Possible worlds are one of the foundational concepts in modal and intensional logics. Formulas in these logics are used to represent statements about what might be true, what should be true, what Barack Obamabelieves to be true and so forth. To give these statements a formal interpretation, logicians use structures containing possible worlds. For instance, in the relational semantics for classical propositional modal logic, the formula is actually true iff is true at some world which is accessible from the actual world. Possible worlds play a central role in the work of both linguists and philosophers working in formal semantics. Contemporary formal semantics is couched in formal systems rooted in Montague grammar, which is itself built on Richard Montague's intensional logic. Contemporary research in semantics typically uses possible worlds as formal tools without committing to a particular theory of their metaphysical status. The term possible world is retained even by those who attach no metaphysical significance to them.
Philosophical issues and applications
Metaphysics
The ontological status of possible worlds has provoked intense debate. David Lewis famously advocated for a position known as modal realism, which holds that possible worlds are real, concrete places which exist in the exact same sense that the actual world exists. On Lewis's account, the actual world is special only in that we live there. This doctrine is called the indexicality of actuality since it can be understood as claiming that the term "actual" is an indexical, like "now" and "here". Lewis gave a variety of arguments for this position. He argued that just as the reality of atoms is demonstrated by their explanatory power in physics, so too are possible worlds justified by their explanatory power in philosophy. He also argued that possible worlds must be real because they are simply "ways things could have been" and nobody doubts that such things exist. Finally, he argued that they could not be reduced to more "ontologically respectable" entities such as maximally consistent sets of propositions without rendering theories of modality circular. Modal realism is controversial. W.V. Quine rejected it as "metaphysically extravagant". Stalnaker responded to Lewis's arguments by pointing out that a way things could have been is not itself a world, but rather a property that such a world can have. Since properties can exist without them applying to any existing objects, there's no reason to conclude that other worlds like ours exist. Another of Stalnaker's arguments attacks Lewis's indexicality theory of actuality. Stalnaker argues that even if the English word "actual" is an indexical, that doesn't mean that other worlds exist. For comparison, one can use the indexical "I" without believing that other people actually exist. Some philosophers instead endorse the view of possible worlds as maximally consistent sets of propositions or descriptions, while others such as Saul Kripke treat them as purely formal devices.
Explicating necessity and possibility
At least since Aristotle, philosophers have been greatly concerned with the logical statuses of propositions, e.g. necessity, contingency, and impossibility. In the twentieth century, possible worlds have been used to explicate these notions. In modal logic, a proposition is understood in terms of the worlds in which it is true and worlds in which it is false. Thus, equivalences like the following have been proposed:
True propositions are those that are true in the actual world.
False propositions are those that are false in the actual world.
Possible propositions are those that are true in at least one possible world. This includes propositions which are necessarily true, in the sense below.
Impossible propositions are those that are true in no possible world.
Necessarily true propositions are those that are true in all possible worlds.
Contingent propositions are those that are true in some possible worlds and false in others.
The idea of possible worlds is most commonly attributed to Gottfried Leibniz, who spoke of possible worlds as ideas in the mind ofGod and used the notion to argue that our actually created world must be "the best of all possible worlds". Arthur Schopenhauer argued that on the contrary our world must be the worst of all possible worlds, because if it were only a little worse it could not continue to exist. Scholars have found implicit earlier traces of the idea of possible worlds in the works of René Descartes, a major influence on Leibniz, Al-Ghazali, Averroes, Fakhr al-Din al-Razi and John Duns Scotus. The modern philosophical use of the notion was pioneered by David Lewis and Saul Kripke.
