A possible world is a complete and consistent way the world is or could have been. They are widely used as a mathematical tool in logic, philosophy, and linguistics in order to provide a semantics for intensional and modal logic. Their metaphysical status has been a subject of controversy in philosophy, with modal realists such as David Lewis arguing that they are literally existing alternate realities, and others such as Robert Stalnaker arguing that they are not.
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 Obama believes 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 (read as "possibly P") 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
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. (He referred to these theories as "ersatz modal realism" which try to get the benefits of possible worlds semantics "on the cheap".)
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 (i.e. mathematical) 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 (for example: "Richard Nixon became president in 1969").
- False propositions are those that are false in the actual world (for example: "Ronald Reagan became president in 1969").
- Possible propositions are those that are true in at least one possible world (for example: "Hubert Humphrey became president in 1969"). (Humphrey did run for president in 1968, and thus could have been elected.) This includes propositions which are necessarily true, in the sense below.
- Impossible propositions (or necessarily false propositions) are those that are true in no possible world (for example: "Melissa and Toby are taller than each other at the same time").
- Necessarily true propositions (often simply called necessary propositions) are those that are true in all possible worlds (for example: "2 + 2 = 4"; "all bachelors are unmarried").
- Contingent propositions are those that are true in some possible worlds and false in others (for example: "Richard Nixon became president in 1969" is contingently true and "Hubert Humphrey became president in 1969" is contingently false).
Possible worlds play a central role in many other debates in philosophy. These include debates about the Zombie Argument, and physicalism and supervenience in the philosophy of mind. Many debates in the philosophy of religion have been reawakened by the use of possible worlds.
History of the concept
The idea of possible worlds is most commonly attributed to Gottfried Leibniz, who spoke of possible worlds as ideas in the mind of God 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 (The Incoherence of the Philosophers), Averroes (The Incoherence of the Incoherence), Fakhr al-Din al-Razi (Matalib al-'Aliya) and John Duns Scotus.
- Standard translation, an embedding of modal logics into first-order logic which captures their possible world semantics
- Modal fictionalism
- Impossible world
- "Formal Semantics: Origins, Issues, Early Impact". Baltic International Yearbook of Cognition, Logic and Communication. This Proceeding of the Symposium for Cognition, Logic and Communication. 6. 2011.
- Lewis, David (1973). Counterfactuals. John Wiley & Sons.
- Lewis, David (1986). On the plurality of worlds. Wiley-Blackwell.
- W. V. O. Quine, "Proportional Objects" in Ontological Relativity and Other Essays', 1969, pp.140-147
- Stalnaker, Robert (1976). "Possible worlds". Noûs. 10 (1).
- Kripke, Saul (1972). Naming and necessity. Harvard University Press.
- See "A Priori and A Posteriori" (author: Jason S. Baehr), at Internet Encyclopedia of Philosophy: "A necessary proposition is one the truth value of which remains constant across all possible worlds. Thus a necessarily true proposition is one that is true in every possible world, and a necessarily false proposition is one that is false in every possible world. By contrast, the truth value of contingent propositions is not fixed across all possible worlds: for any contingent proposition, there is at least one possible world in which it is true and at least one possible world in which it is false." Accessed 7 July 2012.
- Arthur Schopenhauer, "De Welt als Wille and Vorstellung," supplement to the 4th book "Von der Nichtigkeit und dem Leiden des Lebens" p. 2222, see also R.B. Haldane and J. Kemp's translation "On the Vanity and Suffering of Life" pp 395-6
- "Nor could we doubt that, if God had created many worlds, they would not be as true in all of them as in this one. Thus those who could examine sufficiently the consequences of these truths and of our rules, could be able to discover effects by their causes, and, to explain myself in the language of the schools, they could have a priori demonstrations of everything that could be produced in this new world." -The World, Chapter VII
- Taneli Kukkonen (2000), "Possible Worlds in the Tahâfut al-Falâsifa: Al-Ghazâlî on Creation and Contingency", Journal of the History of Philosophy, 38 (4): 479–502, doi:10.1353/hph.2005.0033
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