In what way, if any, do possible worlds ground the notions of
necessity and possibility?
The concepts of necessity and possibility are inter-definable. Thus we have
“necessarily P” is equivalent to “not possible that not-P”, and
“possibly P” is equivalent to “not necessarily not-P”.
One can take either as basic, and define the other in terms of the first. Generally, it is necessity that is taken as basic, and possibility defined in terms of necessity.
In his 1973 work “Counterfactuals“, David Lewis provided a rigorous definition of “possible worlds”. His main proposal was that possible worlds should be the proper basis from which to understand counterfactual conditionals. Along the way, however, he defined “necessity” as follows:
“For any possible world (i) and sentence P, the sentence “necessarily P” is true at the world (i) if and only if for every world (j), such that (j) is accessible from (i), P is true at (j).”
[I have altered the symbols for reasons of reproductive simplicity]
David Lewis, Counterfactuals, 1973, pg 5.
Or, in terms of “Spheres of Accessibility” which Lewis defines quite technically –
“A sentence ‘necessarily P’ is true at the world (i) if and only if P is true throughout the sphere of accessibility S(i) around (i).”
[symbols altered for reasons of reproductive simplicity]
David Lewis, Counterfactuals, 1973, pg 7.
Since that seminal work, the language of “possible worlds” has become the standard way of speaking about necessity and possibility. Commonly, in the literature, you see possible worlds used to define (logical) necessity and possibility as follows –
“necessarily P” is true iff P is true in all (logically) possible worlds, and
“possibly P” is true iff P is true in at least one (logically) possible world.
Certainly, the possible worlds semantics has made it much easier to discuss necessity and possibility. But I think, however, that this easy simplicity masks a number of key underlying assumptions. Therefore, to regard possible worlds language as the proper way to ground our understanding of the concept of necessity is getting the cart before the horse. For a number of reasons.
Firstly, possible worlds are not inspectable. Whether they are abstract concepts as many maintain, or real existents as Lewis maintains, they are not causally connected with this, the actual world. We too easily forget that this means that we cannot really inspect any other possible world than our current world. So the question of whether some statement is true in some possible world, or in all possible worlds, is not a question of empirical investigation. Whether the statement is necessarily or possibly true must be determined on the basis of factors that are known to us here in this actual world. What is in fact true in any other world can have no actual bearing on the answer, regardless of how we choose to express the results of our determination. Given that all we have epistemic access to is the actual world, how could we know anything about these other possible worlds or their contents? How are we supposed to know what is possible or not possible, what is universal and what is not universal? How could we even guess? Obviously, we can’t. What we wish to claim is true or not in one, some, or all possible worlds is so purely by stipulation. Whatever the answer, the grounding for our concepts of necessity and possibility has to be right before us in this actual world.
Secondly, the problem of trans-world identity is not definitively resolved. When trying to decide whether a statement like “Possibly, Aristotle had six fingers” is true or not, the possible worlds semantics would propose (in principle at least) that we examine every possible world to see if in some world where Aristotle exists, Aristotle had six fingers. Obviously, from the above argument, we do not do this examination in anything like an empirical sense. Rather we do it conceptually, on the basis of what we stipulate about these possible worlds. But just what is it that (conceptually) picks out an instance of “Aristotle” in any other possible world than this actual world. Some philosophers (Kripke in particular) have suggested that trans-world identity is established only by identical origins. So any existent in some possible world that had the same parents (same sperm and egg? same genetic code?) would be the “Same” Aristotle. But this runs into difficulties. One is that the hypothesis leaves problematic the question of the identicality of the origins – and so forth with infinite circularity. Another is that usually we would intuitively judge the statement “possibly, Aristotle had six fingers” as true. But if the trans-world counterpart had the same parents, and human developmental processes are the same, then contrary to our intuitive judgement, it would in fact not be possible. On the other hand, if human developmental processes are allowed to freely flex, then it would also be equally possible that Aristotle was very much like a wombat. But this is a possibility we intuitively judge as false, on the assumption that any wombat-like thing originating from the “Same” origin as Aristotle, would not be the same “Aristotle” that we refer to in this world. Kripkean rigid designation not withstanding, we intuitively withhold “Sameness” from something that is so different from the Aristotle in this world, that he/it would be unrecognizable in another. Alternatively, if we allow the human development processes to flex only enough to render false the statement “possibly, Aristotle was very much like a wombat” while allowing the sentence “possibly Aristotle had six fingers” to be true, then we are letting our intuitive responses govern the sense of trans-world identity. And if that is the case, then clearly our judgements of necessity and possibility are prior to, and not dependent on the possible worlds semantics.
Thirdly, necessity is a matter of linguistic definition, not a matter of how things are. The statement “necessarily, all bachelors are male” is true because of how we define our terms, independently of how the rest of the world may be. Likewise, “necessarily, 2 plus 2 equals 4” is true by definition. Statements which are necessary, which are true in all possible worlds, are true because that is the way that we have defined the concepts we are using. The question of whether “Aristotle is necessarily human” is true or not depends on how we have defined the words “Aristotle” and “human”. If we conceive of “Aristotle” as a collection of descriptions that includes the proposition that he was a human being, then it is true that “Aristotle was necessarily human”. But if we conceive of “Aristotle” as a cluster of descriptions that includes say “The greatest Greek philosopher”, but does not include that he was a human being, then we allow it to be possible that Aristotle was not human (possibly he was very much like a wombat). Determining whether any statement “necessarily P” is true or not is thus a matter of investigating the meaning of the words employed in P (factors which exists in this world) to see if P is “True by definition”. This is certainly not a trivial exercise, and it must sometimes be resolved by fiat. Rational people, with equivalent information, can reach different intuitive judgements on the truth of necessity and possibility statements, when the intentional contents of their respective concepts differ. If I regard Aristotle as essentially human, while you regard Aristotle as essentially a philosopher, then we can reach different conclusions as to whether “possibly, Aristotle was very much like a wombat”.
Fourthly, we simply do not think in terms of possible worlds. Necessity and possibility are notions that are in common use in the general populations. To be sure, philosophy has formalized the notions to be much more specific that those in general use. But the intent of that formalization is not to create new and exotic concepts, but rather to retain as grounds for that formalization the manner of common usage of the notions. The significance of this source of grounding is that, with the exception of trained philosophers, no one thinks of necessity and possibility in terms of possible worlds. Although possible worlds semantics has proved very useful in talking about necessity and possibility in any non-specific sense, in terms of common usage, whatever is necessary is so because it simply has to be that way to make sense of the words that we use. And whatever is possible, is so because the meaning of the words we use imply that it is not impossible.
And finally, we had the concepts of necessity and possibility long before the theory of possible worlds arose. Indeed, the theory of possible worlds is parasitical on the notion of possibility. Hence the notion of “possible” worlds. And hence Lewis’s notion of relative “closeness” of possible worlds. We may not have been able to talk about necessity and possibility as easily, prior to the advent of the possible world semantics. But the literature, prior to David Lewis and Robert Stalnaker, is full of discussions of necessity de re and necessity de dicto. Discussions that demonstrate a sound conception of necessity and possibility independent of any possible worlds language.
It is therefore obvious that possible worlds semantics is parasitic on the notions of necessity and possibility. They are not the grounds on which these concepts are based.