Category Archives: Metaphysics

Regimentation (x-post).

Here’s something you frequently hear said about ontological commitment. First, that to determine the ontological commitments of some sentence S, one must look not at S, but at a regimentation or paraphrase of S, S*. Second (very roughly), you determine the ontological commitments of S by looking at what existential claims follow from S*.

Leave aside the second step of this. What I’m perplexed about is how people are thinking about the first step. Here’s one way to express the confusion. We’re asked about the sentence S, but to determine the ontological commitments we look at features of some quite different sentence S*. But what makes us think that looking at S* is a good way of finding out about what’s required of the world for S to be true?

Reaction (1). The regimentation may be constrained so as to make the relevance of S* transparent. Silly example: regimentation could be required to be null, i.e. every sentence has to be “regimented” as itself. No mystery there. Less silly example: the regimentation might be required to preserve meaning, or truth-conditions, or something similar. If that’s the case then one could plausibly argue that the OC’s of S and S* coincide, and looking at the OC’s of S* is a good way of figuring out what the OC’s of S is.

(The famous “symmetry” objections are likely to kick in here; i.e. if certain existential statements follow from S but not from S*, and what we know is that S and S* have the same OC’s, why take it that S* reveals those OC’s better than S?—so for example if S is “prime numbers exist” and S* is a nominalistic paraphrase, we have to say something about whether S* shows that S is innocent of OC to prime numbers, or whether S shows that S* is in a hidden way committed to prime numbers).

Obviously this isn’t plausibly taken as Quine view—the appeal to synonymy is totally unQuinean (moreover in Word and Object, he’s pretty explicit that the regimentation relationship is constrained by whether S* can play the same theoretical role as we initially thought S played—and that’ll allow for lots of paraphrases where the sentences don’t even have the appearance of being truth-conditionally equivalent).

Reaction (2). Adopt a certain general account of the nature of language. In particular, adopt a deflationism about truth and reference. Roughly: T- and R-schemes are in effect introduced into the object language as defining a disquotational truth-predicate. Then note that a truth-predicate so introduced will struggle to explain the predications of truth for sentences not in one’s home language. So appeal to translation, and let the word “true” apply to a sentence in a non-home language iff that sentence translates to some sentence of the home language that is true in the disquotational sense. Truth for non-home languages is then the product of translation and disquotational truth. (We can take the “home language” for present purposes to be each person’s idiolect).

I think from this perspective the regimentation steps in the Quinean characterization of ontological commitment have an obvious place. Suppose I’m a nominalist, and refuse to speak of numbers. But the mathematicians go around saying things like “prime numbers exist”. Do I have to say that what they say is untrue (am I going to go up to them and tell them this?) Well, they’re not speaking my idiolect; so according to the deflationary conception under consideration, what I need to do is figure out whether there sentences translate to something that’s deflationarily true in my idiolect. And if I translate them according to a paraphrase on which their sentences pair with something that is “nominalistically acceptable”, then it’ll turn out that I can call what they say true.

This way of construing the regimentation step of ontological commitment identifies it with the translation step of the translation-disquotation treatment of truth sketched above. So obviously what sorts of constraints we have on translation will transfer directly to constraints on regimentation. One *could* appeal to a notion of truth-conditional equivalence to ground the notion of translatability—and so get back to a conception whereby synonymy (or something close to it) was central to our analysis of language.

It’s in the Quinean spirit to take translatability to stand free of such notions (to make an intuitive case for separation here, one might, for example, that synonymy should be an equivalence relation, whereas translatability is plausibly non-transitive). There are several options. Quine I guess focuses on preservation of patterns of assent and dissent to translated pairs; Field appeals to his projectivist treatment of norms and takes “good translation” as something to be explained in projective terms. No doubt there are other ways to go.

This way of defending the regimentation step in treatments of ontological commitment turns essentially on deflationism about truth; and more than that, on a non-universal part of the deflationary project: the appeal to translation as a way to extend usage of the truth-predicate to non-home languages. If one has some non-translation story about how this should go (and there are some reasons for wanting one, to do with applying “true” to languages whose expressive power outstrips that of one’s own) then the grounding for the regimentation step falls away.

So the Quinean regimentation-involving treatment of ontological commitment makes perfect sense within a Quinean translation-involving treatment of language in general. But I can’t imagine that people who buy into to the received view of ontological commitment really mean to be taking a stance on deflationism vs. its rivals; or about the exact implementation of deflationism.

Of course, regimentation or translatability (in a more Quinean, preservation-of-theoretical-role sense, rather than a synonymy-sense) can still be significant for debates about ontological commitments. One might think that arithmetic was ontologically committing, but the existence of some nominalistic paraphrase that was suited to play the same theoretical role gave one some reassurance that one doesn’t *have* to use the committing language, and maybe overall these kind of relationships will undermine the case for believing in dubious entities—not because ordinary talk isn’t committed to them, but because for theoretical purposes talk needn’t be committed to them. But unlike the earlier role for regimentation, this isn’t a “hermeneutic” result. E.g. on the Quinean way of doing things, some non-home sentence “there are prime numbers” can be true, despite there being no numbers—just because the best translation of the quoted sentence translates it to something other than the home sentence “there are prime numbers”. This kind of flexibility is apparently lost if you ditch the Quinean use of regimentation.

Aristotelian indeterminacy and partial beliefs

I’ve just finished a first draft of the second paper of my research leave—title the same as this post. There’s a few different ways to think about this material, but since I hadn’t posted for a while I thought I’d write up something about how it connects with/arises from some earlier concerns of mine.

The paper I’m working on ends up with arguments against standard “Aristotelian” accounts of the open future, and standard supervaluational accounts of vague survival. But one starting point was an abstract question in the philosophy of logic: in what sense is standard supervaluationism supposed to be revisionary? So let’s start there.

