What kind of problems is PWS particularly suited that e.g. FOL couldn't address? I'm just wondering when is a good time to use PWS and for what kinds of questions.
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The only approach to modality available in any standard First (or Higher-) Order Logic is the approach that attaches an entire theory with a complete axiomatization for every model of every mood.
To discuss, for instance Aristotle's famous naval battle argument, we would need to define an entire system of terms that encompass all of the rules and considerations about what 'must' and 'might' mean in an Aristotelian context.
You can add symbols for 'qualifiers' like the quantifiers that represent the modality. People do it, and it remains a standard logic with a few extra rules about 'box' and 'diamond' operators. For given moral moods or moods of desire, belief, etc, you can start from the box/diamond rules and extend them to handle the extra parameters. These have first-order instantiations by models.
For any given mood, the odds are this cannot be done at all in only first-order terms. So then this whole consideration would have to be thrown out, if you want to tick to FOL as your entire logical theory. But standard higher-order logic is necessary for a lot of things, including real arithmetic.
But this is not a particularly tractable way of addressing the problem. It is very hard to work out the axioms, in a way that would reflect normal usage, without examples. And examples can really only be had by falling back onto a model theory, involving some ad-hoc variant of the possible worlds perspective.
You need some notions of 'what-if-scenario', and how scenarios are and are not seen to be compatible with the same sets of facts.
Kripke's insight was basically "Why not start from a version that is common, and not ad-hoc?" So he injected one very basic modality (possibility) into set theory itself, and found that most variant modalities can be modeled in the resulting enriched world.