Saturday, July 28, 2007

Assignment Part 1

I'm going to post what I originally submitted (invalid warts and all) for the first part of the assignment here (sorry if it looks messy... blogger did not like the formatting from the word doc.), along with your recommendations as comments to this post. I hope y'all don't mind, but it does seem easier to blog these works in progress then e-mailing things back and forth.

So here is the first part:

“Lewis proposes that a world w may represent de re of an individual x (even when x is not part of w) that x has a certain feature F by w having as a part an individual y that is a suitable simulacrum – a counterpart – of x and which is F.” (Divers 122) So, if the facts about counterparts are relevant, then they are relevant to the modal truth about an individual.

Irrelevance objection from Salmon:

  1. If an ordinary possibility sentence of modal English is about a counterpart (i.e. facts about an individual y having a certain feature F, at possible world, w), then that counterpart is relevant to the truth of a sentence of ordinary modal English which is about an individual x, which shares the same feature F, as y, at another world. [P → Q]
  2. (P5) [R]
  3. (P6) [S]
  4. (2) & (3) [R & S (2), (3) CONJ]
  5. If (2) & (3), then the relevant modal truth about an individual, x, is its own counterpart in its own world (i.e. not another counterpart, y, having a certain feature F, at possible world, w)[(R&S) → ~Q]
  6. So, the relevant modal truth about an individual, x, is its own counterpart in its own world (i.e. not another counterpart, y, having a certain feature F, at possible world, w). [~Q (5), (4) MP]
  7. Therefore, that counterpart is not relevant (irrelevant) to the modal truth about an individual, x. [~P (1), (6) MT]

Justifications:

(1) According to the counterpart-theoretic specifications for truth-conditions (CT-P) x is possibly F, just in case there is a world in which x has a (relevant) counterpart which is F. It is the counterpart’s connection in the left side of the conditional that figures in (i.e. is relevant) to part of the left side’s truth conditions, which is about the individual.
(2) Postulate (P5) of CT stipulates that: “nothing is a counterpart of anything else in this world” (Divers 124)
(3) Postulate (P6) of CT stipulates that: “anything in a world is a counterpart of itself” (Divers 124)
(4) Conjunction of premises (2) and (3).
(5) Because premises (2) and (3) fix identity as an intra-world counterpart relation, then the relevant modal truth about an individual, x, is its own counterpart in its own world. As Divers’ puts it loosely, x = y. So, the separation, and thereby the relevancy of y’s identity to that of x, is mute. This can be seen when considering the sentence examples of ordinary modal English (e.g. (1) and (2)) and their respective translations (e.g. (1*) and (2*)) in Divers pages 125-126.
Where the sentence:

(1) Humphrey might have won.

and, its translation:

(1*) ∃w[∃x[Pxw & Cxh & Vx]]

Is considered with respect to the sentence involving Humphrey’s counterpart:

(2) There might have been a Humphrey counterpart who won.

and, its translation:

(2*) ∃w[∃x[∃y[Pxw & Pyw & Cyh & Cxy & Vx]]]

When the postulates (P5) and (P6) are taken into consideration, then as mentioned the identity of x = y, become fixed, and thus gives:

(1*) ∃w[∃x[Ixw & Cxh & Vx]]

as a valid translation for (2*). Notice that this translation is the same as that given to the sentence of (1) above.
(7) Which gives the conclusion, that the counterpart, y, is irrelevant to the truth of a sentence about an individual, x.

Argument Against Salmon's Objection (i.e. denying (P5) of CT):

  1. If (P5) & (P6), then the relevant modal truth about an individual, x, is its own counterpart in its own world.
  2. If (P5), then, if (P6), then the relevant modal truth about an individual, x, is its own counterpart in its own world.
  3. But, it is not the case that if (P6), then the relevant modal truth about an individual, x, is its own counterpart in its own world.
  4. So, it is not the case that (P5)

Justifications:
(1) See premises (2)-(5) of Salmon's objection above for justifications.
(2) Exportation from premise (1).
(3) It cannot be the case that (P6) alone gives the relevant modal truth about an individual, x, as being its own counterpart in its own world. The identity fixing axioms (P5) and (P6), can only do this in conjunction, not on their own.
(4) Therefore it is not the case that (P5). Which means that the argument from Salmon above starting with premise (4) cannot follow through.

2 comments:

Chelsey said...

