Friday, December 4, 2009

Substitutional Quantification and Supervaluations

(Cross-posted at Metaphysical Values.)

Let U be the (universal) substitutional quantifier: its truth-conditions are
"UxF(x)" is true iff, for every name n, "F(n)" is true.
(Normal quotes are doing double-duty as quasi-quotes here.)

Peter van Inwagen has an argument that we can't understand substitutional quantification. It goes like this:
(1) We can't understand a sentence unless we can specify what proposition it expresses.
(2) The only proposition we know of with the right truth-conditions to be expressed by "UxF(x)" is the proposition that, for every name n, "F(n)" is true. (Call this proposition "UU".)
(3) Friends of substitutional quantification say that UU is not what is expressed by "UxF(x)".
(4) There are no other candidates to be the proposition expressed by "UxF(x)".
(5) So if friends of substitutional quantification are right, we can't understand "UxF(x)".

I want to respond to this argument, but I don't know whether my response rejects premise (1) or (4). So I'll outline the basic idea, and then maybe someone can help me know which premise I'm rejecting.

Suppose some sort of supervaluationism is the right treatment of vagueness, and set aside higher order vagueness. Then a sentence like "Fido is red" doesn't express a proposition simpliciter; rather, it expresses a proposition relative to every precisification of "red".

(Since we can understand "Fido is red", this alone might be enough to lead us to deny (1). But it's not clear how this denial gives us any positive reason to think we should be able to understand substitutional quantification. I want to aim higher. So let's press on.)

The truth-conditions for this sentence with the determinacy operator are:
"Det(Fido is red)" is true iff "Fido is red" is true on every precisification of "red".
Now, we can think about precisifications in a number of ways. One of them is an explicitly semantic way: the precisifications of a term are the precise meanings it can have. But another is a bit more syntactic, relating more precise terms to less. If we have semantic precisifications, we can easily define syntactic ones as follows: T is a syntactic precisification of T* iff T's semantic value is a semantic precisification of T*. If we don't have semantic precisifications, we might take the syntactic ones as primitive, or we might be able to define them some other way (maybe by appealing to metalinguistic predicates like "admits of borderline cases" and some others).

If we have the syntactic understanding of precisification, then we have the truth-conditions
"Det(Fido is red)" is true iff "Fido is R" is true for every term R that is a precisification of "red",
which look remarkably similar to the ones we had for the substitutional quantifier.

So here's my basic idea: think of "x" as a maximally vague name --- a name such that every precise name is a (syntactic) precisification of it. Then think of "U" as a determinacy operator. This gives us essentially the truth-conditions we want.

How does van Inwagen's argument look now, with this understanding of the substitutional quantifiers? That depends, I think, on what we say about the proposition expressed by "Det(Fido is red)". I think there are very good reasons to think that this sentence does not express the proposition that "Fido is R" is true for every term R that is a precisification of red. (One very good reason is that it won't embed right at all --- it might be necessary, say, that Det(Fido is red), even though it certainly isn't necessary that "red" is even a word, much less that it has precisifications. And these thoughts extend to the truth-conditions that go via semantic precisifications, too.) But are we in any position at all to specify a proposition it expresses?

Here I don't know what to say, and this is why I don't know which premise I reject in van Inwagen's argument. On the one hand, maybe we have some recipe for specifying a proposition expressed by "Det(Fido is red)". If so, then we can use the same recipe to specify one expressed by "UxF(x)", and I deny premise (4). Maybe we think "Det(Fido is red)" expresses the conjunction of all the propositions expressed by "Fido is R", where R is a (syntactic) precisification of "red", for instance. If so, then we can say that "UxF(x)" expresses the conjunction of all propositions expressed by sentences of the form "F(a)" for some name "a".