The basic result—allegedly—is that while all classical tautologies are supervaluational tautologies, certain classical rules of inference (such as reductio, proof by cases, conditional proof, etc) fail in the supervaluational setting.

Now I’ve argued previously that one might plausibly evade even this basic form of revisionism (while sticking to the “global” consequence relation, which preserves traditional connections between logical consequence and truth-preservation). But I don’t think it’s crazy to think that global supervaluational consequence is in this sense revisionary. I just think that it requires an often-unacknowledged premise about what should count as a logical constant (in particular, whether “Definitely” counts as one). So for now let’s suppose that there are genuine counterexamples to conditional proof and the rest.

The standard move at this point is to declare this revisionism a problem for supervaluationists. Conditional proof, argument by cases: all these are theoretical descriptions of widespread, sensible and entrenched modes of reasoning. It is objectionably revisionary to give them up.

Of course some philosophers quite like logical revisionism, and would want to face-down the accusation that there’s anything wrong with such revisionism directly. But there’s a more subtle response available. One can admit that the letter of conditional proof, etc are given up, but the pieces of reasoning we normally call “instances of conditional proof” are all covered by supervaluationally valid inference principles. So there’s no piece of inferential practice that’s thrown into doubt by the revisionism of supervaluational consequence: it seems that all that happens is that the theoretical representation of that practice has to take a slightly more subtle form than one might except (but still quite a neat and elegant one).

One thing I mention in that earlier paper but don’t go into is a different way of drawing out consequences of logical revisionism. Forget inferential practice and the like. Another way in which logic connects with the rest of philosophy is in connection to probability (in the sense of rational credences, or Williamson’s epistemic probabilities, or whatever). As I sketched in a previous post, so long as you accept a basic probability-logic constraint, which says that the probability of a tautology should be 1, and the probability of a contradiction should be 0, then the revisionary supervaluational setting quickly forces you to a non-classical theory of probability: one that allows disjunctions to have probability 1 where each disjunct has probability 0. (Maybe we shouldn’t call such a thing “probability”: I take it that’s terminological).

Folk like Hartry Field have argued completely independently of this connection to Supervaluationism that this is the right and necessary way to handle probabilities in the context of indeterminacy. I’ve heard others say, and argue, that we want something closer to classicism (maybe tweaked to allow sets of probability functions, etc). And there are Dutch Book arguments to consider in favour of the classical setting (though I think the responses to these from the perspective of non-classical probabilities are quite convincing).

I’ve got the feeling the debate is at a stand-off, at least at this level of generality. I’m particularly unmoved by people swapping intuitions about degrees of belief it is appropriate to have in borderline cases of vague predicates, and the like (NB: I don’t think that Field ever argues from intuition like this, but others do). Sometimes introspection suggests intriguing things (for example, Schiffer makes the interesting suggestion that one’s degree of belief in a conjunction of two vague propositions is typically matches one’s degree of belief in the propositions themselves). But I can’t see any real dialectical force here. In my own case, I don’t have robust intuitions about these cases. And if I’m to go on testimonial evidence on others intuitions, it’s just too unclear what people are reporting on for me to feel comfortable taking their word for it. I’m worried, for example, they might just be reporting the phenomenological level of confidence they have in the proposition in question: surely that needn’t coincide with one’s degree of belief in the proposition (thinking of an exam you are highly nervous about, but are fairly certain you will pass… your behaviour may well manifest a high degree of belief, even in the absence of phenomenological trappings of confidence). In paradigm cases of indeterminacy, it’s hard to see how to do better than this.

However, I think in application to particular debates we might be able to make much more progress. Let us suppose that the topic for the day is the open future, construed, minimally, as the claim that while there are definite facts about the past and present, the future is indefinite.

Might we model this indefiniteness supervaluationally? Something like this idea (with possible futures playing the role of precisifications) is pretty widespread, perhaps orthodoxy (among friends of the open future). It’s a feature of MacFarlane’s relativistic take on the open future, for example. Even though he’s not a straightforward supervaluationist, he still has truth-value gaps, and he still treats them in a recognizably supervaluational-style way.

The link between supervaluational consequence and the revisionionary behaviour of partial beliefs should now kick in. For if you know with certainty that some P is neither true nor false, we can argue that you should invest no credence at all in P (or in its negation). Likewise, in a framework of evidential probabilities, P gets no evidential probability at all (nor does its negation).

But think what this says in the context of the open future. It’s open which way this fair coin lands: it could be heads, it could be tails. On the “Aristotelian” truth-value conception of this openness, we can know that “the coin will land heads” is gappy. So we should have credence 0 in it, and none of our evidence supports it.

But that’s just silly. This is pretty much a paradigmatic case where we know what partial belief we have and should have in the coin landing heads: one half. And our evidence gives exactly that too. No amount of fancy footwork and messing around with the technicalities of Dempster-Shafer theory leads to a sensible story here, as far as I can see. It’s just plainly the wrong result. (One doesn’t improve matters very much by relaxing the assumptions, e.g. taking the degree of belief in a failure of bivalence in such cases to fall short of one: you can still argue for a clearly incorrect degree of belief in the heads-proposition).

Where does that leave us? Well, you might reject the logic-probability link (I think that’d be a bad idea). Or you might try to argue that supervaluational consequence isn’t revisionary in any sense (I sketched one line of thought in support of this in the paper cited). You might give up on it being indeterminate which way the coin will land—i.e. deny the open future, a reasonably popular option. My own favoured reaction, in moods when I’m feeling sympathetic to the open future, is to go for a treatment of metaphysical indeterminacy where bivalence can continue to hold—my colleague Elizabeth Barnes has been advocating such a framework for a while, and it’s taken a long time for me to come round.

All of these reactions will concede the broader point—that at least in this case, we’ve got an independent grip on what the probabilities should be, and that gives us traction against the Supervaluationist.

