Objection from Spatiotemporal Analogy

I'm thinking of doing something or other with this objection, so if anyone has anything to add please do!

as always:
<> is the possibility daimond
[] is the necessity box
A is the universal quantifier
E is the existential quantifier

Let (ST) be a second order predicate that has all and only spatiotemporal relations in its extension. Given that there are multiple spatiotemporal relations, a natural definition of worldmateism is as follows:

defn': Two things x and y are worldmates iff ER((ST)R&xRy)

Too avoid sticky issues, lets assume all worlds discussed are chaotic (no causal relations involved). This isn't cheating since causal relations aren't necessary for spatiotemporal relations.

The complaint is that Lewis now owes us some necessary and sufficient conditions for being a spatiotemporal relation. I'll argue that he can't, in principle give one.
Consider two relations that must count as spatiotemporal relations: being spacially related, and being temporally related. These can be conceptually divided. Moreso, being related to something in a world by either means is sufficient for being in that world. For isntance, if there's two spacially isolated worlds, but each moment in time in one is the very same moment in time in the other, they would be one world on Lewis's view. Indeed, he gives these kind of compromises when he argues that the impossibility of island universes is not so bad.

So if we propose any necessary conditions for being a spatiotemporal relation, both space and time must have them. Maybe some philosophers of time will help sort this out. What I will argue is that all the features that space and time have in common do not jointly make sufficient conditions for being a spatiotemporal relation. If the sum of all necessary conditions do not make for sufficient conditions, there is not set of necessary&sufficient conditions. If a condition is added so the the set is sufficient for being spatiotemporal, the resulting set would not be necessary since either space or time (or both) does not have that condition. Thus, necessary and sufficient conditions can not be given, and Lewis's fallback definition of worldmate gainst the status of "bogus".

So, space and time are conceptually different. One may say qualitatively different. By this I mean there's no basic unexpressable quality they have in common (there's no way to describe time to a timeless person in terms of space, or space to an intelligent volumeless spirit in terms of time). Barring that, there are a few similarities that may be candidates for necessary conditions of spatiotemporallness.
note: when "R" contains no ' after it, it denotes a general relation "sharing a space" if you will. R', R'' denote specific relations within the general one ex. "being 3 feet to the left of", "being 5 seconds after".
1) conditional reflexivity: Ax(Ey(xRy)->xRx)
2) symetry: AxAy(xRy->yRx)
3) transitivity: AxAyAz((xRy&yRz)->xRz)
4) occupancy: Ax(Ey(xR'y&xR''y&xR'''y...)->Az((~z=x)->~(zR'y&zR''y...)))
5) absolutism: AR'ExEy(xR'y)
6) capacity: Ax(Ey(xRy)->Px)

I assume the first 3 are self explanitory. Occupancy simply states that if one "point" in the "world" is properly situated, no other point can share that space. That is to say, if a world has n many spatiotemporal relation, and x bears a specific relation to y with respect to each of them, no z may bear the exact same relations to y. Absolutism would state that for any specific spatiotemporal relation (ex. being 3 cm to the left of), you'll find a couple points that meet it. Capacity requires some explanation. I'll define Px as "x can be assigned a value". I mean this only to say some points can be considered "occupied" while others can be considered "unoccupied". I wanted to leave it open whether a point could be more or less occupied on a scale. I figure, those are the important similarities between space&time (although there are more I'm sure). Which of those merit being necessary conditions for being a spatiotemporal relation? Well, for arguments sake let's suppose all of them are (that will only make my conclusion stronger). Now I simply need to ask, is there any relations that satisfies these which is not a spatiotemporal relation? Of course there is! There's lots! Consider the relation (xRy iff x and y both hold world records). Say x holds a world record if for some value, x holds more of that value than anything else. The universe would probably get the world record for biggest individual, perhaps Carl Friedrich Gauss would hold the world record for greatest mathematical ability. To produce the sub-relations R', R'' etc. you could just arrange an arbitrary ordering of the world record holders, and xR'y iff y succeeds x in the ordering.