On the other hand, maybe we can't specify any proposition expressed by "Det(Fido is red)". (Maybe we dislike the conjunction proposal for both the "Det" and "U" cases because we think it misses out on the "that's-all"-ish nature of the quantifications involved in the truth-conditions.) Nonetheless, I think it's entirely clear that we understand "Det(Fido is red)". And I also think (but I haven't argued for it) that one way we can come to understand a vague term by learning a recipe for figuring out what its precisifications are, so we can understand what the "x" in "UxF(x)" is doing. But in this case, "UxF(x)" is essentially just "Det F(x)"

There's a lot of details I've left out --- stuff about variable-binding, the viability of the syntactic characterization of precisifications, how to think of modally embedded substitutional quantifications, and so on. But setting these techy details aside, I'm wondering what the right thing to say about the argument is. Or, more to the point, I'm wondering what we should deny when we run a parody argument for our inability to understand the sentence "Det(Fido is red)".

Thoughts, anyone?


  1. I don't think this comparson to vagueness is helpful. There's no doubt what proposition "Ux Fx" expresses (assuming F non-vague). For that matter, even if you do make that comparison, "Det(Fido is red)" isn't , or isn't obviously, vague, even if "Fido is red" is. Van Inwagen could just weaken his argument to non-vague sentences and the argument would go through (assuming 'F' were non-vague.).

    I doubt the existence of propositions wholescale, which makes me doubt (1), but let me put that aside, and pretend I believed in propositions. In that case...

    IMHO, the error in Van Inwagen's argument is premise (2) (well, it infects (4) as well...).

    People fundamentally misunderstand, and overinflate, the importance of a semantic metatheory for language. A proposition does not state that its own truth conditions are fulfilled. (Actually, Wittgenstein makes this point in the Tractatus.)

    Compare disjunction. The normal clause for disjunction in a logical metatheory is:

    (Disj) "p v q" is true if and only if "p" is true or "q" is true.

    But it would absurd to conclude from this that "p v q" asserted the proposition `"p" is true or "q" is true'. "p v q" is not about its disjuncts, not about language, nor about truth. It's just about what its disjuncts are about. The clause (Disj) just explains that the truth-conditions of the disjunction depend recursively on certain simpler sentences. It doesn't make them "about" those disjuncts, since the proposition isn't about its truth-conditions.

    So someone who accepts:

    (SQ) "Ux Fx" is true if and only if for every name n, "U(n/x)" is true.

    Is not committed to the thesis that "Ux Fx" expresses the proposition that for every name n, "U(n/x)" is true either. (SQ) just fixes the recusion; it doesn't tell us what "Ux Fx" is about, or what proposition it expresses.

    So what proposition does "Ux Fx" express? If it expresses a proposition, it expresses the proposition that UxFx. Yeah, that's not very helpful, but it would be a complete mistake to think we could do better. The purpose of a logical metatheory is not to tell us what a sentence "reall means". That's impossible. It's to compare, or construct models of one language in terms of another (say, the language of set theory), which we already understand. To suggest we don't understand THAT language until we give it its truth conditions in yet another formal semantics leads to an infinite regress.

    Sub. Quan. is a theory is a theory about how to do formal semantics; it isn't a theory about "the real meaning" of language.

  2. Hi Kevin, thanks for the comments. I agree with almost everything you say. Van Inwagen agrees with a lot of it, too. He agrees, for instance, that (Disj) doesn't tell you what proposition "p v q" expresses, that truth-conditions aren't in the meaning-giving business, etc. But he also thinks that he has an independent grip on what "p v q" means (that is, he already understands it). His point is that he doesn't have a prior understand of "UxF(x)", (and simply disquoting it won't help, any more than telling someone that "vorpal" means vorpal will help them understand "vorpal"). He does understand the truth-conditions, but since the truth-conditions don't give the meaning (as you say), he doesn't know what does.

    My thought was this: I presume van Inwagen already understands "Det"; if I could help him understand a vague term "x" in the way described above, then (since there's a candidate with the right truth-conditions), why I couldn't I tell him what "UxF(x)" means in terms he understands? Just say that "UxF(x)" means that Det F(x)?