I think there are other cases where we can find similar grounds for rejecting the structure of partial beliefs/evidential probabilities that supervaluational logic forces upon us. One is simply a case where empirical data on folk judgements has been collected—in connection with indicative conditions. I talk about this in some other work in progress here. Another which I talk about in the current paper, and which I’m particularly interested in, concerns cases of indeterminate survival. The considerations here are much more involved than in indeterminacy we find in connection to the open future or conditionals. But I think the case against the sort of partial beliefs supervaluationism induces can be made out.

All these results turn on very local issues. None, so far as see, generalizes to the case of paradigmatic borderline cases of baldness and the rest. I think that makes the arguments even more interesting: potentially, they can serve as a kind of diagnostic: this style of theory of indeterminacy is suitable over here; that theory over there. That’s a useful thing to have in one’s toolkit.

Structured propositions and metasemantics

Here is the final post (for the time being) on structured propositions. As promised, this is to be an account of the truth-conditions of structured propositions, presupposing a certain reasonably contentious take on the metaphysics of linguistic representation (metasemantics). It’s going to be compatible with the view that structured propositions are nothing but certain n-tuples: lists of their components. (See earlier posts if you’re getting concerned about other factors, e.g. the potential arbitriness in the choice of which n-tuples are to be identified with the structured proposition that Dummett is a philosopher.)

Here’s a very natural way of thinking of what the relation between *sentences* and truth-conditions are, on a structured propositions picture. It’s that metaphysically, the relation of “S having truth-conditions C” breaks down into two more fundamental relations: “S denoting struc prop p” and “struc prop p having truth-conditions C”. The thought is something like: primarily, sentences express thoughts (=struc propositions), and thoughts themselves are the sorts of things that have intrinsic/essential representational properties. Derivatively, sentences are true or false of situations, by expressing thoughts that are true or false of those situations. As I say, it’s a natural picture.

In the previous posting, I’ve been talking as though this direction-of-explanation was ok, and that the truth-conditions of structured propositions should have explanatory priority over the truth-conditions of sentences, so we get the neat separation into the contingent feature of linguistic representation (which struc prop a sentence latches onto) and the necessary feature (what the TCs are, given the struc prop expressed).

The way I want to think of things, something like the reverse holds. Here’s the way I think of the metaphysics of linguistic representation. In the beginning, there were patterns of assent and dissent. Assent to certain sentences is systematically associated with certain states of the world (coarse-grained propositions, if you like) perhaps by conventions of truthfulness and trust (cf. Lewis’s “Language and Languages”). What it is for expressions E in a communal language to have semantic value V is for E to be paired with V under the optimally eligible semantic theory fitting with that association of sentences with coarse-grained propositions.

That’s a lot to take in all at one go, but it’s basically the picture of linguistic representation as fixed by considerations of charity/usage and eligibility/naturalness that lots of people at the moment seem to find appealing. The most striking feature—which it shares with other members of the “radical interpretation” approach to metasemantics—is that rather than starting from the referential properties of lexical items like names and predicates, it depicts linguistic content as fixed holistically by how well it meshes with patterns of usage. (There’s lots to say here to unpack these metaphors, and work out what sort of metaphysical story of representation is being appealed to: that’s something I went into quite a bit in my thesis—my take on it is that it’s something close to a fictionalist proposal).

This metasemantics, I think, should be neutral between various semantic frameworks for generating the truth conditions. With a bit of tweaking, you can fit in a Davidsonian T-theoretic semantic theory into this picture (as suggested by, um… Davidson). Someone who likes interpretational semantics but isn’t a fan of structured propositions might take the semantic values of names to be objects, and the semantic values of sentences to be coarse-grained propositions, and say that it’s these properties that get fixed via best semantic theory of the patterns of assent and dissent (that’s Lewis’s take).

However, if you think that to adequately account for the complexities of natural language you need a more sophisticated, structured proposition, theory, this story also allows for it. The meaning-fixing semantic theory assign objects to names, and structured propositions to sentences, together with a clause specifying how the structured propositions are to be paired up with coarse-grained propositions. Without the second part of the story, we’d end up with an association between sentences and structured propositions, but we wouldn’t make connection with the patterns of assent and dissent if these take the form of associations of sentences with *coarse grained* propositions (as on Lewis’s convention-based story). So on this radical interpretation story where the targetted semantic theories take a struc prop form, we get a simultaneous fix on *both* the denotation relation between sentences and struc props, and the relation between struc props and coarse-grained truth-conditions.

Let’s indulge in a bit of “big-picture” metaphor-ing. It’d be misleading to think of this overall story as the analysis of sentential truth-conditions into a prior, and independently understood, notion of the truth-conditions of structured propositions, just as it’s wrong on the radical interpretation picture to think of sentential content as “analyzed in terms of” a prior, and independently understood, notion of subsentential reference. Relative to the position sketched, it’s more illuminating to think of the pairing of structured and coarse-grained propositions as playing a purely instrumental role in smoothing the theory of the representational features of language. It’s language which is the “genuine” representational phenomenon in the vicinity: the truth-conditional features attributed to struc propositions are a mere byproduct.

Again speaking metaphorically, it’s not that sentences get to have truth-conditions in a merely derivative sense. Rather, structured propositions have truth-conditions in a merely derivative sense: the structured proposition has truth-conditions C if it is paired with C under the optimal overall theory of linguistic representation.

For all we’ve said, it may turn out that the same assignment of truth-conditions to set-theoretic expressions will always be optimal, no matter which language is in play. If so, then it might be that there’s a sense in which structured propositions have “absolute” truth-conditions, not relative to this or that language. But, realistically, one’d expect some indeterminacy in what struc props play the role (recall the Benacerraf point King makes, and the equally fitness of [a,F] and [F,a] to play that “that a is F” role). And it’s not immediately clear why the choice to go one way for one natural language should constrain way this element is deployed in another language. So it’s at least prima facie open that it’s not definitely the case that the same structured propositions, with the same TCs, are used in the semantics of both French and English.

