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A propositional formula is something like this, A&~B, which uses letters to represent propositions. The letters are called propositional variables.

Compare the following two sets of terminologies related to propositional formulas:

A valuation is an assignment of truth values to propositional variables. A model of a formula is a valuation in which the formula is true. If the formula is true in all valuations, it is a tautology. If one formula x is true in all models of another formula y, then we say that y entails x.

A possible world is such that all propositional variables are either true or false in that world. There is a set of possible worlds in which a given formula is true. If the formula is true in all possible worlds, it is necessarily true. If one formula x is true in all possible worlds in which another formula y is true, then we say that y strictly implies x.

There is a parallel here between valuation and possible world, between model and the set of possible worlds in which a formula is true, between tautology and necessity, and between entailment and strict implication. I have a vague sense that the difference involves the fact that model theory is explicitly extensional and deals with formulas (that is, purely syntactic objects) while possible worlds semantics is intensional and deals generally with propositions (semantic objects), although here, I don't see how you can eliminate talk of formulas.

Can anyone elucidate this relationship?

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  • A side comment that doesn't answer your question, "possible worlds" is quite slippery, and abused routinely. It is too tempting to conjure up "possible worlds" that will come in handy to support whatever argument. What is possible and what is not?
    – Frank
    Feb 20, 2023 at 23:14
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    Valuations are purely combinatorial objects, possible worlds are typically metaphysically loaded and somehow pre-defined. A possible world will valuate all propositional variables, including those not in your formula. For propositional logic, if you are using ersatz approach, i.e. building possible worlds by simply compiling all possible valuations for a set of formulas, there wouldn't be much of a difference when you take the set to have a single formula. In predicate logic it gets hairier.
    – Conifold
    Feb 21, 2023 at 0:24
  • There's a bridge between mathematical logic and natural language semantics by way of Montague grammar.
    – J D
    Feb 21, 2023 at 0:46

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Broadly speaking, I think your sensibility about extensional formulas being contrasted with intensional features that are represented in possible world semantics is correct. The question opens up a bunch of interesting issues.

It is worth mentioning that there are several different conceptions of possible worlds. Also, there are many different kinds of necessity and we can use modal logics and possible worlds with any of them if we choose. The question of the relationship between sentences and possible worlds depends on what we are trying to do. Are we aiming to express the relationship of logical consequence between formulas in a simple artificial language like first-order classical logic? Or a more complex language that has modal extensions? Or are we aiming for something much more ambitious such as capturing the meaning of modal statements in a natural language?

Starting with the first of these. The standard approach to logical consequence is to divide it into two kinds, syntactic or proof-theoretic, and semantic or model-theoretic. The former is concerned with what we can derive using formal methods, and the latter with what lacks a falsifying interpretation. There is fair amount of literature on how to represent these using modal logic, and while there is no general concensus, John Burgess makes a good case for saying that we can represent the syntactic kind using S4, and the semantic kind using S5. Since your first piece of quoted text uses model-theoretic terminology, this indicates that it is no surprise that there is a parallel between logical entailment and strict implication. Logical entailment in simple cases is correctly represented by S5 strict implication.

Moving on to more complex modal logics. There is expressive value in adding an 'actually' operator to modal logic and distinguishing between what is true from the perspective of the actual world and what is true from the perspective of any world. This allows us to distinguish two kinds of necessity. There is a kind that is expressed by saying that P is necessary at the actual world if it holds in all worlds that are accessible to it, and a kind that is expressed by saying that P holds in every world, no matter which one we designate as actual. Davies and Humberstone maintain that the latter provides a better understanding of logical entailment, particularly if we wish to follow Kripke and others in holding to a theory of names as rigid designators and to allow the existence of a priori contingent propositions and a posteriori necessary propositions.

As an interesting corollary, Brian Weatherson shows how we might use the same distinction to create a unified account of the logic of indicative and subjunctive conditionals. The very short and simplified version is that an indicative conditional is concerned with what holds from the perspective of the actual world, while a subjunctive conditional is concerned with what holds if some other world were the actual world. It is a neat idea, though not without its difficulties.

It is fairly common that talk of possible worlds brings with it some metaphysical assumptions. This widens the distinction between logical entailment and necessity, since metaphysical necessity is quite different in many respects. Moving further in this direction takes us into the realm of two-dimensional semantics. There is a great deal of recent literature on this. Lloyd Humberstone provides a survey of some of the approaches to its logic.

John Burgess, "Which Modal Logic is the Right One?", Notre Dame Journal of Formal Logic, Vol. 40, (1999), pp. 81-93.
Martin Davies and Lloyd Humberstone, "Two Notions of Necessity" Philosophical Studies, Vol. 38, (1980), pp. 1-30.
Brian Weatherson, "Indicative and Subjunctive Conditionals", Philosophical Quarterly, Vol. 51, (2001), pp. 200-216.
Lloyd Humberstone. "Two-Dimensional Adventures", Philosophical Studies, Vol. 118, (2004), pp. 17-65.

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