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This is a concept in logic and analytic philosophy that is designed to work with modal statements. Modal statements contain words that express modality (modal operators) – for example, the modality of truth (possible, impossible, necessary…), time (someday, sometime, always…), etc.

In semantics, there is a distinction between extensionality and intensionality. The extension of a name is its referent (the object it refers to), the extension of a predicate is the set of things it applies to, and the extension of a statement is its truth value. The intensity can be called the” meaning ” of an expression, indicating its extension.

Logic is extensional if the truth of each of its statements is determined only by its form and the extensionals of its constituent components-statements, predicates, and names. Therefore, in extensional logic, a set of valid substitution principles applies : if two expressions have the same extension, then they are interchangeable in any statement, without changing the truth value of the statement. A famous example of Frege: the ancient Greeks called the planet Venus ” Morning Star “and” Evening Star”, considering it two different celestial bodies. In this case, the extension is the planet Venus, and the Morning and Evening star are two different intensities. In the statement “The Morning Star has risen”, you can freely replace the Morning Star with the Evening Star and vice versa – the meaning may change, but the truth value does not.

Extensionality is a well-known feature of classical logics. In intensional logic, however, to determine the truth value of certain statements, it is not enough to know their form and the extensionals of their components. Sometimes the truth value of a statement is determined by a “meaning” that is not formally expressed in the framework of extensional logic. As a result, one or more substitution principles are no longer valid. Modal logic is inten-sional. In order to more strictly and fully comprehend and systematize ideas about the properties of modal logic, a convenient tool was needed, and the idea of possible worlds became such a tool.

In any possible world, a statement can be true or false. The semantics of possible worlds considers modal operators as quantifiers of possible worlds, that is, they list in which possible worlds the modal statement in question is true and in which it is false. For example, a true statement is true in the actual (our) world. A false statement is false in the actual world. A necessary true statement (analytic truth/tautology) is true in all possible worlds. A possible statement is true in at least one possible world. An impossible statement (for example, one that contains a logical contradiction) is not true in any world. Etc.

There are different points of view on the ontological status of possible worlds, that is, whether they really exist. Proponents of concretism (in particular, Lewis himself) believed that yes-the set of possible worlds is a set of physical contexts, concrete worlds. Abstract artists (Stalnaker et al.) hold the view that possible worlds can be considered as potential states of the actual world. Kripke believes that possible worlds are purely logical constructs. There is also combinatorialism – a rather complex metaphysical system, akin to Russell's logical atomism and the metaphysics of early Wittgenstein, but this is too voluminous and not very important for the question.

For a more detailed acquaintance with modal logic and the semantics of possible worlds, it is better to refer to the textbook, there is a lot of formalism there.