It’s entirely in the spirit of the current proposal that we think of we identify [a,F] with the structured proposition that a is F only relative to a given natural language, and that this creature only has the truth-conditions it does relative to that language. This is all of a piece with the thought that the structured proposition’s role is instrumental to the theory of linguistic representation, and not self-standing.

Ok. So with all this on the table, I’m going to return to read the book that prompted all this, and try to figure out whether there’s a theoretical need for structured propositions with representational properties richer than those attributed by the view just sketched.

[update: interestingly, it turns out that King’s book doesn’t give the representational properties of propositions explanatory priority over the representational properties of sentences. His view is that the proposition that Dummett thinks is (very crudely, and suppressing details) the fact that in some actual language there is a sentence of (thus-and-such a structure) of which the first element is a word referring to Dummett and the second element is a predicate expressing thinking. So clearly semantic properties of words are going to be prior to the representational properties of propositions, since those semantic properties are components of the proposition. But more than this, from what I can make out, King’s thought is that if there was a time where humans spoke a language without attitude-ascriptions and the like, then sentences would have truth-conditions, and the proposition-like facts would be “hanging around” them, but the proposition-like facts wouldn’t have any representational role. Once we start making attitude ascriptions, we implicitly treat the proposition-like structure as if it had the same TCs as sentences, and (by something like a charity/eligibility story) the “propositional relation” element acquires semantic significance and the proposition-like structure gets to have truth-conditions for the first time.

That’s very close to the overall package I’m sketching above. What’s significant dialectically, perhaps, is that this story can explain TCs for all sorts of apparently non-semantic entities, like sets. So I’m thinking it really might be the Benacerraf point that’s bearing the weight in ruling out set-theoretic entities as struc propns—as explained previously, I don’t go along with *that*.]

Structured propositions and truth conditions.

In the previous post, I talked about the view of structured propositions as lists, or n-tuples, and the Benacerraf objections against it. So now I’m moving on to a different sort of worry. Here’s King expressing it:

“A final difficulty for the view that propositions are ordered n-tuples concerns the mystery of how or why on that view they have truth conditions. On any definition of ordered n-tuples we are considering, they are just sets. Presumably, many sets have no truth conditions (eg. The set of natural numbers). But then why do certain sets, certain ordered n-tuples, have truth-conditions? Since not all sets have them, there should be some explanation of why certain sets do have them. It is very hard to see what this explanation could be.”

I feel the force of something in this vicinity, but I’m not sure how to capture the worry. In particular, I’m not sure whether the it’s right to think of structured propositions’ having truth-conditions as a particularly “deep” fact over which there is mystery in the way King suggests. To get what I’m after here, it’s probably best simply to lay out a putative account of the truth-conditions of structured propositions, and just to think about how we’d formulate the explanatory challenge.

Suppose, for example, one put forward the following sort of theory:

(i) The structured proposition that Dummett is a philosopher = [Dummett, being a philosopher].
(ii) [Dummett, being a philosopher] stands in the T relation to w, iff Dummett is a philosopher according to w.
(iii) bearing the T-relation to w=being true at w


(i) For all a, F, the structured proposition that a is F = [a, F]
(ii) For all individuals a, and properties F, [a, F] stands in the T relation to w iff a instantiates F according to w.
(iii) bearing the T-relation to w=being true at w

In a full generality, I guess we’d semantically ascend for an analogue of (i), and give a systematic account of what structured propositions are associated with which English sentences (presumably a contingent matter). For (ii), we’d give a specification (which there’s no reason to make relative to any contingent facts) about which ordered n-tuples stand in the T-relation to which worlds. (iii) can stay as it is.

The naïve theorist may then claim that (ii) and (iii) amount to a reductive account of what it is for a structured proposition to have truth-conditions. Why does [1,2] not have any truth-conditions, but [Dummett, being a philosopher] does? Because the story about what it is for an ordered pair to stand in the T-relation to a given world, just doesn’t return an answer where the second component isn’t a property. This seems like a totally cheap and nasty response, I’ll admit. But what’s wrong with it? If that’s what truth-conditions for structured propositions are, then what’s left to explain? It doesn’t seem that there is any mystery over (ii): this can be treated as a reductive definition of the new term “bearing the T-relation”. Are there somehow explanatory challenges facing someone who endorses the property-identity (iii)? Quite generally, I don’t see how identities could be the sort of thing that need explaining.

(Of course, you might semantically ascend and get a decent explanatory challenge: why should “having truth conditions” refer to the T-relation. But I don’t really see any in principle problem with addressing this sort of challenge in the usual ways: just by pointing to the fact that the T-relation is a reasonably natural candidate satisfying platitudes associated with truth-condition talk.)

I’m not being willfully obstructive here: I’m genuinely interested in what the dialectic should be at this point. I’ve got a few ideas about things one might say to bring out what’s wrong with the flat-footed response to King’s challenge. But none of them persuades me.

Some options:

(a)Earlier, we ended up claiming that it was indefinite what sets structured propositions were identical with. But now, we’ve given a definition of truth-conditions that is committal on this front. For example, [F,a] was supposed to be a candidate precisification of the proposition that a is F. But (ii) won’t assign it truth conditions, since the second component isn’t a property but an individual.

Reply: just as it was indefinite what the structured propositions were, it is indefinite what sets have truth-conditions, and what specification of those truth-conditions is. The two kinds of indefiniteness are “penumbrally connected”. On a precisification on which the prop that a is F=[a,F], then the clause holds as above; but on a precisification on which that a is F=[F,a], a slightly twisted version of the clause will hold. But no matter how we precisify structured proposition-talk, there will be a clause defining the truth-conditions for the entities that we end up identifying with structured propositions.

