# 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

Generalizing,

(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.

### 7 responses to “Structured propositions and truth conditions.”

1. Kenny

This all looks relatively appealing to me, though I wouldn’t want to commit to it without thinking through whether it would entail a sort of deflationism about truth.

But at the end you start worrying about truth-conditions for pictures – why should we think they have truth conditions? Can’t we just give purely causal stories about the representational status of pictures, perhaps supplemented with the intentions of the draw-er?

2. Robbie

Hey Kenny, thanks for this.

I don’t think it need entail anything particularly scary about *sentence*-truth… does that help at all? All the cool stuff can be packed into the S-expresses-prop-p relation.

I’m not particularly opinionated about pictures (maybe I came across as more opinionated than I was). The generic point is: unless only things that stand in relations to struc props have TCs, then there may be a worry around here. Pictures were a sort of stalking horse.

I’m not sure what it means for pictures to have rep properties but not TCs—I’d need to hear more about that (as I was using it, TCs just meant an association with a set of worlds: the worlds where the picture represents accurately.

But I’m inclined to agree that it’s ok to respond like this. Many things, including (say) pictures and struc props represent the world. That (let’s suppose) at least involves them having TCs in the minimal sense of being associated with a coarse grained propn that is their representational content. But the *representational relation* that links the bearer to the representational content can be very different in the different cases: sometimes definitionally (struc props) sometimes indirectly via expressing-prop (sentences) perhaps sometimes (say) through some Fodor-esque story (pictures). That’s in effect to deny what I labeled (A) in the above. And I’m happy with that.

I do think that if I go that way I probably owe some account of in virtue of what the various relations are all “representational”. I don’t know whether I’ll get anything like that to cover the propns case.

(Ooops, I promised not to do philosophy for the next week… resolution broken already).

3. Kenny

That’s some sort of end-of-year “new week’s resolution” or something? 🙂

Anyway, I somehow didn’t think about the fact that I think meaning (and thus representation) has to be tightly tied up with truth-conditions (though I’m not sure if I have very good reasons for thinking that other than that studying truth-conditions has greatly helped us understand meaning).

But my first thought was that a photograph is just a photograph of something – it can’t be correct or incorrect. I suppose it can be misleading or not, depending on whether there is strange lighting or something else, or if it was developed incorrectly, or photoshopped or something. Then I was thinking of a drawing from life in the same way, just with a less tight causal connection between the scene the and the picture (especially if the person doesn’t know how to draw perspective, or something like that).

The next question is what happens with drawings where the person either intentionally introduces features that aren’t there, or just isn’t trying to render some visible scene at all, but just drawing from the imagination. My intuition is that in these cases, the drawing isn’t itself correct or not – but the person’s claim “this drawing is what I saw” or “this drawing is what you will find if you look in there” can be true or false. Maybe there’s some notion of accuracy that can be given, but this might not be anything deep and significant – to say that a drawing is accurate might be ambiguous between saying that it’s close to the relevant photograph, or to say that it produces in the viewer some similar phenomenal aspects to being at the scene. (My boyfriend and I had a long discussion about this the other day when I claimed that impressionism was the first time that painting moved intentionally away from accuracy, and he pointed out that it might have been moving towards accuracy in a phenomenal sense and not a photographic sense.)

Anyway, I’m sure plenty of other people have already thought and written about this, but this is just where I was led by thinking of truth-conditions for non-mental and non-linguistic entities.

4. Robbie

Hmmm. I’d guess I’d thought of photos as paradigmatically representational entities (in the sense of representing some state of the world). Intuitively, photos can e.g. show me stealing chocolate from the fridge, and might be used in court as evidence, etc (chocolate-trials!) It’s sort of hard to figure out how they could be used like that if they weren’t representing states of the world. There’s some hard cases where the it’s a bit dodgier to know whether there’s rep content or if so what it is (and I wouldn’t want to say that representational content is essential to paintings, for example). But the paradigmatic cases of pictures and photos do seem naturally classified as depicting some state of the world.

If paintings or even photos worry you, what about other kinds of pictures: e.g. anatomical diagrams, maps, etc. The first don’t seem to be de re about anything, but still represent generic facts about physiology. The job-description of the second seems to rely on representing features of the world (relative distances between towns, say).

5. Kenny

But can’t all sorts of non-representational things be used as evidence in court? For instance, your fingerprints can be used as evidence, or a piece of hair found at the crime scene. I would think that in fact being used as non-testimonial evidence would mean that it doesn’t have any propositional content.

But I definitely feel the intuition that pictures are representing (“depicting” especially) parts of the world, and even more so when you bring up diagrams and maps and the like. It also seems more right to say that a map is correct or incorrect than to say that a picture is. But I’m not totally convinced. I now recall reading a paper by Achille Varzi and Philip Kitcher called “Sometimes a Picture is Worth 2^{aleph_0} Words”, where they argue that a map has lots of propositional content. But now I can’t recall what their argument was like, other than that the correctness of the map entailed continuum-many logically independenct propositions.

6. Nick Shackel

Maybe I’m missing something obvious, but I’m not following this claim: ‘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.’

‘1 instantiates 2 according to w’ doesn’t seem to me to be meaningless, but rather meaningful and false. But if that is right then [1,2] is a proposition that is not true.

7. Robbie

Hey Nick… Happy new year!

Yeah, that was badly expressed. My thought was that the definition of the truth-conditions has something like the following clause: [x,y] (where x is some object, and y some property) is true iff x instantiates y, and false otherwise.

And I guess what I want to say here is that the statement “[1,2] is true” has the same sort of status as “Robbie is true”. I’m inclined to say that it’s false (not meaningless), but I can imagine someone saying it’s meaningless and I wouldn’t object too hard. What’s important is that neither “[1,2] is false” or “Robbie is false” get to be true.

I guess you’d get something likethe picture you sketch—where [x,y] turns out to be a false proposition—if e.g. any ordered pair [x,y] is said to be false iff x doesn’t instantiate y. But we do need some sort of restrictions here, since there’ll be lots of true propositions that are of this form. For example, the proposition expressed by “It’s not the case that Jill is foolish” might be an ordered pair consisting of (1) Negation (2) the structured proposition that Jill is foolish.

In this case, the second element isn’t instantiated by the first element, but still the proposition may be true.

Moral: there’s got to be more sensitivity built into the assignment of TCs to props. (Even more than indicated above, since we prsumably want to allow singular propositions whose object-component is a set, and we’ve got to ensure that this sort of thing is handled adequately). There’s various ways to do this, no doubt, but I reckon the result will be that [1,2] won’t end up being either true or false.

Of course, that’s not what I *said* in the passage you quote! But it’s the thinking that was behind it. The point was that the TCs assigned to pairs, and the propositions themselves, is imagined to be fixed by a stipulative definition, so various requests for explanation— why X has the TCs it does, rather than some other, or why Y doesn’t have TCs at all—aren’t really well-formed.