Here's what Dan had to say:

Evaluation of Chelsey on GR

As it stands, this argument is not valid. The antecedent of (1) is “an ordinary possibility sentence of modal English is about a counterpart”. If the conclusion were changed to “It is not the case that an ordinary possibility sentence of modal English is about a counterpart” the argument would be valid. The two conclusions are similar in nature, and both damaging to GR. The argument contains no idle premises.

I do not think that Salmon would accept premise (5). Salmon’s argument involves a dis-analogy between the semantics of GR and that of English. That is, he asserts that “there may have been a Humphrey counterpart who won” entails “Humphrey might have won” in GR, but it does not in English. This lack of analogy is meant to support the claim that concerns about our counterparts are different from, and irrelevant to, concerns about our modal properties. Salmon would agree that premise (5) holds for certain individuals & propositions meant to produce counter-examples, but I do not think he would have it be a fully general rule. He would reject (6) on the same grounds. To retain validity, I’ll also modify premise (1) in the reconstruction.

(1)If ordinary possibility sentences of modal English are about counterparts (i.e. facts about an individual y having a certain feature F, at possible world W) then entailment relations that hold in CT will hold in ordinary modal English.
(2)(P5)
(3)(P6)
(4)(2)&(3)
(5)If (4) then there is some entailment relation that holds in CT but not in ordinary modal English
(6)There is some entailment relation that holds in CT but not in ordinary modal English
(7)It is not the case that ordinary modal sentences are about counterparts

After this, we note that 2 – 5 are superfluous. Indeed, the argument may be better without them. (P5) and (P6) need not be true for the argument to go through, they merely need to be true according to GR. If we hold them true simpliciter, then the argument entails things about counterparts. However, it’s best if an argument against counter part theory does not entail things about counterparts. So, simply put, the argument can be reconstructed like so:
(8)If ordinary possibility sentences of modal English are about counterparts (i.e. facts about an individual y having a certain feature F, at possible world W) then entailment relations that hold in GR will hold in ordinary modal English.
(9)There is some entailment relation that holds in GR but not in ordinary modal English
(10)It is not the case that ordinary modal sentences are about counterparts

The justification for (9) will have to invoke (P5) and (P6) when asserting which entailments hold under GR, but not outside the scope of GR.

(1) is adequately defended. Modified (1) would need to be defended in a slightly different way. Assume that ordinary possibility sentences of modal English are about counterparts. Then an English possibility sentence can be paraphrased to be explicitly referring to the appropriate counterparts. If every possibility sentence in English can be paraphrased as a counter-part theoretic sentence, then all inferences valid of the counter-part theoretic sentences should be valid with the English sentences as well.

(5) is defended at length. However, the proof runs over a specially chosen sentence of English, and is not fully general. If the claim “the relevant modal truth about an individual, x, is its own counterpart in its own world” then the sentence “Humphrey might have won” would be true iff Humphrey won (if the only relevant counter-part was himself). This is not the case. To support the modified (5) I would proceed pretty much as Chelsey did. I would add that (2*) -> (1*) while (2) does not entail (1). This would be the counter-example generated.
In the counter argument, Chelsey argues against premise (2). This counter argument takes the form:
(A&B)->C
A->(B->C)
~(B->C)
~A

This is a pretty slick move, but I don’t think it holds (even though it’s valid). If A and B are together sufficient for C. The tricky part is saying that since B alone is not sufficient for C, ~(B->C). Note, this argument can be reconstructed for any A,B,C in which A and B are jointly sufficient for C. But surely being jointly sufficient for something doesn’t entail falsehood. So what’s gone wrong? Not being sufficient for something does not entail the negative of the pertinent conditional. That is, ~(B is sufficient for C) does not imply ~(B->C) (at least when using the material conditional).
Furthermore, (P5) is stipulated in CT. So if denial of (P5) is the only way to object to an argument against CT then the argument has still gained a victory of sorts. It has, in a sense, forced CT to reformulate itself anyway.
The premise doing the heavy lifting, and the one most vulnerable would be premise (1). In the original version of the argument it invokes vague notions or relevance which may be exploited. In the new version some controversy may ensue about what exactly the similarities between English and CT would have to be in order for CT to be a correct theory of modal English.
As a proponent of the original argument, I would deny premise (3) in the counter-argument for the reasons given above.

Chelsey said...

Here's what Adam had to say:

A.