(b) You can’t just offer definitional clauses or “what it is” claims and think you’ve evaded all explanatory duties! What would we think of a philosopher of mind who put forward a reductive account whereby pain-qualia were by definition just some characteristics of C-fibre firing, and then smugly claimed to have no explanatory obligations left.

Reply: one presupposition of the above is that clauses like (ii) “do the job” of truth-conditions for structured propositions, i.e. there won’t be a structured proposition (by the lights of (i)) whose assigned “truth-conditions” (by the lights of (ii)) go wrong. So whatever else happens, the T-relation (defined via (ii)) and the truth-at relation we’re interested in have a sort of constant covariation (and, unlike the attempt to use a clause like (ii) to define truth-conditions for sentences, we won’t get into trouble when we vary the language use and the like across worlds, so the constant covariation is modally robust). The equivalent assumption in the mind case is that pain qualia and the candidate aspect of C-fibre firing are necessarily constantly correlated. Under those circumstances, many would think we would be entitled to identify pain qualia and the physicalistic underpinning. Another way of putting this: worries about the putative “explanatory gap” between pain-qualia and physical states are often argued to manifest themselves in a merely contingent correlation between the former and the latter. And that’d mean that any attempt to claim that pain qualia just are thus-and-such physical state would be objectionable on the grounds that pain qualia and the physical state come apart in other possible worlds.
In the case of the truth-conditions of structured propositions, nothing like this seems in the offing. So I don’t see a parody of the methodology recommended here. Maybe there is some residual objection lurking: but if so, I want to hear it spelled out.

(c)Truth-conditions aren’t the sort of thing that you can just define up as you please for the special case of structured propositions. Representational properties are the sort of things possessed by structural propositions, token sentences (spoken or written) of natural language, tokens of mentalese, pictures and the rest. If truth-conditions were just the T-relation defined by clause (ii), then sentences of mentalese and English, pictures etc couldn’t have truth-conditions. Reductio.

Reply: it’s not clear at all that sentences and pictures “have truth-conditions” in the same sense as do structured propositions. It fits very naturally with the structured-proposition picture to think of sentences standing in some “denotation” relation to a structured proposition, through which may be said to derivatively have truth-conditions. What we mean when we say that ‘S has truth conditions C’ is that S denotes some structured proposition p and p has truth-conditions C, in the sense defined above. For linguistic representation, at least, it’s fairly plausible that structured propositions can act as a one-stop-shop for truth-conditions.

Pictures are a trickier case. Presumably they can represent situations accurately or non-accurately, and so it might be worth theorizing about them by associating them with a coarse-grained proposition (the set of worlds in which they represent accurately). But presumably, in a painting that represents Napolean’s defeat at waterloo, there doesn’t need to be separable elements corresponding to Napolean, Waterloo, and being defeated at, which’d make for a neat association of the picture with a structured proposition, in the way that sentences are neatly associated with such things. Absent some kind of denotation relation between pictures and structured propositions, it’s not so clear whether we can derivatively define truth-conditions for pictures as the compound of the denotation relation and the truth-condition relation for structured propositions.

None of this does anything to suggest that we can’t give an ok story about pairing pictures with (e.g.) coarse-grained propositions. It’s just that the relation between structured propositions and coarse-grained propositions (=truth conditions) and the relation between pictures and coarse-grained propositions can’t be the same one, on this account, and nor is even obvious how the two are related (unlike e.g. the sentence/structured proposition case).
So one thing that may cause trouble for the view I’m sketching is if we have both the following: (A) there is a unified representation relation, such that pictures/sentences/structured propositions stand in same (or at least, intimately related) representation relations to C. (B) there’s no story about pictorial (and other) representations that routes via structured propositions, and so no hope of a unified account of representation given (ii)+(iii).

The problem here is that I don’t feel terribly uncomfortable denying (A) and (B). But I can imagine debate on this point, so at least here I see some hope of making progress.

Having said all this in defence of (ii), I think there are other ways for the naïve, simple set-theoretic account of structured propositions to defend itself that don’t look quite so flat-footed. But the ways I’m thinking of depend on some rather more controversial metasemantic theses, so I’ll split that off into a separate post. It’d be nice to find out what’s wrong with this, the most basic and flat-footed response I can think of.

Structured propositions and Benacerraf

I’ve recently been reading Jeff King’s book on structured propositions. It’s really good, as you would expect. There’s one thing that’s bothering me though: I can’t quite get my head around what’s wrong with the simplest, most naïve account of the nature of propositions. (Disclaimer: this might all turn out to be very simple-minded to those in the know. I’d be happy to get pointers to the literature (hey, maybe it’ll be to bits of Jeff’s book I haven’t got to yet…)

The first thing you encounter when people start talking about structured propositions is notation like [Dummett, being a philosopher]. This is supposed to stand for the proposition that Dummett is a philosopher, and highlights the fact that (on the Russellian view) Dummett and the property of being a philosopher are components of the proposition. The big question is supposed to be: what do the brackets and comma represent? What sort of compound object is the proposition? In what sense does it have Dummett and being a philosopher as components? (If you prefer a structured intension view, so be it: then you’ll have a similar beast with the individual concept of Dummett and the worlds-intension associated with “is a philosopher” as ‘constituents’. I’ll stick with the Russellian view for illustrative purposes.)

For purposes of modelling propositions, people often interpret the commas as brackets as the ordered n-tuples of standard set theory. The simplest, most naïve interpretation of what structured propositions are, is simply to identify them as n-tuples. What’s the structured proposition itself? It’s a certain kind of set. What sense are Dummett and the property of being a philosopher constituents of the structured proposition that Dummett is a philosopher? They’re elements of the transitive closure of the relevant set.

