A place in a sentence is extensional if words with the same extension can always be substituted into it without changing the truth-value of the whole sentence. (That definition is a little too crude in about three ways, but bear with me.) A place in a sentence is intensional, in one sense of “intensional”, when words that necessarily share the same extension can always be substituted into it without changing the truth-value of the whole sentence.
It has become increasingly clear since the 1970s that we need to carve meanings more finely than by “intensions” in the sense associated with the specification above. Call the sorts of intensions employed, for example, by Richard Montague possible worlds intensions. Handling belief clauses by insisting that anyone who believes something believes everything necessarily equivalent to it has always caused problems. Once we accept that names are rigid designators, allowing their substitution in all sorts of representational and psychological contexts causes trouble: the Sheriff of Nottingham can be hunting for Robin Hood without hunting for Robin of Locksley, or so it seems.
There seem to be places outside our psychological talk that require hyperintensionality. Talk of entailment in the sense of logical consequence, for example: it does not logically follow from apples being red that all bachelors are unmarried, let alone that water is H2O, even though it does follow that either apples are red or apples are not red. Use of counter-possible conditionals is another example: two conditionals can have necessarily false antecedents but differ in truth-value. Talk about moral obligation and permission seems to be hyperintensional, as anyone struggling with substituting logical equivalents in the scope of deontic operators may have seen. I’m just back from a conference in Colorado where people were insisting that “in virtue of”, “because”, and other explanatory expressions were hyperintensional. (Benjamin Schnieder, Gideon Rosen and Kit Fine were three in particular.) Once you look around you see quite a bit of hyperintensionality.
There’s a piece of rhetoric I associate with Richard Sylvan about this. He was fond of suggesting that there would be a move from using possible-worlds intensions to using hyperintensional resources that would parallel the move made from extensionalism to possible-worlds intensionalism. In the nineteen-sixties, the big goal was to be able to do philosophy of language while treating language extensionally: think of Davidson’s project in particular, though Quine was also a big booster of the extensionalist program. I guess it was typical of that project to assign extensions to categories of expressions, and then have some syncatogramatic expressions that operated on extensions to yield other extensions. (E.g. “all” did not get an extension, but (All x)(Fx) operated on the extension of “F” to yield a sentence-extension, i.e. a truth-value)
There are still people trying to carry out that extensionalist project, but it came under increasingly severe attack since the early 1970s. (And maybe earlier: I think Carnap might be an important precursor here, along with Prior, and perhaps many others). The extensional programme was not very satisfying in its treatment of propositional attitude reports, entailment, normative discourse such as the use of “ought”, and a number of other areas. But the star witness against the extensional programme was modal vocabulary. Treating “necessarily” extensionally does not get you very far, and after Saul Kripke popularised possible-worlds semantics for “necessarily”, the floodgates started to open. Richard Montague and David Lewis were among the vanguard of those arguing for a systematic, intensional treatment of natural language, arguing that it handled all sorts of constructions that extensional treatments faced serious difficulty with.
The intensions that Montague and Lewis relied upon were set-theoretic constructions out of possible worlds and possible individuals. (Not just sets of possibilia or functions from possibilia to possiblia, but also sets of those sets, functions from those functions to other functions, etc. etc.) The Montague project of trying to handle all of language with these possible-worlds intensions is alive and well today: I take Robert Stalnaker to be one of its prominent contemporary philosophical defenders, though I haven’t scrutinised his recent work to see if any weakening has happened.
But I think that project is doomed. There is too much work that needs to be done that requires hyperintensional distinctions, and those trying to hold the line that everything can be done with possible-worlds intensions will look as outdated in thirty years as the extensionalists look to the intensionalists today.
Of course, even if we decided we wanted to do more justice to hyperintensional phenomena than standard possible-worlds semantics, we have several options about how to go on from here. The response that is perhaps closest to the standard possible-worlds tradition is to let the semantic value of a piece of language be a pair of a possible-worlds-intension plus some kind of constituent tree, that serves as a logical form or otherwise conveys information about the internal linguistic structure of the expression. Alternatively, we could let the semantic value of a complex expression be a tree whose nodes are possible-worlds intensions: Lewis discusses this way of going, for example, in OTPW p 49-50.
Another response that is close to the possible-worlds tradition is to use impossible worlds as well as possible ones. Since things that do not vary across possible worlds can vary across impossible worlds, impossible worlds give us finer-grained distinctions. If we allow logically impossible worlds, we can even get the effect of places in sentences where substitution of logical equivalents fail, since for example the worlds where (p or not-p) obtain need not be the ones where (q or not-q) obtain. I take it that semantics using situations instead of worlds is often a close cousin of this.
More radical responses to hyperintensionality include moving to an algebraic semantics, such as the sort advocated by George Bealer. Even these can be seen as successors to the possible-worlds tradition, since the structures of the algebras are often inspired by the structural relationships possible-worlds intensions stand in to each other. No doubt philosophers will come up with other approaches too - some revert to talking about Fregean senses and functions on them, though whether this is much more than a cosmetic difference from algebraic approaches I’m not sure.
Why does this matter for metaphysics? Well, one immediate reason it matters is that the metaphysics of language had better be able to cope with hyperintensionality and hyperintensions. One place that disputes in the philosophy of language often spill over is into the metaphysics of meaning, of truth (or at least truth-conditions), of propositions and so on.
A connected reason is that respect for hyperintensionality might go along with more warmth towards hyperintensional entities. We may be less likely to smile on the demand that properties that necessarily have the same instances are identical, for example. This in turn may motivate rejecting the picture of properties as sets of their actual and possible instances. Indeed, set theory might be of less use in metaphysics in general once we want to individuate things hyperintensionally.
There are other ways the hyperintensional turn could affect metaphysics. It might make us more sympathetic to impossible worlds, for example: I’ve argued elsewhere that counter-possible conditionals give us a good reason to postulate impossible worlds. It might make us think that some relational predicates are not associated with relations, or maybe are associated with finer-grained relata than they appear to be associated with: see Carrie Jenkins’s post about grounding. Modal analyses of hyperintensional pieces of language seem unappealing, since modal analyses are normally only intensional not hyperintensional. I could go on.
So, metaphysicians, join the hyperintensional revolution! You have nothing to lose but your coarse grains!