Although the inferences all work together, the argument is not formally valid. Here’s the form:

(1)P→Q
(2)R
(3)S
(4)(R&S) (2,3)
(5)(R&S)→~Q
(6)~Q (4,5)
(7)~P (1,6)

The English content of consequent of the conditional in (5) should match the content of the negation of the consequent of the conditional in (1), but in the original, it doesn’t. Also, the conclusion has different content from the negation of the antecedent in (1). But the content of the premises should match across the argument. If this could be done, the argument would be valid. The best way to do this would probably be to fix the content of the premises as follows:

P: An ordinary sentence of modal English is about a counterpart.
Q: A counterpart is relevant to the truth of an ordinary modal English sentence.
R: Nothing is a counterpart of anything else in this world.
S: Anything in a world is a counterpart of itself.

I believe this could be shortened to the following and still remain valid:

(1)R
(2)S
(3)(R&S)
(4)(R&S)→~Q
(5)~Q→~P
(6)~P (1-3 HS)

and still further by one step if the conjunction in (3) can be suppressed. I don’t believe that this conclusion can be obtained with fewer premises than the minimal form just presented above.
Neither the first or second versions of the argument presented above seem faithful to the argument presented by Divers, since both of the above versions have as their conclusion that it is not the case that an ordinary sentence of modal English is about a counterpart. But Divers argues that, insofar as counterpart theoretic truth-conditions specified for our ordinary modal English sentences misrepresent our pre-theoretical intuitions, they are not relevant to the modal truth of the individuals the are about. These two conclusions seem different. I think a more faithful version of (Divers’s version of) Salmon’s objection would look like this:

C: The non-modal (metalinguistic) truth-conditions specified by (CT) for ordinary (object language) sentences of modal English correctly represent our pre-theoretical modal intuitions.

R: The non-modal (metalinguistic) truth-conditions specified by (CT) for ordinary (object language) sentences of modal English are relevant to the modal truth of the individuals they are about.

M: An (object language) sentence of modal English about a counterpart of an individual a is intuitively weaker than a sentence of modal English about a itself.

U: The (CT) truth-conditions specified for a modal English sentence about a counterpart of an individual a logically entail the (CT) truth-conditions specified for a modal English sentence about a itself.

(1)~C→~R
(2)M
(3)M→(U→~C)
(4)(U→~C) (2,3)
(5)U
(6)~C (4,5)
(7)~R (1,6)

I think this is a valid and faithful presentation of Divers’s version of the Salmon objection from irrelevance. I won’t present a justification here of each premise, but only the section of the argument doing the work- i.e., the move from (2)-(4) and from (4)-(6).
(2) is an expression of what Divers calls our pre-theoretical modal intuition concerning the strength of modal claims about ordinary, this worldly individuals vs. modal claims about counterparts of those individuals. According to Divers, we have it pre-theoretically that a modal claim about a counterpart of an individual is weaker than a modal claim about that same individual itself. And (3) just says that, if that’s right, then if (CT) specifies truth-conditions for the former type of modal claim that logically entail the latter, then it is not the case that (CT) correctly represents our pre-theoretical modal intuitions.
Granting the truth of (2), we have the conditional in (4), which does most of the work in the argument. Divers establishes (5) via the following considerations: Consider a modal English sentence like:

(1)Humphrey might have won.

and its (CT)-specified truth-conditions:

(1*) ∃w[∃x[Pxw & Cxh &Vx]].

Since it is the case that:

(2)There might have been a Humphrey counterpart who won.

is intuitively weaker than (1), Divers argues, it had better not be the case that the (CT)-specified truth-conditions for (2) logically entail (1*). But they do, according to Divers: (2) may be interpreted as either an ordinary possibility claim, or an extraordinary possibility claim. If we opt with the former, we have the (CT) truth-conditions for (2) specified as:

(2*) ∃w[∃x[∃y[Pxw & Pyw & Cyh & Cxy & Vx]]].

Given (P5) and (P6) of (CT), (2*) entails (1*). If we opt with the latter, we have the (CT) truth-conditions:

(2**) ∃x[Cxh & Vx].

(2**) also entails (1*), thus premise (5) and the step to the conclusion.

B.

