Here are two questions I’d appreciate any thoughts on. Firstly, I’ve recently detected an apparent tension in my beliefs. In my paper for Phil Compass on the grounds of necessity, I argue that the Lewisian realist needn’t be worried about the epistemological objection. The objection goes: how could we know what’s (merely) possible if what’s possible is what’s true at a spatially-temporally isolated concrete world – such worlds do not interact with us causally, so how can we come to know what they’re like? Lewis responds by saying that causal interaction is necessary only when the subject matter is a contingent truth – when the claim to be known is a non-contingent matter, causal acquaintance with what the claim is about is not necessary even when the claim is about the realm of concreta. It’s not obvious to me that this is a good reply, but I thought Lewis had a simpler reply available: metaphysical priority is not conceptual priority. To say that what it is for it to be possible that p is for p to be true at some world does not commit us to saying that our epistemic access to the fact that p is possible must go via epistemic access to there being a world at which p. The Lewisian realist needn’t claim we have any way of discovering what’s true at a world independently of discovering what’s possible. Why can’t the Lewisian simply say she knows there’s a world where there’s a talking donkey because she knows that there could be talking donkeys (here appealing to whatever story about modal epistemology that any realist appeals to), and she knows that everything that could be the case is the case at some world (and here she cites the familiar Lewisian reasons for believing that claim)? What’s the problem?

That still seems convincing to me. Here’s my problem. I also find convincing an epistemological objection to consequentialism: were consequentialism true we couldn’t know what’s right or wrong because we can’t know what the full consequences of our actions would be. And it doesn’t seem to me in the least bit satisfying for the consequentialist to say: I know that murdering X will have the worst consequences because I know that murder is wrong – metaphysical priority isn’t epistemic priority, so my knowledge that it is wrong can ground my knowledge about the consequences even though what it is for it to be wrong is for it to have the worst consequences.

What I’d like is for the two cases to be disanalogous so I can consistently do what seems to me intuitive: hold the epistemological objection to consequentialism and reject the epistemological objection to Lewisian modal realism. I haven’t been able to convince myself that they’re analogous yet, so any thoughts on this are welcome (even if they’re of the form: they’re obviously analogous, and you’re wrong about the epistemological objection to ____). (Incidentally, I barely know the literature on consequentialism, so if anyone knows what consequentialists say about the epistemological objection, please enlighten me!)

Question 2. I was reminded by Brian’s post about the autonomy in logic issue. There’s a thought that every logical truth should be provable using only the rules governing the connectives in that truth. This is meant to be bad for classical logic because there are classical tautologies like Pierce’s law where the only connective is the conditional but one can’t prove Pierce’s law using only the rules for the conditional. I was thinking about this briefly, and I couldn’t see how the objection could possibly be right. We can do classical logic with just one logical connective: the Sheffer stroke, e.g. Every wff of classical logic – a fortiori every theorem – has a translation into a sentence statable using only the Sheffer stroke, and the translations of the theorems will be provable using only the rules governing the Sheffer stroke, as those are the only rules you have. But it can’t be the case that the acceptability of a logic depends on what connectives you allow yourself to use to state its theorems. The defenders of the objection are obviously going to be unimpressed with such a simplistic response, so my question to those who know more about this than me (= those who know than is written in this paragraph!) is: why not?

## Friday, December 11, 2009

## Friday, December 04, 2009

### Substitutional Quantification and Supervaluations

(Cross-posted with Matters of Subtsance.)

Let U be the (universal) substitutional quantifier: its truth-conditions are

Peter van Inwagen has an argument that we can't understand substitutional quantification. It goes like this:

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:

If we have the syntactic understanding of precisification, then we have the truth-conditions

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

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?

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 isnotwhat 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?

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