So all that is nice and familiar. So what’s the problem? In his ch 1. (and, in passing, in the SEP article here) King mentions two concerns. In this post, I’ll just set the scene by talking about the first. It’s a version of a famous Benacerraf worry, which anyone with some familiarity with the philosophy of maths will have come across (King explicitly makes the comparison). The original Benacerraf puzzle is something like this: suppose that the only abstract things are set like, and whatever else they may be, the referents of arithmetical terms should be abstract. Then numerals will stand for some set or other. But there are all sorts of things that behave like the natural numbers within set theory: the constructions known as the (finite) Zermelo ordinals (null, {null}, {{null}}, {{{null}}}…) and the (finite) von Neumann ordinals (null, {null}, {null,{null}}…) are just two. So there’s no non-arbitrary theory of which sets the natural numbers are.

The phenomenon crops up all over the place. Think of ordered n-tuples themselves. Famously, within an ontology of unordered sets, you can define up things that behave like ordered pairs: either [a,b]={{a},{a,b}} or {{{a},null},{{b}}}. (For details see It appears there’s no non-arbitrary reason to prefer a theory that ‘reduces’ ordered to unordered pairs one way or the other.

Likewise, says King, there looks to be no non-arbitrary choice of set-theoretic representation of structured propositions (not even if we spot ourselves ordered sets as primitive to avoid the familiar ordered-pair worries). Sure, we *could* associate the words “the proposition that Dummett is a philosopher” with the ordered pair [Dummett, being a philosopher]. But we could also associate it with the set [being a philosopher, Dummett] (and choices multiply when we get to more complex structured propositions).

One reaction to the Benacerrafian challenge is to take it to be a decisive objection to an ontological story about numbers, ordered pairs or whatever that allows only unordered sets as a basic mathematical ontology. My own feeling is (and this is not uncommon, I think) that this would be an overreaction. More strongly: no argument that I’ve seen from the Benacerraf phenomenon to this ontological conclusion seems to me to be terribly persuasive.

What we should admit, rather, is that if natural numbers or ordered pairs are sets, it’ll be indefinite which sets they are. So, for example, [a,b]={{a},{a,b}} will be neither definitely true nor definitely false (unless we simply stipulatively define the [,] notation one way or another rather than treating it as pre-theoretically understood). Indefiniteness is pervasive in natural language—everyone needs a story about how it works. And the idea is that whatever that story should be, it should be applied here. Maybe some theories of indefiniteness will make these sort of identifications problematic. But prominent theories like Supervaluationism and Epistemicism have neat and apparently smooth theories of what it we’re saying when we call that identity indefinite: for the supervaluationist, it (may) mean that “[a,b]” refers to {{a},{a,b}} on one but not all precisifications of our set-theoretic language. For the epistemicist, it means that (for certain specific principled reasons) we can’t know that the identity claim is false. The epistemicist will also maintains there’s a fact of the matter about which identity statement connecting ordered and unordered sets is true. And there’ll be some residual arbitrariness here (though we’ll probably have to semantically ascend to find it)—but if there is arbitriness, it’s the sort of thing we’re independently committed to to deal with the indefiniteness rife throughout our language. If you’re a supervaluationist, then you won’t admit there’s any arbitriness: (standardly) the identity statement is neither true nor false, so our theory won’t be committed to “making the choice”.

If that’s the right way to respond to the general Benacerraf challenge, it’s the obvious thing to say in response to the version of that puzzle that arises for the Benacerraf case. And this sort of generalization of the indefiniteness maneuver to philosophical analysis is pretty familiar, it’s part of the standard machinery of the Lewisian hoardes. Very roughly, the programme goes: figure out what you want the Fs to do, Ramsify away terms for Fs and you get a way to fix where the Fs are amidst the things you believe in: they are whatever satisfy the open sentence that you’re left with. Where there are multiple, equally good satisfiers, then deploy the indefiniteness maneuver.

I’m not so worried on this front, for what I take to be pretty routine reasons. But there’s a second challenge King raises for the simple, naïve theory of structured propositions, which I think is trickier. More on this anon.

Two problems of the many.

Here’s a paradigmatic problem of the many (Geach and Unger are the usual sources cited, but I’m not claiming this to be exactly the version they use.) Let’s take a moulting cat. There are many hairs that are neither clearly attached, nor clearly unattached to the main body of the cat. Let’s enumerate them 1—1000. Then we might consider the material objects which are the masses of cat-arranged matter that include half of the thousand hairs, and exclude to the other half. There are many ways to choose the half that’s included. So by this recipe we get many many distinct masses of cat-arranged matter, differing only over hairs. The various pieces of cat-arranged matter change their properties over time in very much the way that cats do: they are now in a sitting-shape, now in a standing-shape, now in a lapping-milk shape, now in an emitting-meows configuration. They each seem to have everything intrinsically required for being a cat.

If you’re inclined to think (and I am) that a cat is a material object identical to some piece of cat-arranged matter, then the problem of the many arises: which of the various distinct pieces of cat-arranged matters is the cat? Various answers have been suggested. Some of the most obvious (though not necessarily the most sensible) are: (i) nihilism: none of the cat-candidates are cats. (ii) brutalism: exactly one is a cat, and there is a brute fact of the matter which it is; (iii) vague cat: exactly one is a cat, and it’s a vague matter which it is; (iii) manyism: lots of the cat-candidates are cats.

(By the way, (ii) and (iii) may not be incompatible, if you’re an epistemicist about vagueness. And those who are fans of many-valued logics for vagueness should have a think about whether they can really support (iii). Consider the best candidates to be a cat, c1….c1000. Suppose these are each cats to an equal degree. Then “one of c1…c1000 is a cat” will standardly have a degree of truth equal to the disjunction=the maximum of the disjuncts=the degree of truth of “c1 is a cat”. And the degree of truth of the conjunction: “all of c1…c1000 is a cat” will standardly have a degree of truth equal to the conjunction=the minimum of the conjuncts=the degree of truth of “c1 is a cat”. So to the extent that the (determinately distinct) best candidates aren’t all cats, to exactly that extent there’s no cat among them (and since we chose the best candidates, we won’t get a higher degree of truth for “the cat is present” by including extra disjuncts. Conclusion: if you’re tempted by response (iii) to the problem of the many, you’ve got strong reason not to go for many-valued logic. [Edit (see comments): this needs qualification. I think you’ve reason not to go for many-valued logics that endorse the (fairly standard, but not undeniable) max/min treatment of disjunction/conjunction; and in which the many values are linearly arranged].)