    (I was not, incidentally, intending the "comparison with vagueness" to suggest that "UxF(x)" is anything like vague. The thought was, rather, that vague sentences are not unlike open expressions, so we might get some understanding by comparing determinacy operators (which, in the absence of higher-order determinacy) remove vagueness, with quantifiers.)

  3. Yeah, I should read what he wrote, but I can't help but think he's been disingenuous if he claims to understand "or" but not "all" pre-theoretically.

  4. I think the structure of van Inwagen's argument is slightly different from the structure you give it. It's based on the principle that we if we understand a sentence s1, while there is a sentence s2 that has the same truth conditions as s1, and nothing more is known (that is relevant) about s2 besides the fact that s2 expresses a proposition different from s1, then we do not understand s2.

    So, now I am thinking about your claim that Det(Fido is red) doesn't express the related objectually quantified metalanguage proposition. The one argument you give is that it doesn't embed right modally. Suppose that argument is sound. Then by parallel you also have a direct argument that the objectual quantification truth-conditions are extensionally wrong in some worlds. But if so, then this destroys van Inwagen's argument, because it is no longer the case that s1 and s2 have the same truth conditions.

    However, I am not convinced by the modal argument. So the worry is that Necessarily(Det(Fido is red)) might be true even though possibly "red" isn't a word. Surely that's easily fixed by indexing with L, where L rigidly refers to a sufficiently rich language: Det(Fido is red) iff (R)(Precification(R,"red",L) → True("Fido is R",L)).

  5. I'm not quite sure I'm following the indexing-to-L move. So I'm clear: are we individuating languages finely, so that different interpretations lead to different languages? Otherwise I'm worried about the possibility that "red" is a word, but means "green". (And either way I'm worried about the possibility that there be no languages...)

  6. Hi Jason,

    Why should PVI agree that he antecedently understands 'Det'? I'd have expected him to react to your point by saying that 'Det' is in the same boat as 'Ux'.

    I've always wondered why one can't simply respond to his argument, though, by saying that 'UxFx' expresses the proposition that everything is F.

  7. Sure, he might be careful to not understand anything whereby "U" might be explained. I was thinking, though, that he'll get less of a sympathetic ear if he has to start saying he doesn't understand "it's determinately (or definitely) the case that..." (I was also thinking that, once we see why the "det" move works in the supervaluational case, then even if he's not a supervaluationist, so long as he understands "it's determinately the case that" (and is not an epistemicist) we can give the same explanation.)

    As for "everything is F": he's (more or less) taking it for granted that the natural language quantifier has objectual truth-conditions (see the very long next-to-last footnote in the paper). This might be dialectically odd if he were arguing against someone who thought otherwise; but there are good reasons to think he's right about this (e.g., "everything has a name" seems like it should be false, but ought to be true if "everything" has a substitutional interpretation).

  8. I was thinking of L along the lines of Lewis's "thesis" in "Languages and Language". L is an abstract entity--say, a function from strings of symbols to meanings--and it is impossible for "red" to mean anything but red in L. What is contingent is not what "red" means in L, but whether L is actually used by a given population or by any population. Even in a world where there are no languages, "red" means red in L, just as 7 is a member of the set of all primes even in a world with no mathematicians.

  9. Thanks for this -- I think I see what's going on now. One residual worry is that, if we thought that only words could be precisifications of other words, then in a world where there were no words,

    (R)(Precification(R,"red",L) → True("Fido is R",L))

    would count as trivially true, and so would "Det(Fido is red)". But maybe we could get around this by allowing that L assigns semantic values to some non-words (numbers, maybe), too.

    I was also thinking there was a second, non-modal argument in the offing, though (although I never mentioned it). It runs something like this: "Det(Fido is red)" is not about words. But the proposition that (R)(Precification(R,"red",L) → True("Fido is R",L)) is (partly) about a word --- it's partly about "red". So, assuming that sentences are about what the propositions they express are about, the latter proposition isn't expressed by the former sentence.