So, finaly, I can set up an argument:

1) Space and time must both meet all necessary conditions for being a spatiotemporal relation
2) If (1) then (3)
3) If there are necessary&sufficient conditions for being a spatiotemporal relation, space and time must have them as well
4) ~(space and time jointly have sufficient conditions for being a spatiotemporal relation)
5) if (4) then (6)
6) ~(consequent of 3)
7) ~(there are necessary & sufficient conditions for being a spatiotemporal relation)

if (7) is right, Lewis can't say much about what constitutes a world aside from "gather the stuff I want to be worldmates together, and call that a world". We need more than that!!!

8) if(7) then (9)
9) ~(there are necessary&sufficient conditions for two things being worldmates)

it's 4:30 AM, so I'll add some sarcasm here

10) if (9) then (11)
11) only magic can bind worldmates together! (that's kind of a sweet thought)

I wanted to argue that Lewis get the wrong result as far as saying what's a world-mate of what, but I've argued that he doesn't get any result. Well, I'll stop here. Please comment, especially time folks.

Assignment Part 3

Sorry I know this is long, and repeats a number of things, but I wanted to address both Dan and Adam's comments point by point.

Both Dan and Adam noted that my reconstruction of Salmon's irrelevance objection, as it is presented in Divers (8.1), was invalid. Which I do agree with. So first, I reconstructed the argument, which appears as follows:

Irrelevance objection from Salmon (via Divers) made valid:

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.
2. Nothing is a counterpart of anything else in its world (P5).
3. Anything in a world is a counterpart of itself (P6).
4. (2) & (3)
5. If (4), 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).
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).
7. Therefore, it is not the case that 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).

I'll first deal with Dan's, then Adam's objections to my reconstruction of the irrelevance argument presented above, then rewrite the above argument to better reflect their suggestions.
Dan on Chelsey, on Divers, on Salmon:

As Dan notes in his evaluation, there are two significant changes that he would make to the irrelevance argument which he reconstructs as follows:

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.

First, Dan argues that Salmon would not accept premise (5) and (6) in the above argument. There are two ways this can be taken, either Dan thinks that I am being unfaithful in reconstructing the argument presented in Divers, or my reconstruction is faithful to Divers but Divers is unfaithful to Salmon. In either case he thinks (5) and (6) should be changed in order to deliver a better argument. His justification for this is as follows:

“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.”

Second, Dan suggests that (2) to (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:

9. 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.

10. There is some entailment relation that holds in GR but not in ordinary modal English

11. It is not the case that ordinary modal sentences are about counterparts.”

Here, Dan makes two particular points about the argument's construction which I will address in turn.

First, I think Dan is jumping the gun a bit with Salmon's argument. As it is presented in Divers, it is meant as an irrelevance objection, that is, counterparts are irrelevant to the truth of an ordinary possibility sentence of modal English. In other words, that counterparts are irrelevant, for certain purposes. As it is presented, it is not an argument against counterparts, it is simply to show that they suck in a particular way. So, incorporating counterparts by way of premises (2) to (5), that is to hold them true simpliciter is required. This may just be a preference on my part, but I think stripping Salmon's argument of premises (2) to (5) is to eliminate one of the awesome characteristics that it has as an argument type. Essentially what Salmon's irrelevance argument is saying to the genuine realist is this, okay fine I'll accept your counterpart theory, and even better I'll accept the identity fixing postulates that you give which accompany it, but even when I play your game by your rules, counterparts are still useless at doing what it is that you want them to do. To Dan's credit, I do believe that he is correct, in that this argument lends to the further argument against CT and GR. In which case, it would be best if the argument does not entail things about counterparts, but that would be an additional argument, which would require additional premises, and not that present in Divers as Salmon's irrelevance objection.

The second point is that for the reconstruction, (8) to (10), that Dan offers, he has essentially chosen to eliminate the premises (2) to (5) because they are plainly superfluous. But as he notes later, the justification for (9) will require invoking the postulates (P5) and (P6) when asserting which entailment relations hold under GR. So, I suppose (and Chris your input on this particular point would be appreciated), should the postulates (P5) and (P6) be written as explicit premises within the argument's reconstruction, or taken as the implicit justification for premise (9)? In either case they are still being used, as it were, by and in the argument. Personally, I would prefer to keep them as explicit premises. It seems only fair to have them as individual premises, so that if an objector to the argument wishes to deny either (for what it is worth denying either postulate, in the hopes of denying the conclusion, seems like an untenable and misguided way to object to this argument) they can do so.