What I’d really like to emphasize is the above leaves open the following question: Is there a super-cat-candidate, i.e. a piece of cat-arranged matter of which every other cat-candidate is a proper part? Take the Tibbles case above, and suppose that the candidates only differ over hairs. Then a potential super-cat-candidate would be the piece of matter that’s maximally generous: that includes all the 1000 not-clearly-unattached hairs. If this particular fusion isn’t genuinely a cat-candidate, then it’s open that if you arrange the cat-candidates by which is a part of which, you’ll end up with multiple maximal cat-candidates none of which is a part of the other. Perhaps they each contain 999 hairs, but differ amongst themselves which hair they don’t include.

If there is a super-cat-candidate, let’s say the problem of the many is of type-1, and if there’s no super-cat-candidate, let’s say that the problem of the many is of type-2.

My guess is that our description of cases like Tibbles leaves is simply underspecified as to whether it’s of type-1 or type-2. But I certainly don’t see any principled reason to think that the actual cases of the POM we find around us are always of type-1. There’s certainly no a priori guarantee that the sort of criterion that rules in some things as parts of a cat won’t also dismiss other things as non-parts. So for example, perhaps we can rank candidates for degrees of integration: some unintegrated parts are ok, but there’s some cut-off where an object is just too unintegrated to count as a candidate. One cat-candidate includes some borderline-attached skin cells, and is to that extent unintegrated. Another cat-candidate includes some borderline-attached teeth, and is to that extent unintegrated. But plausibly the fusion that includes both skin cells and teeth is less integrated: enough to disqualify it from being a cat-candidate. It’s hard to know how to argue the case further without going deeply into feline biology, but I hope you get the sense of why type-2 POM need to be dealt with.

Now, one response to the standard POM is to appeal to the “maximality” allegedly built into various predicates (like “rock”, “cat”, “conscious” etc): things that are duplicates of rocks, but which are surrounded by extra rocky stuff, become merely parts of rocks (and so forth). There are presumably intrinisic duplicates of rocks embedded as tiny parts at the centre of large boulders: but there’s no intuitive pressure to count them as rocks. Likewise a cat might survive after it’s limbs are destroyed by a vengeful deity, but it’s unintuitive to think of the duplicate head-and-torso part of Tibbles as itself a cat-candidate. So there’s some reasons independently of paradigmatic problem of the many scenarios to think of “cat” and “rock” etc as maximal. (For more discussion of maximality, see Ted Sider’s various papers on the topic).

If we’ve got a type-1 problem of the many, then one might think that the maximality of “cat” or “rock” or whatever gives a principled answer to our original question: the super-cat-candidate (/super-rock-candidate) is the one uniquely qualified to be the cat (/rock). For we’ve then got an explanation for why all the others, though intrinsically qualified just like cats, aren’t cats: being a cat is a maximal property, and all the rival cat-candidates are parts of the one true cat in the vicinity.

But the type-2 problem of the many really isn’t addressed by maximality as such. There’s no unique super-cat-candidate in this setup, rather a range of co-maximal ones. So maximality won’t save our bacon here.

The difference between the two cases is important when we consider other things. For example, in the light of the (fairly widely accepted) maximality of “house” and “cat” and “rock” and the like, few would say that any duplicate of a house must be a house (even setting aside extrinsicality due to social setting). But there’s an obvious fall back position, which is floating around the literature: that any duplicate of a house must be a (proper or improper) part of a house (holding fixed social setting etc). That is, any house possesses the property of being part of a house intrinsically (so long as we hold fixed social setting etc). And the same goes for cat: at least holding fixed biological origin, it’s plausible that any cat is intrinsically at least part of a cat, and any rock is intrinsically at least part of a rock.

These claims aren’t threatened by maximality. But appealing to them in a type-2 problem of the many gets us an argument directly for response (iv): manyism. For plausibly if you took a duplicate of one of the co-maximal cat candidates, T, while eliminating from the scene those bits of matter that are not part of T but are part of one of the other co-maximal cat candidates, then you get something T* that’s (determinately) a cat. And so, any duplicate of T* must be at least part of a cat. And since T is a duplicate of T*, T must be at least part of a cat. But T isn’t proper part of anything that’s even a cat-candidate. So T must itself be a cat.

So the type-2 POM is harder to resolve than the type-1 kind. Maybe some extra weakening of the properties a cat-candidate has intrinsicality are called for. Or maybe (very surprisingly) type-2 POMs never arise. But either way, more work is needed.

Nihilism, maximality, problem of the many

Does nihilism about ordinary things help us out with puzzles surrounding maximal properties and the problem of the many? It’s hard to see how.

First, maximal properties. Suppose that I have a rock. Surprisingly, there seem to be microphysical duplicates of the rock that are not themselves rocks. For suppose we have a microphysical duplicate of the rock (call it Rocky) that is surrounded by extra rocky stuff. Then, plausibly, the fusion of Rocky and the extra rocky stuff is the rock, and Rocky himself isn’t, being out-competed for rock-status by his more extensive rival. Not being shared among duplicates, being a rock isn’t intrinsic. And cases meeting this recipe can be plausibly constructed for chairs, tables, rivers, nations, human bodies, human animals and (perhaps) even human persons. Most kind-terms, in fact, look maximal and (hence) extrinsic. Sider has argued that non-sortal properties such as consciousness are likewise maximal and extrinsic.