All technical terms appear to be defined. However, one worry is the introduction of the term ‘counterpart’ in the first premise. Here it is said that “a counterpart is relevant to the truth of a sentence of modal English which is about an individual x, which shares the same feature F, as y, at another possible world.” Assuming that ‘y’ names the appropriate counterpart of x, this sentence makes it unclear as to what is meant. Assume that x exists at w and y at w’. The sentence is then indeterminate between (i) x and y share some feature F at a third world, w’’, at which they both exist, and (ii) x and y share some feature F at w’ (the world y exists in). Both (i) and (ii) are false specifications of what (CT) truth-conditions would be given for a sentence like “possibly x is F.”
Each premise is accompanied by a justification. Premise (1) requires further disambiguation if its justification is to work. Consider the antecedent in (1). Are we concerned here with (i) ordinary possibility sentences about counterparts in the sense in which an assertion about some object may be a disguised or abbreviated assertion about some other object, or (ii) instances of modal English sentences which modalize about counterparts, as in ‘A counterpart of Conrad Black could have been less of a jerk.”(?). These two types of claim are each compatible with the antecedent in (1), so a disambiguation is needed. Likewise, the justification for (1) could be interpreted as a justification for either reading. The first sentence of the justification appears to support reading (i). The second supports (ii), for why else would a counterpart figure in the left side of the conditional at all?

The objection is aimed at blocking the move from (5)-(7) in the original argument. The exportation move is interesting, but I think the first thing that should be noted is that the counterargument could be blocked by a supporter of the original by simply switching (2) and (3) in the order of premise introduction. That alone might say something about the strength of the objection. A couple of further points: the objection has as its conclusion that it is not the case that nothing is a counterpart of anything else in this world. I see how the move can block the conclusion of your version of the Salmon objection, but I don’t see how this objects to the argument that counterparts are irrelevant to the truth of object language modal sentences. Also, this argument would block only the first route taken by Divers to establish his conclusion. As he shows on 126, it is legitimate to take modal sentences about counterparts as extraordinary possibility claims, in which case there is no need to invoke (P5) or (P6).

Given the way Divers sets up the objection, it seems the quickest route in terms of (CT) defense is just to deny (5) (in the version of the argument I presented). Divers presents (2*) as the (ordinary) (CT) specification of the truth-conditions for (2). Since identity is the unique intra-world counterpart relation, this forces an interpretation of (2*) which entails (1*). But it seems clear that (2*) is not the only (CT) interpretation of (2) available. For it seems equally plausible that, in uttering a sentence like (2), we are predicating de re of some Humphrey counterpart that it (he) might have won the election. And, if that is right, then the proper (CT) truth-condition for (2) should be

(2***) ∃w ∃w*[∃x[∃y[Pxw & Pyw* & Cxh & Cyx & Vy]]].

(2***) does not entail (1*), blocking the move from (4)-(6) in the revised version of the argument.

Incidentally, it appears that Divers misrepresents the original irrelevance objection from Salmon. Salmon is concerned with the fact that (CT) truth-conditions do not accurately reflect the differences in truth-conditions we pre-theoretically assign to (object language) sentences of the following two types: (i) those that ‘modalize’ about the origins of a world-bound individual, as in (3):

(3)“It might have been the case that this be made from hunk H.”

and (ii) those that ‘modalize’ about counterparts of world-bound individuals and counterparts of the substances from which they originate, as in (4):

(4)“It might have been the case that a counterpart of this be made from a counterpart of H.”

Salmon’s argument can be presented as follows:

I: Intuitively, modal English sentences about the origins of an individual a differ in truth-value from modal English sentences about a counterpart of an individual a and a counterpart of the object from which a (in fact) originated.

C: (CT)-specified truth-conditions for sentences of modal English correctly represent our pre-theoretical modal intuitions.

R: (CT)-specified truth-conditions for sentences of modal English distinguish between the truth-conditions of modal English sentences about the origins of an individual a and those of modal English sentences about a counterpart of an individual a and a counterpart of the object from which a (in fact) originated.

T: There are no plausible grounds for prohibiting the counterpart notion from occurring in (object language) sentences of modal discourse.

P: (Object language) sentences about counterparts are legitimate occurrences of modal English discourse.

E: (CT) truth-conditions are relevant to the modal truth of the individuals they are about.

(1)I
(2)I→[(C & P)→ R]
(3)(C & P)→ R (1,2)
(4)~ R
(5)~C v ~P (3,4 MT & DEM)
(6)T
(7)T→P
(8)P (6,7)
(9)~C (5,8)
(10)~C→~E
(11)~E (9,10)