Adam on Chelsey, on Divers, on Salmon:

Adam makes one main recommendation for the reconstruction of my Divers on Salmon (other then making it valid). He thinks a more faithful reconstruction would look something 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)

Adam argues that the conclusion, that it is therefore not the case that an ordinary possibility sentence of modal English is about a counterpart, in my reconstruction is unfaithful to that presented in Divers. In Adam's reconstruction the conclusion states that, it is not the case that 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. I both agree and disagree with Adam on this point (as is my usual m.o.). I agree that my previous reformulation does not quite capture faithfully the conclusion intended by the irrelevance argument. The problem does not lie with English not being about couterparts, which is the conclusion that my reconstruction delivers. But, I disagree that his reformulation is any better. For the argument to be faithful to one presented in Divers, it must be the case that the individuals (or counterparts) are doing the work. That is, that the roles expressed in the conclusion of his reconstruction need to be reversed and reformulated as such:

7. It is not the case that the modal truth of the individuals are relevant to the non-modal (metalinguistic) truth-conditions specified by (CT) for ordinary (object language) sentences of modal English, in which they are about.

Adam also suggests that the argument can be shortened to the following:

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)

I must admit, I'm a little confused here. Premises (1) to (3) cannot derive the conclusion in (6) using hypothetical syllogism. (4) and (5) can because they are conditionals. But, then the conclusion in (6) would read ~P → (R&S). Which is not the conclusion that is needed for the irrelevance argument. If I am mistaken here please let me know.

The other points that are brought up by Adam pertain to the fact that the argument as it is presented in Divers is not faithful to Salmon. I think he is correct in this analysis. Unfortunately, for the present purposes of this assignment I am unsure where my loyalties are supposed to lay. Am I supposed to reconstruct the argument as faithfully to Divers' text, or should I be going to the original source and be faithful to that presented by Salmon?

Sticking with the Divers text and taking the previously mentioned, new and hopefully more faithful conclusion, the irrelevance argument can be reformulated once again as follows:

Irrelevance objection from Salmon (via Divers) made valid:

1. If the modal truth of a counterpart (i.e. facts about an individual y having a certain feature F, at possible world, w) are relevant to the non-modal (metalinguistic) truth-conditions specified by (CT) for ordinary (object language) sentences of modal English, in which they are about, 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.
2. Nothing is a counterpart of anything else in its world (P5).
3. Anything in a world is a counterpart of itself (P6).
4. (2) & (3)
5. If (4), then it is not the case that 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. (i.e. not another counterpart, y, having a certain feature F, at possible world, w).
6. So, it is not the case that 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. (i.e. not another counterpart, y, having a certain feature F, at possible world, w).
7. Therefore, it is not the case that the modal truth of a counterpart (i.e. facts about an individual y having a certain feature F, at possible world, w) are relevant to the non-modal (metalinguistic) truth-conditions specified by (CT) for ordinary (object language) sentences of modal English, in which they are about.

Sorry for the wordiness, but hopefully it's v.f.a. (valid, faithful and awesome).

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 &amp;amp; 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.

Comment on "Counterparts and Actuality"

This paper was posted a while back, here's the link:
http://www.princeton.edu/%7Efara/papers/counterparts.pdf
for this post:
<> is the possibility daimond
[] is the necessity box
A is the universal quantifier
E is the existential quantifier

Williamson and Fara argue that any modal theory needs an actuality operator to handle sentences like this:

(4) it might have been that everyone who is in fact rich was poor

Without an actuality operator, CT would translate it as follows:
(4*) <>Ax(Rx -> Px)
or
(4**) Ax(Rx -> <>Px)

(4*) says that possibly all rich people are poor. That's not a good translation of (4) because (4*) excludes the possibility of some people who are in fact poor being rich while (4) does not. (4**) is not a good translation because it asserts that each individual who is rich may have been poor. This is different from asserting that it could've been the case that they're all poor.