Second, the problem of the many. In its strongest version, suppose that we have a plentitude of candidates (sums of atoms, say) more or less equally qualified to be a table, cloud, human body or whatever. Suppose further that both the sum and intersection of all these candidates isn’t itself a candidate for being the object. (This is often left out of the description of the case, but (1) there seems no reason to think that the set of candidates will always be closed under summing or intersection (2) life is more difficult–and more interesting–if these candidates aren’t around.) Which of these candidates is the table, cloud, human body or whatnot?

What puzzles me is why nihilism—rejecting the existence of tables, clouds, human bodies or whatever—should be thought to avoid any puzzles around here. It’s true that the nihilist rejects a premise in terms of which these puzzles would normally be stated. So you might imagine that the puzzles give you reason to modus tollens and reject that premise, ending up with nihilism (that’s how Unger’s original presentation of the POM went, if I recall). But that’s no good if we can state equally compelling puzzles in the nihilist’s preferred vocabulary.

Take our maximality scenario. Nihilists allow that we have, not a rock, but some things arranged rockwise. And we now conceive of a situation where those things, arranged just as they actually are, still exist (let “Rocky” be a plural term that picks them out). But in this situation, they are surrounded by more things of a qualitatively similar arrangement. Now are the things in Rocky arranged rockwise? Don’t consult intuitions at this point—“rockwise” is a term of art. The theoretical role of “rockwise” is to explain how ordinary talk is ok. If some things are in fact arranged rockwise, then ordinary talk should count them as forming a rock. So, for example, van Inwagen’s paraphrase of “that’s is a rock” would be “those things are arranged rockwise”. If we point to Rocky and say “that’s a rock”, intuitively we speak falsely (that underpins the original puzzle). But if the things that are Rocky are in fact arranged rockwise, then this would be paraphrased to something true. What we get is that “are arranged rockwise” expresses a maximal, extrinsic plural property. For a contrast case, consider “is a circle”. What replaces this by nihilist lights are plural predicates like “being arranged circularly”. But this seems to express a non-maximal, intrinsic plural property. I can’t see any very philosophically significant difference between the puzzle as transcribed into the nihilists favoured setting and the original.

Similarly, consider a bunch of (what we hitherto thought were) cloud-candidates. The nihilist says that none of these exist. Still, there are things which are arranged candidate-cloudwise. Call them the As. And there are other things—differing from the first lot—which are also arranged candidate-cloudwise. Call them the Bs. Are the A’s or the B’s arranged cloudwise? Are there some other objects, including many but not all of the As and the B’s that *are* arranged cloudwise? Again, the puzzle translates straight through: originally we had to talk about the relation between the many cloud-candidates and the single cloud; now we talk about the many pluralities which are arranged candidate-cloudwise, and how they relate to the plurality that is cloudwise arranged. The puzzle is harder to write down. But so far as I can see, it’s still there.

Pursuing the idea for a bit, suppose we decided to say that there were many distinct pluralities that are arranged cloudwise. Then “there at least two distinct clouds” would be paraphrased to a truth (that there are some xx and some yy, such that not all the xx are among the yy and vice versa, such that the xx are arranged cloudwise and the yy are arranged cloudwise). But of course it’s the unassertibility of this sort of sentence (staring at what looks to be a single fluffy body in the sky) that leads many to reject Lewis’s “many but almost one” response to the problem of the many.

I don’t think that nihilism leaves everything dialectically unchanged. It’s not so clear how many of the solutions people propose to the problem of the many can be translated into the nihilist’s setting. And more positively, some options may seem more attractive once one is a nihilist than they did taken cold. Example: once you’re going in for a mismatch between common sense ontology and what there really is, then maybe you’re more prepared for the sort of linguistic-trick reconstructions of common sense that Lewis suggests in support of his “many but almost one”. Going back to the case we considered above, let’s suppose you think that there are many extensionally distinct pluralities that are all arranged cloudwise. Then perhaps “there are two distinct clouds” should be paraphrased, not as suggested above, but as:

there are some xx and some yy, such that almost all the xx are among the yy and vice versa, such that the xx are arranged cloudwise and the yy are arranged cloudwise.

The thought here is that, given one is already buying into unobvious paraphrase to capture the real content of what’s said, maybe the costs of putting in a few extra tweaks into that paraphrase are minimal.

Caveats: notice that this isn’t to say that nihilism solves your problems, it’s to say that nihilism may make it easier to accept a response that was already on the table (Lewis’s “many but almost one” idea). And even this is sensitive to the details of how nihilism want to relate ordinary thought and talk to metaphysics: van Inwagen’s paraphrase strategy is one such proposal, and meshes quite neatly with the Lewis idea, but it’s not clear that alternatives (such as Dorr’s counterfactual version) have the same benefits. So it’s not the metaphysical component of nihilism that’s doing the work in helping accommodate the problem of the many: it’s whatever machinery the nihilist uses to justify ordinary thought and talk.

There’s one style of nihilist who might stand their ground. Call nihilists friendly if they attempt to say what’s good about ordinary thought and talk (making use of things like “rockwise”, or counterfactual paraphrases, or whatever). I’m suggesting that friendly nihilists face transcribed versions of the puzzles that everyone faces. Nihilists might though be unfriendly: prepared to say that ordinary thought and talk is largely false, but not to reconstruct some subsidiary norm which ordinary thought and talk meets. Friendly nihilism is an interesting position, I think. Unfriendly nihilism is pushing the nuclear button on all attempts to sort out paradoxes statable in ordinary language. But they have at least this virtue: the puzzles they react against don’t come back to bite them.

[Update: I’ve been sent a couple of good references for discussions of nihilism in a similar spirit. First Matt McGrath’s paper “No objects, no problem?” argues that the nihilist doesn’t escape statue/lump puzzles. Second, Karen Bennett has a forthcoming paper called “Composition, Colocation, and Metaontology” that resurrects problems for nihilists including the problem of the many (though it doesn’t now appear to be available online).]