    You can take this argument or leave it; let's set it aside for now and look at the substitutional case. The parallel move there gives truth-conditions:

    "UxF(x)" is true iff (n)[name(n,L) --> True("F(n)",L)]

    Consider the proposition that (n)[name(n,L) --> True("F(n)",L)]. Is that the proposition that, according to friends of substitutional quantification, is not expressed by "UxF(x)"? If their reason for denying the earlier equation of truth-conditions with proposition expressed was something like a modal worry, then presumably not. In which case, they might accept this as the proposition expressed by "UxF(x)", and van Inwagen's complaint will be answered.

    If their reason for denying this earlier equation was instead something like a worry about what a sentence is "about", then they won't like this identification. But they also won't like the identification of the proposition expressed by "Det(Fido is red)" with its (indexed-to-L) truth-conditions, either.

    (Incidentally, I'm thinking van Inwagen would accept the modal argument in the case of objectual quantification. "Ex(x is a dog)" could be true in a world w where there are no variables, and allowing for the truth of "Variable("x",L)" in a world where "x" doesn't exist would pull him away from the sort of serious actualism that (if I recall correctly) he accepts. And I'm also thinking he would "aboutness" arguments against certain identifications, even in the case of objectual quantification.

    I might be wrong about either of those. But I'm pretty confident that he doesn't in general think that we can get from a sentence's truth-conditions to the proposition that it expresses. He's pretty hot that we can't come to know what the objectual quantifier means by looking at truth-conditions; he thinks the only way is by direct translation into natural language (or a stilted idiom thereof).)

  10. I posted a long comment, but it got eaten up. So let me be brief.

    1. Van Inwagen is a Platonist so he will say: names are types, and types are necessary beings. Variables are an artefact of notation. In any case, the propositions expressed by sentences involving variables do not contain variables. (One can, in fact, have a variable-free notation for quantification, e.g., writing "Ex(Dog(x))" as "E(Dog( ))" and drawing an arrow from the "E" to the blank.) If one doesn't like propositions but prefers sentences, nonetheless one needs to distinguish "Necessarily(p)" from "Necessarily(T('p'))". The latter, when coupled with a serious actualism, requires 'p' and its components to exist in all worlds; the former does not.

    2. There are different kinds of substitutional quantification: over sentences, names, verbs, adverbs, etc. Plausibly, substitutional quantification over sentences is about sentences, and substitutional quantification over names is about names. If so, and if (and I think you're right about that) "Det" isn't about names, then "U" and "Det" are not analogous.

    3. Suppose you're right that "UxFx" isn't about names. I think van Inwagen can say that this doesn't help him understand what UxFx is about--if anything, it makes it more mysterious. After all, that it's not about names is a negative claim.

    4. There are two other reasons people may have for denying that "UxFx" expresses the same proposition as '(n)(Name(n) → T("Fn"))':
    i. Substitutionalists think that objectual quantification is to be explained in terms of substitutional quantification, rather than the other way around.
    ii. Some people want to make use of substitutional quantification within a theory of truth, whether as a translation of "Everything George says is true" (Ux(Says(George,x) → x), with the substitutional quantification being over sentences) or as a way of giving the T-schema (Ux(T("x") iff x)). Neither works if substitutional quantification presupposes the concept of truth.

  11. Thanks for this. On 1: I'm not going to pursue the modal argument any further. I am convinced that van Inwagen does not want to say that the proposition expressed by "Ex(x is a dog)" is not just its Tarskian truth-conditions. I'm convinced because he all but says this in note 4 to the substitutional quantification paper. I'm less concerned if his reasoning is that it wouldn't embed right modally or something entirely different.

    On 4: That's helpful. I hadn't been thinking about deflationary theories of truth before, so I should look into that literature further. (I know Belnap and co had some thoughts along those lines, but I hadn't really been thinking about it.)

    I'm losing the plot a bit with your 2 and 3. At 2, you say that "Det(...)" isn't about words, and that this makes it disanalogous to "U(...)", but then in 3 you consider my proposal that "U(...)" isn't about words and say this provides fodder for van Inwagen. But the point I was trying to press was that, if "Det(...)" isn't about words (and van Inwagen understands it), then we could propose a reading of "U(...)" analogous to "Det(...)" which also wouldn't be about words, which van Inwagen might also be able to understand. When you say that the fact that "Det(...)" isn't about words makes it disanalogous to "U(...)", you seem to be presupposng exactly what I'm suggesting the friend of substitutional quantification deny.