Williamson and Fara introduce the actuality operator to solve this problem:
for any sentence S:
ACT S is true at a world iff S is true at the actual world

This provides a correct translation of (4):
(4A) <>Ax(ACT(Rx) -> Px)

The problems arise when the question is raised of how this should be implemented into counterpart theory. W&F propose a few formulations:
(L1) ACT S(a)w is Ex(Ix@&Cxa&S@(x))
(L2) ACT S(a)w is Ax[(S@(x">Ix@&Cxa)->S@(x)]
These fail because they validate known contradictions. For instance, L1 translates
(12) <>Ex(ACT Fx iff ACT~Fx)
into
(13) ExEw(Ixw&[Ey(Iy@&Cyx&Fy)iffEy(Iy@&Cyx&~Fy)]).
(12) is inconsistent while (13) can be satisfied in certain models of CT. Bad translation. To see how (13) is satisfiable, just picture a situation in which one has more than one actual world counterpart. (L2) and the rest of W&F's proposals have the same drawback. They find, for each one, some logical contradiction that is translated into a satisfiable sentence.
W&F are very thorough, and ultimately correct (I think) about the formulations of the CT actuality operator they come up with.
My confusion is what stops CT from adopting a more neutral ACT operator that doesn't invoke counterparts? Consider:
(AD) ACT S(a)w iff Ex(Ix@&x=a&S(x))
(L1) & (L2) as well as the other suggestions in the paper try and make it so you can speak of non-actual individuals (that are counterparts of actual individuals) and say true things of how they actually are. Consider the hypothetical Humphrey at world w that didn't win. With the actuality operator (L1) you could truely say of him that he actually lost. That is:
(H) Humphrey(w) actually lost
This would come out false on (AD) translation:
(HAD) Ex(Ix@&x=h&L(x))
This may be seen as a drawback of (AD), but I don't see it as such. It's perfectly reasonable to say that the non-actual winner Humphrey actually won. That is to say, evaluated at this world, the proposition about non-actual Humpthrey to the effect that he won is true. Another worry may be that (H2) would get the wrong result.
(H2) Necessarily Humphrey actually lost.
(HAD) would translate that to:
(H2*) AwAx((Cxh&Ixw)->Ey(Iy@&y=x&L(y))
But (H2*) is false. So we get the right result again. This fits for Lewis's indexical account of "Actually". What about such sentences evaluated at other worlds? Surely at world w, all ordinary sentences involving the actuality operator shouldn't come at false! To this I'd answer that when evaluating a sentence at another world, substitute that world in for @. That's not ad-hoc, after all according to CT according to that world w IS the actual world.
I haven't checked (AD) thoroughly, but I believe the problems involving the other trials for an actuality operator were problems in invoking the counterpart relation. (AD) does not.

New Readings

For the next meeting please read Scott Soames's "Actually". There should be enough overlap with views discussed in Divers for you to formulate objections to and questions about Soames's view. So please bring objections to and questions about Soames's view to the next meeting.

After Soames, I had previously proposed we read a paper by Kit Fine. But there's a brand new paper by Otavio Bueno and Edward Zalta, "A Defense of Actualist Realism", that is a direct response to Divers. So I think we should look at it before looking at Fine (and before we forget too much of Divers).

Then I think we should read Kit Fine's "The Problem of Possibilia".

Homework 3

Here's the revised argument against GR. I've re-ordered the premises as to make it a valid IP. I've also added a new premise and new conclusion so that the addition of 013 makes it possible to object to a premise. A quick note about Adam's criticism. He claimed that the definition given of an alien natural property can be generalized as a definition for what it means for a natural property to be alien of a world. This is true. And if we were to generalize the definition, his criticism would be correct. However, for the purposes here I need not, and did not make this definition general. Given that "alien natural property" as defined here is Adam's "@ alien natural property", that criticism is avoided and no revision is required. However the criticism is relevant to the response I gave. I did invoke Adam's generalized definition when stating the principle 013. I didn't define what it means for a property to be "alien to a world". For the purposes of stating 013, let the definition be the same, except replace @ with w*.

