Another kind of sentence that is fun to think about is the following Partee sentence:
(1) If you were king, you would cut off the head of everyone who offended you.
Partee thinks that (1) is ambiguous between a transparent interpretation where 'everyone who offended you' denotes everyone who offended you in the past in the actual world, and an opaque interpretation where it denotes everyone who offended you in a non-actual world where you are king. On Partee's proposal, we can get the transparent reading by assuming that the past tense refers to some actual past time. So Partee predicts the following two readings ([]--> is the subjunctive conditional, @ is a past time in the actual world, K is 'is king', O is 'offend', C is 'cut off the head of', and 'y' refers to you). I am ignoring the past tense on the Opaque reading:
Opaque: Ky []--> (x)(xOy --> yCx)
Transparent: Ky []--> (x)(xOy@ --> yCx)
What is special about Partee's analysis is that she takes the tenses to be similar to pronouns, and on the transparent reading the past tense refers to an actual past time. Given a Prior-style approach, on the other hand, we should probably assume that there is (optional) movement of the nominal + relative clause. So we get (P is past):
Opaque: Ky []--> (x)(xOy --> yCx)
Transparent: (x)(PxOy --> (Ky []--> yCx))
The two formulations of the transparent reading differ in that the Prior-style formalization entails that for everyone in the actual world @ who offended you, you cut off their heads in the king-world w, while the Partee-style formalization entails only that for everyone in the actual world @ who offended you and who exists in w, you cut off their heads in w. As Dave has pointed out to me, these analyses are not equivalent unless the converse Barcan formula obtains (the converse Barcan formula entails that nothing in the actual world could have failed to exist).
Tuesday, August 08, 2006
Partee Sentences
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9 comments:
Brit,
I'm not sure you need the strength of converse Barcan to get that equivalence. What you need is just those offensive individuals to exist in every world. So, for instance, if every offending being were a necessarily existing being then the transparent readings would be equivalent (I think). That would allow for otherwise varying domains world to world. You'd have the equivalence without converse Barcan. But that's minor.
I'm a little confused in the transparent readings. Here's the second one,
T. (x)(PxOy --> (Ky []--> yCx))
(T) implies that even in worlds where no one offended me (so the antecedent is false), if I were King, I would kill them all! Yikes! It would be good to have the consequent false in every world where the antecedent is. Or maybe conjoining (Ex)PxOy? Maybe I'm misreading (T).
Hi Mike! I think you're right that we'd get the right result in models in which every actual offender exists necessarily. But to get the formal equivalence don't we need to include models in which some actual offender exists contingently (assuming the converse Barcan formula does not hold)?
Of course, in S5 the Barcan formula will give us the converse Barcan formula.
I think the second reading says:
For all x, if x offended you in the past, then (if you were king, then you would cut off x's head)
If no one offerended you, this would be vacuously true, but I think that is fine. There will still be worlds where you are king and cut off x's head but you do not cut it off because x actually offended you.
"If no one offerended you, this would be vacuously true, but I think that is fine. There will still be worlds where you are king and cut off x's head but you do not cut it off because x actually offended you"
Actually, I would be cutting off the heads of *all* the x's that did not actually offend me. Weird that not actually offending me should be enough to cost you your head (whether or not there are other reasons). But then, as Mel says, "It's good to be the King".
"But to get the formal equivalence don't we need to include models in which some actual offender exists contingently (assuming the converse Barcan formula does not hold)?"
I don't know, Brit. Whether some actual offenders exist contingently is a metaphysical question. It would be odd if it weren't true. But I don't think the formal equivalence depends on it. Imagine the strange discovery that those are the guys that necessarily exist. Go figure.
o.k. suppose the metaphysicians (or maybe the meta-physicists :-) find out that those are the guys that necessarily exist (perhaps only gods and demons offend me here in the actual world). Then relative to the actual world, the content of the two sentences will have the same truth-values. But there will still be worlds where the offenders' existence is not necessary (just not metaphysically possible worlds). So if the sentences are uttered at those worlds then we don't get this result.
Also, suppose we keep the context of use constant, and suppose quantifiers do not make reference to the actual world. Then the propositions expressed by the sentences will have different truth-values at logically possible but metaphysically impossible worlds (worlds where the offenders do not exist necessarily).
Is that right?
Brit, I guess my inclination is to agree, but I think it turns on some tricky questions. Here are the transparent theses.
T1. Ky []--> (x)(xOy@ --> yCx)
T2. (x)(PxOy --> (Ky []--> yCx))
To keep it simple, take a Stalnaker model on which there is a uniquely close world to each world. When are T1 and T2 are going to diverge in truth-value? It will be just in those cases where (Ky []--> yCx) is false. If that proposition is true, then both T1 and T2 are true. So let w be thw unique closest world to @ and at which Ky is true. Here's a model where they diverge:
w ={Ky, ~xCy, ~xOy@}
@= {PxOy, T1, ~T2}
We would have no divergence if it were true that were I king, there would still exist all of the beings that actually offended me. So, if there is no world w in which it is true that were I king, then some people who offended in me in w would not exist, then T1 and T2 are equivalent. Here I've made no assumptions about necessary beings or the domains of worlds. The interesting thing, I think, is that that may well be true. There may well be no world w in which it is true that were I king, then some people who offended in me in w would not exist. I just don't know. So the logical equivalence question seems to turn on some subtle worries in modal epistemology.
Hi Mike. Well, but we do know that the beings in questions do not exist necessarily in all logically possible worlds, right? What I mean by that is that there will be logically possible worlds that are not metaphysically possible but in which the beings in question are metaphysically contingent beings. That, of course, does not entail that they are metaphysically contingent here as well, since those worlds will not be accessible to this world. So, I think the two sentences will not be logically equivalent, even if they are metaphysically equivalent.
Brit,
Suppose it is true at @ that the closest world in which x is king is a world in which everyone who actually offended x exists. So under the counterfactual assumption we have just the right set of individuals (i.e., the actual offending individuals) existing. In that case the principles will have the same truth-value, right? Now generalize that point to every world. Then you get the equivalence: they will have the same truth-value in every world. The only question open is whether it is true that for all w, the closest world to w at which x is king is a world in which everyone who offended x exists. We don't know that that's not true, do we? If so, how?
Well, what you say about propositions and subjunctive conditionals seems right. But I still think there is a case to be made for non-equivalence. For doesn't the world that is closest to w (for any w) have to be one that is accessible from the w? If yes, then here is a rough argument (assuming that the Barcan formula and its converse do not hold):
(1) For any being x, 'x exists' is not a logical theorem.
(2) For any being x, it is not logically necessary that x exists.
(3) So, for any being x, there is a logically possible world in which x's existence is a contingent matter
(4) The relevant sentences will have different truth-conditions when uttered at any world w at which beings who offend me in w exist only contingently.
(5) What was said in (4) is consistent with the possibility that anyone who offends me here in the actual world exists necessarily, in a metaphysical sense.
Here I am assuming that for ANY logically possible context c, a declarative sentence S1 and another declarative sentence S2 are logically equivalent iff the proposition expressed by S1 and the proposition expressed by S2 have the same truth-values at the world determined by c.
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