  12. Re. 2 and 3: Point 3 was made in case point 2 didn't convince. :-) Hence the apparent conflict between the two points. In 3, I concede that Det(...) isn't about words, and I concede that U(...) is proposed not to be about words.

    I think van Inwagen's response will be something like this. Suppose I understand a sentence L, which is about Fs. You introduce a new sentence L*, and you tell me the following things about it:

    1. Necessarily, L* iff L.
    2. L* is not synonymous with L.
    3. L* is not about Fs.

    But, van Inwagen will insist, doing this is not going to suffice for me to understand L*. But that's what your modified story about substitutional quantification did. I had a locution I understood--objectual quantification over names--and you gave me a new locution, UxFx, which you told me had the same truth conditions, but was not synonymous with the objectually quantified sentence and was not about words. This is not enough to make me understand the locution UxFx.

    For instance, suppose L is "George is wearing a hat". This locution is (inter alia) about hats. You introduce a new locution L*: "George is wearing a cyt". You tell me that, necessarily, L* holds iff L does. You also tell me that L* is not synonymous with L, and that L* is not about hats.

    At this point I will be puzzled. When you told me that L* had the same truth conditions as L, I formed as my simplest explanatory hypothesis that L* was synonymous with L, and "cyt" was a synonym of "hat". But then you denied this, and you even denied that L* is about hats.

    Now, granted, I can coherently believe you that L* satisfies all of these constraints, but believing that L* satisfies the constraints is not sufficient for understanding L*. After all, there may be a multitude of readings of L* on which the constraints are satisfied--maybe a "cyt" is any object which God believes to fall under the concept hat (this isn't about hats, but about the concept hat and about God's beliefs), or maybe "to wear a cyt" is a metaphor for a certain arrangement of molecules on which, as a matter of fact, the wearing of a hat supervenes).

  13. Good -- this looks like van Inwagen's original argument, I agree. But I don't see how it's responding to my proposal. Perhaps my proposal has been unclear up 'till this point, so I'll see if I can make it clearer. I want to suggest the following strategy for "teaching" someone like van Inwagen to understand "UxFx". Here's the first-pass proposal:

    (1) Teach them the meaning of a new vague name, "x". I am supposing that it is possible to learn the meaning of a vague term by learning all the terms that are precisifications of it. The precisifications of "x" are all and only the names of the language.

    (2) Say that "UxFx" is synonymous with "Det(Fx)"

    (3) Since (we are supposing) we do understand "Det(...)", and if by (1) we now understand "x", we should now be in a position to understand "UxFx", too.

    The idea was that, since "Det(Fx)" has truth-conditions that look roughly like the ones proposed for "UxFx", but has the advantage that all sides will (should!) agree it's not about words, this makes "Det(Fx)" a good candidate for what was meant by "UxFx" all along.

    Now, the first-pass proposal won't quite work, because (i) "F" might be vague or indeterminate itself, and (ii) even if it isn't, we'll get the wrong results when multiple variables are involved. So the next pass is to see if we can understand a "selective" determinacy operator, "Det__". The rough idea is that Det(bald)[Some bald man is tall] means roughly something like "Some definitely bald man is tall". But I hadn't gotten that far in the post yet; I was still considering the first-pass proposal.

    One thing I was wondering (this was the question of the original post) was whether people thought that (i) there was a proposition expressed by "Det(a is F)", and (ii) if so, whether the proposition was the same as the proposition that "a is F1 and a is F2 and..." where "F1", "F2", etc. are all the precisifications of "F". The question was partly because, if the answer to both was "yes", then my proposal would collapse into the proposal that "UxFx" expressed the proposition that "a is F and b is F and ...", where we get a fix on that proposition by running through the names.

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