Def’n:“An alien natural property is a natural property that is not instantiated by any individual in @, and is not analyzable as a conjunctive or structural property built up from constituents that are all instantiated by parts of @”Divers p.115 quoting Lewis (1986a: 91)Let’s assume that Lewis can give a good account of natural properties, since if he can’t that is an objection in and of itself. The objection can proceed as follows:
1) It’s possible for alien natural properties to be instantiated
2) If (4) then (5)
3) If (5) then (6)
4) Postulates O1 – O12 exhaustively represent reality (as required by GR to give a complete analysis of modal concepts). (assume for reductio)
5) No individuals (speaking un-restrictedly) instantiate alien natural properties
6) ~(1)
7) (1)&(6)
8) ~(4)
9) if (8) then (10)
10) GR does not give a complete account of modal concepts

In support of (1), we intuitively think it’s possible that there are ways things could have been that are different than ways things are on a fundamental level. That is, things could have been so different that Humean recombination cannot account for that difference. For instance, things could have been massless, colourless, shapeless, but still had properties we can’t conceive of. Alternately, the universe could have been Newtonian.In support of (2), postulates O1 – O12 postulate the actual world, and us. It then uses postulates O8-O10 and O12 to generate the rest of the worlds. Any individuals generated in this manner can’t be fundamentally different from actual individuals in the way previously described.In support of (3), according to GR P is possible iff P at some world. If no individual at any world instantiates natural property A, then A is impossible by the lights of GR. In support of (9), if the ontological componend of GR does not include all things we think are possible, then by CT not all things we think are possible are possible. If by GR's lights some pre-theoretic possibilities are impossible, then GR is does not give a complete account of possibility.
The most assailable principle is (9). A proponent of GR may argue that while 01-012 are inadequate it may be supplemented, and promise a suitable addition to the postulates O13 - On that may account for alien natural properties. Here’s a possible O13 that may account for alien properties:
O13: For any world w and for any number n, there exists a world w* such that w* has n number of instantiated natural properties that are alien to w AND w* has parts that instantiate every natural property in w.
O13 will guarantee that there is a plenitude of alien natural properties that are instantiated in other worlds. It’s also more defensible in the face of the argument in divers on p 118. That objection tries to set up a model which satisfies (GR) + (OAN), and then set up another model that also satisfies (GR) + (OAN) but fails to include all of the alien natural properties of the first model (thus saying GR + OAN is apt to lose out on some). That is more difficult with (GR) + (O13). Should you generate a w* with respect of @, such that it excludes some number of alien natural properties, O13 posits another world w** with properties alien to both w* and @ that may include them after all. O13 will posit a world that includes n alien natural properties as well as all the natural properties of @. It will in turn posit a world with all of those natural properties as well as other natural properties not yet accounted for, and iterate. Recombination (O12) guarantees that there are worlds with only alien properties, and different combinations of those properties. Note also that alien properties that only come in pairs (or triplets, or quadruplets etc.) can be generated using O13. While it’s still true that no particular alien property is guaranteed to be accounted for, none are explicitly un-accounted for. The drawback is that by O13, it’s impossible for a world to instantiate all of the natural properties. O12 would like there to be such a world, but the qualification “if there is a spacetime big enough to hold them all” would have to be used to say that this is an exception.Should that fail, the next most assailable principle is (3). One would need a good account of natural properties to ensure that O1-O12 don’t in fact capture them all. In other words, this objection may be a little premature

Homework V.3

First of all, sorry to Dan for bumping down his new and excellent post. Here's my homework for the week:

Here is a revised version of the objection from alien natural properties. I believe I’ve incorporated the changes suggested by Dan and Chelsey (with respect to the original version of the objection I had presented). Here goes:

(1) There is at least one world in which there is an individual that instantiates an α-alien natural property.
(2) If there is at least one world in which there is an individual that instantiates an α-alien natural property, then there is an infinite sequence of α-alien instantiable natural properties.
(3) If there is an infinite sequence of α-alien instantiable natural properties, then it is not the case that the GR analysis of the modal concepts is complete.
(4) If it is not the case that the GR analysis of the modal concepts is complete, then it is not the case that the ontological component of GR (postulates O1-O11 and the principle of recombination) provides an accurate analysis of the modal concepts.
(5) It is not the case that the ontological component of GR (postulates O1-O11 and the principle of recombination) provides an accurate analysis of the modal concepts.(1-4)*

Divers provides a justification of the move from (1) above to the conditional in (2) with the following sub-argument. Let ‘L1’ designate the exhaustive list of actually instantiated natural properties.

