What are "Possible Worlds"? A "possible world" is the label given to a concept of how the Universe might have otherwise been. By "The Universe" I mean the collection of all actually existing things in our present reality. The Universe that we exist in is called the "actual world". Every time a writer pens a fictional tale of how it is in some imaginary world - say that of Sherlock Holmes - the writer is describing a "possible world". Every time someone entertains a notion of how reality might otherwise be if something were different, the thinker is describing a "possible world".
The extent to which the definition of "possible worlds" is formalised depends on the purpose that the concept is going to serve. Some philosophers have identified "possible worlds" with "maximal consistent sets of sentences" when they wish to investigate the logical relationships between and consequences of consistent sets of sentences. Some philosophers have identified "possible worlds" as actually existing parallel universes when they wish to explore the epistemological issues of the meaning of truth in various scenarios. Or where they are concerned about the meaning of "exists" if the "possible world" is not actually existing. Some philosophers have identified "possible worlds" as an interpretation of a first order logical semantics when they wish to investigate the logical consequences of such systems of semantics. But the "Standard", "casual", or "man-in-the-street" meaning of "possible worlds" is just an alternate way that things might be, given some specific difference with the way things actually are, and leaving only vaguely conceived the details of just exactly what it means to be "an alternate way that things might be".
When employed in specifically philosophical contexts, a generally accepted constraint on "possible worlds" is the requirement that they be logically consistent. Interestingly, since we cannot prove that our actually existing reality is logically consistent, it leaves open the possibility that our actual world does not qualify as a philosopher's "possible world". However, to prevent this possibility from causing any confusion, it is generally assumed that our actual reality is indeed consistent, and thus a "possible world".
Another logical characteristic that philosophers apply to the concept of "possible worlds" is that a definition, established by us in this actual world, is assumed to apply to all possible worlds. The definition is interpreted as a feature of how we conceive of matters, and is thus applicable to our discussion of any possible world. Hence a bachelor is an unmarried male in all possible worlds, including those where there is no entity that could be male or unmarried. Similarly, the definitional structures of mathematics are considered to hold in all possible worlds, including those containing no consciousness to be aware of mathematical truths.
A proposition is considered "logically possible" if there is at least one logically possible world where the proposition is true. In other words, where one can imagine a logically consistent way in which the Universe might be different than it is, and allow the proposition to be true. A proposition is considered "logically necessary" if it is true in all possible worlds. In other words, where one can not imagine a logically consistent way in which the Universe might be different than it is, without the proposition being true. And a proposition is considered "contingent" if it is possible (true in at least one possible world) but not necessary (not true in all possible worlds).
There are different kinds, or sets, of "possible worlds" depending on the various constraints that are accepted as necessary for the purposes at hand. The largest of these sets is the set of "logically possible worlds" - where anything is possible that is not logically contradictory. If one requires the stipulation that "possible worlds" are, by definition, logically consistent, then this set is the set of "all possible worlds".
Some philosophers maintain that there is a set of "metaphysically possible worlds" which is different from the set of "logically possible worlds". The distinction between the two sets draws upon the difference between possibilities that are not logically contradictory, and possibilities that would violate one's metaphysical assumptions. (For example, having a mind without a brain is not (according to some thinkers) a logical contradiction. Yet a materialist might maintain that such separation is metaphysically impossible.)
Then there is the set of "physically possible worlds" where it is presumed that our current physical laws obtain. This set is much smaller than the set of "logically possible worlds" because it is logically possible that the physical laws that obtain in some possible worlds are much different than the laws that obtain in this actual world. How large the set actually is will depend on one's metaphysical assumptions about the extent of Determinism. A "Super-hard" determinist might maintain, for example, that once the physical laws are specified, then only one possible world is inevitable -- our actual world. Alternatively, a quantum physicist might maintain that quantum indeterminancy dictates that the set of physically possible worlds is some higher order of infinite.
What are "possible worlds" used for? As suggested above, the more common use of "possible worlds" is simply to imagine how things might have otherwise been. The more formal employment of "possible worlds", and the reason for the more formal definitions of "possible worlds" mentioned above, is the philosophical exploration of the meaning and consequences of the logical concepts of "necessity", "contingency", and "possibility", the meaning of contra-factual propositions, and the meaning of truth for contra-factual propositions. Some examples may be helpful here.
What does it mean to ask "Do griffons exist?" (where a "griffon" is a fabulous beast with the head and wings of an eagle and the body of a lion)? In terms of "possible worlds" the question is asking whether there is at least one logically possible world where at least one griffon exists. (Without worrying about what it really means for something to "exist" in a possible world.) Since griffons do not exist in our actual world, we can conclude that it is not necessary that griffons exist, and that if they do exist on some possible world, their existence would be contingent. Further, from what we know of eagles, lions, and the processes of evolution, we can conclude that griffons are not physically possible. However, the concept of a griffon is not logically self-contradictory, so it is logically possible that griffons exist on some logically possible world where the physical laws are different. Such as the magical world of Harry Potter, in which the griffon Buckbeak plays an important role in the events related in "The Prisoner of Azkaban".
What does it mean to ask the question "Can a circle be square?" In terms of "possible worlds" it is asking whether there is at least one possible world where a circle (defined as a plane figure of the locus of all points equidistant from another point, the centre) is a square (defined as a plane figure having four equal sides and four equal angles). Notice that the definitions of the two terms both specify that they are figures of the plane. Clearly, given the definitions of the terms being employed, a single plane figure cannot satisfy both definitions. Now, since the mathematical character of the plane is a matter of how "The plane" is defined, the definition will port to all possible worlds. So it is not logically possible for a circle to be square. Notice in this example, that the words being used in the question are assumed to refer to their normal English definitions. It is a separate question to ask whether, in some possible, world there might be a different language where the relationships between written (or spoken) symbols and the concepts to which they refer are such that a "circle" is "Square". Clearly, the concept of such a different language is not logically contradictory. So it is logically possible that in some possible world where there is such a language, a "circle" could be "Square". But this sentence does not express the same proposition as would the same words in English.
What does the proposition "If I had gone to the theatre yesterday, I would have seen the accident." Actually mean? And how would one determine the truth of such a contra-factual proposition? In terms of "possible worlds" the proposition is stating that there is at least one possible world, identical to this actual world in all respects except that in this possible world, I went to the theatre yesterday. It is further stating that in all those possible worlds where I went to the theatre, I also saw the accident. Note that it is not describing the set of possible world where I went to the theatre and saw the accident. It is saying that in all those possible worlds, identical to this one in all respects except that I went to the theatre yesterday, I saw the accident. One would examine whether this contra-factual statement is true by examining these possible worlds to determine whether it would require any additional unspecified differences from our actual world in order to permit me to see the accident. Since it is assumed that all of these possible worlds are identical to this actual world, it is taken as given that all of the usual physical laws apply. It is also taken as given that all of the other events that occurred yesterday in this actual world also occurred in these possible worlds. If, with these givens, there are no additional differences required (other than my attendance at the theatre), then the statement is considered true. Otherwise, it is considered false.
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