place is that working with two (or indeed any finite number of) truth values cannot produce a semantic notion of validity that matches what can be proved using common sets of axioms for the modal operators.

I do not have a Kripke model M for this

You're supposed to come up with a model yourself if you claim that the formula you're looking at is satisfiable (respectively, not valid).

By the way, Kripke models come in a variety of flavors, depending on the properties accessibility relation is required to have. You can't speak about satisfiability or validity before you decide what those requirements are.

There are standard correspondences between particular axiom sets and assumptions about the accessibility relation. Your textbook ought to contain a discussion of them; otherwise see e.g. https://en.wikipedia.org/wiki/Kripke_semantics#Common_modal_axiom_schemata

On the other hand, if the exercise you're solving explicitly asks about "intuivitely satisfiable", then it's plausible that the exercise actually doesn't want you to muck around with Kripke models or any other formalized semantics at all. It might simply be asking for what your, umm, intuitive conception about what the words "necessary", "possible", "impossible", and so forth ought to mean.

That's especially likely because the content of saying "alethic" logic in particular is that $\Box$ is going to represent "neccessary" rather than one of the other concepts (e.g., "known" or "provable" or "morally required") that one can use modal logic to represent. There you be little reason to specify alethic logic in particular if you're just going to follow technical rules that have already been fixed.

The point of such an exercise could be to prepare you for considering which of the several non-equivalent axiom systems for modal operators yield a result that matches your intuition -- or as background for a classroom discussion to drive home the point that different people have different intuitive understandings of the words, which is why we need actual definitions to make clear what we're talking about.

Troposphere
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