(1a) L1 is the list of all the actually instantiated natural properties.

We have it pre-theoretically that, whatever natural properties are included in L1, it could have been the case that L1 contained one more natural property. In other words, (2a) and (3a):

(2a)(1a)-->by ‘alienation’ and recombination there exists a world W1 such that the individuals in W1 instantiate all the properties in L1, plus some natural property X1 not contained in L1. Call this list L2.
(3a)By ‘alienation’ and recombination there exists a world W1 such that the individuals in W1 instantiate all the properties in L1, plus some natural property X1 not contained in L1. Call this list L2.

The move from (1a)-(3a) above establishes (1) in the main argument. Next, since the (individuals in) the actual world could have instantiated L2 (instead of L1), we have:

(4a)(3a)-->the actual world could have instantiated all the properties in L2.
(5a)The actual world could have instantiated all the properties in L2.

However, if the actual world could have instantiated L2, then it is also the case that the actual world could have instantiated some new list of natural properties, distinct from (L1 and L2). In other words, reiteration of the move in (2a) will generate an infinite sequence of worlds containing more and more alien natural properties:

(6a)(5a)-->successive applications of alienation + the principle of recombination will generate an infinite sequence of worlds containing more and more alien natural properties.
(7a)Successive applications of alienation + the principle of recombination will generate an infinite sequence of worlds containing more and more alien natural properties.

And, if that’s right, then there exist infinitely many instantiable alien natural properties:

(8a)(7a)-->(9a)
(9a)There are infinitely many instantiable alien natural properties.

To justify (3) Divers argues that (i) the ontological postulates of GR are unable to generate a set of worlds such that an infinite sequence of α-alien natural properties are instantiated among individuals existing at those worlds and (ii) that completeness requires GR to provide a world to match all of the possibilities we intuitively think there are. And finally, Divers justifies (4) by arguing that, since completeness is one criterion of extensional accuracy, it follows that if GR does not deliver completeness, then it does not deliver an accurate analysis of the modal concepts.

I proposed that we conceive of the natural properties as tropes. This suggestion did not receive a response, so I will just restate it briefly. Please note that I don’t necessarily think this is a satisfying response to the problem, only that it seems perfectly consistent with what Lewis says on the topic of natural properties. For Lewis says the following:

I noted that one way to characterize natural properties was to help ourselves to Armstrong’s theory of universals. But in ‘Against Structural Universals’ I note that one part of that theory gives us trouble: the part about structural universals. Those are universals built up somehow- the problem is how?- out of simpler universals. I note that a similar theory of structural tropes-particular property instances- has no parallel problem; and, further, that accepting a sparse theory of tropes would be another good way to give ourselves the distinction between natural and gerrymandered properties.**

So: Lewis thinks that it is plausible to suppose that the natural properties are tropes. I suggested that the standard metaphysical account of the nature of tropes leads to the following two principles:

(T1)∀x {[Wx-->∃y (Ty & Iyx)] & ∀z (Tz & Izx)--> z = y}
(T2)∀x {[Wx -->∃y (Ty & Iyx)] & ∀z [(Wz & z ≠x)-->~Iyz]}

(where Wx is ‘x is a possible world’, Tx is ‘x is a class of natural-quality tropes’ and Ixy is ‘x is instantiated by y’).

And, if this is right, then the argument given by Divers in support of (2) is blocked, since the conditional in (2a) would be false. I think it would go something like this:

(1t) All and only the perfectly natural properties are tropes.
(2t) (1t)--> ~[(1a)-->(3a)]
(3t) ~[(1a)-->(3a)] (1t,2t)
(4t) ~(1a) v ~(3a) (3t)
(5t) ~ ~ (1a)
(6t) ~ (3a) (4t, 5t)

(5t) seems right if we think it is plausible to suppose that there is some sort of ideal list of the natural properties that are actually instantiated. So it seems like the above is one argument Lewis could give on behalf of GR. It doesn’t seem to be a happy one, but it would work. The view would appear to be one according to which extreme modal haecceitism is true, since this is a consequence of identifying the natural properties with tropes.

