Thursday, March 26, 2009

The Age of Hyperintensionality

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!


  1. A very good survey; I fully agree that the age of hyperintensionality is coming.
    Yet I would like to draw attention to another way hyperintensional semantics can be realized. It is a procedural / computational turn, the first advocate of which was Pavel Tichy in the 1970s and 1980s. Currently Tichy's followers (Materna, Jespersen, Duzi, and many others) develop a realist procedural semantics, which is at variance with denotational semantics (such as model theory) and pragmatist semantics (such as inferentialism). The leading idea is that to analyze a piece of language is to assign an algorithmically structured procedure to it as its meaning. To this end we use Tichý’s Transparent Intensional Logic (TIL), explicating senses in terms of procedures known as TIL constructions. Our vision of semantics comes with a top-down approach, going from procedures to their products, which may be lower-order constructions or non-constructions such as possible-world intensions and extensions (individuals, truth-values, sets, etc). Whereas denotational semantics assigns only procedural products to expressions as their references and pragmatist semantics eschews reference in favour of socially constituted rules, our procedural semantics has expressions refer to constructions, which are higher-order hyperintensions.

  2. - and proof-theoretic semantics shares a procedural conception of meaning, so there's common ground between that approach and realist procedural semantics. What's more, the latter defies Dummett's stifling claim that semantic realism must be truth-conditional. - Whoever joins Nolan's hyperintensional revolution will be exceeding the bounds of set theory. That's none too soon, anyway, since set theory has proved a bit of a straitjacket when doing ontology. Hyperintensionalists, unite!

  3. Pavel Materna said...
    The first project of hyperintensionality was formulated as early as 1968 and 1969 in Tichý´s articles "Sense and Procedure" and "Intension in Terms of Turing machines", both reprinted in "Pavel Tichý´s Collected Papers in Logic and Philosophy", Prague - Dunedin 2004. Tichý´s monograph "The Foundations of Frege´s Logic" (deGruyter 1988) coíntains exact definitions of constructions (not in the intuitionist sense)and of Transparent Intensional Logic (TIL). Since then many articles and two monographs were published. Just now a monograph by Duží, Jespersen and Materna ("Procedural Semantics for Hyperintensional Logic") has been prepared for submission.

  4. Marie, Pavel: I've been meaning to read some of Tichy's work for quite some time, and your posts inspired me to go and read your "Constructions" paper. It looks like Tichy had some interesting ideas about how to handle hyperintensionality, and quite early as well. I'm glad to hear the Tichy program is flourishing! Reading around some more about the history made me wish I'd mentioned Max Cresswell in the early history of hyperintensionality as well, above.

    Bjorn, you're right, proof-theoretic semantic traditions are another approach people interested in hyperintensionality(including me) should be more aware of.

  5. Daniel, let us know whether you need some brand-new (2009) material on Tichy's program about to come out. The 'Constructions' paper may already be a bit dated by now. Where Carnap had intensional isomorphism and Church had synonymous isomorphism, TIL has procedural isomorphism, which slots in somewhere between Church's Alternatives (0) and (1). - We agree with the constructivists (taking Martin-Loef's theory to be the most sophisticated around) that meaning is an itinerario mentis (i.e., semantics is susceptible to epistemic constraints of some sort), but we don't have a notion of judgement to govern when what (hyper-) proposition may correctly be claimed to be true. In general, I would say that Martin-Loef has a more developed theory of mathematical language, where Tichy'/TIL has a more developed theory of empirical/natural language. It's not clear to me what their proofs/proof-objects would be in the empirical case (though Aarne Ranta and Goeran Sundholm have made some suggestions). - When I talk to colleagues of mine these days - whether they're into probability theory, action theory or whatnot - I certainly sense a growing awareness of the need to go hyperintensional. The possible-world paradigm has served us extremely well, but it's a spent force and now the choice is, in my view, between beefing up one's semantics by adding hyperintensions to the catelogue or keeping one's semantics meagre and inflating one's pragmatics.

  6. I would say that Tichý´s (and our) procedural isomorphism is formally Church´s Alternative 1´ minus beta conversion.
    Otherwise I agree with Bjorn (Marie obviously as well).

  7. Sure, I agree with all of you (at least in these points). "Pragmatics is a garbage basket for lasy semantists". Bjorn is right that our "Constructions" from 2000 are a bit dated now. Yet my students still study mainly this material. But we can provide updated new things, for sure.

  8. There is a nice quote at the end of Semantics in Generative Grammar by Irene Heim and Angelika Kratzer about hyperintensional semantics:

    "The good news is, however, that uncertainty in the area of propositional attitudes does not seem to have a lot of repercussions on the way linguists do semantics every day. A slight change led us from an extensional system to an intensional one. The switch to a hyperintensional system should not be much more eventful. What we have learned about particular extensional and intensional phenomena should be adaptable without too much ado." (p.311)

    So it seems that at least some semanticists agree with the sentiment Daniel attributes to Richard Sylvan in his original post.

  9. Hyperintensionality is one thing. Hyperintensions are another.

    If you think there are cardinality problems with possible worlds, and other intensions, wait until you open the pandora's box of hyperintensions. Violations of Cantor's theorem left and right.

    Church's initial formulations of both Alternative (0) and Alternative (1) were inconsistent because of this. (And something in between wouldn't be any better... the only hope would be to go between (1) and (2), or adopt something altogether different.) Even Tony Anderson's consistent version of Alternative (0) blocks the paradoxes in completely ad hoc ways. You're pretty much stuck with some form of ramification with no real philosophical justification other than "or else it'll be inconsistent".

    And there are other problems too... (some of which will be discussed in my paper forthcoming in Bulletin of Symbolic Logic).

    Tichy's stuff does not seem to me to be an improvement. More like an obscurification, though I'm not an expert on it.

    I tried for a long time to figure out answers to these questions, and have ended up just thinking Russell was right and that we need an atomistic way of dealing with (hper)intensionality without the (hyper)intensions.

  10. Dear Kevin, I just downloaded from your webpage the draft version of your Bulletin paper. Once the paper appears in print, I'd be happy if you'd send me an off-print.

    In general: yes, there are set-theoretic concerns, and they need to be addressed more thoroughly. The first step, though, for us propagating hyperintensions as logical objects in their own right has been to conceive and refine a full-fledged theory of this sort of objects. The next step, obviously, will be to run a consistency check on the theory to see whether we perhaps went too far and where modifications are called for.

    Meanwhile, the TIL book is about to appear:

    Best wishes,

  11. For Kevin Klement,
    I think that if you (not only) open our coming book you will first of all see that no ´obscurification´is connected with Transparent Intensional Logic. True, its philosophy is not based on nominalism and is rather a strongly realist philosophy. Yet is this so dangerous as some post-analytic philosophers are convinced?

    Best regards
    Pavel Materna