Alien Hecceatism

I'd just like to recap here the debate that took place in class on teus between Chris & me. The spark was the following argument against linguistic ersatzism:
Consider and alien individual a, and another alien individual b. Consider two worlds that are qualitatively identical, but in w2 a is qualitatively identical to b in w1, and vice versa. The argument can proceed as follows:
1d) Our language (L) cannot distinguish between w1 and w2
2d) w1 and w2 represent to distinct possibilities
3d) (1)&(2)
4d) if (3) then (5)
5d) Linguistic ersatzism conflates some possibilities that should be distinct

After some deliberation, it was noted that a & b must be unnamable, else (1) does not hold.
I (with waving hands) wanted to deny (2). Chris gave a motivation for why (2) should be accepted:
(1) The property of being nameable is ontologically insignificant
(2) Were a and b nameable, then (2d) would hold
(3) (1)&(2)
(4) if (3) then (5)
(5) (2d)

I countered with this:

(6) If a is nameable then a is describable in terms of actual things
(7) being describably in terms of actual things is ontologically significant
(8) (6)&(7)
(9) if (8) then ~(1)
(10) ~(1)

Leaving aside the obvious issue of spelling out what it is to be describably in terms of actual things, or what ontological difference that's supposed to make, let's move on. Let's call things describable in terms of actuals "impure hypotheticals" and things not describable in terms of actuals "pure hypotheticals". So the handsom man resulting from the sperm that Adam came from and the egg that Chelsea came from (call him Chadam) would be an impure hypothetical. Gandalf the White from Tolken's fantasy would be a pure hypothetical. So my position could be called anti-hecceatism about pure hypotheticals.
Chris then posed a different argument:
Consider a purely descriptive world W that is complete in the appropriate ways and conforms to tolken's Lord of the Rings universe. Chris's argument could run like so:
(11) anti-hecceatism about pure hypotheticals
(12) if (1) then (3)
(13) W would be one possibility
(14) if (13) then (15)
(15) W is necessarily exactly one possibility
(16) Necessarily if W were actual W would be many possibilities
(17) if (15) then (18)
(18) necessarily W is not many possibilities
(19) Necessarily W is not actual (16, 18, necessary MT)

(19) is very bad if we want W to be a real possibility. I responded by denying (12). W is not one possibility because any actual individuals could play the individual roles in W, making for distinct possibilities. For instance, Chris could be qualitatively identical to Gandalf in W, or Chelsea could be.

I think I captured most of the main arguments for this issue. A couple motivations for anti-hecceatism about pure hypotheticals:
A) It seems like there's a difference between Chadam and Gandalf (just an intuition)
B) All accidental properties belonging to a pure hypothetical are stipulative. This is not true for impure hypotheticals (for instance, I need not stipulate that Chadam is male). This may support the intuition that there is little more to pure hypotheticals than their stipulated properties.

The Reasoner

The Reasoner contains new, short articles of philosophical interest. Several in the new issue are relevant to modality.

Cantorian Arguments

Divers is pretty slim on how Cantorian arguments are supposed to work. I thought this exchange between Grim and Plantinga might be helpful.

Propositions and Sets

Today we mentioned some reasons for thinking that propositions are not reducible to set-like entities. The objection explicitly considered by Divers is one from Plantinga:

1. Some propositions have truth values and no sets have truth values.
2. If (1), then some propositions are not sets. (by Leibniz's Law)
3. If some propositions are not sets, then not all propositions are sets.
4. If not all propositions are sets, then propositions are not reducible to sets.
5. So if (1), then propositions are not reducible to sets. (2-4)
6. So propositions are not reducible to sets. (1,5)

Plantinga contends that (1) is obvious. Divers thinks it is not obvious. Let's concede the point to Divers unless someone can come up with another way to support (1).

Another objection, from Jeff King's SEP entry on structured propositions, is as follows:

7. If some sets are propositions, then some sets have truth values (modal properties, etc) and others do not.
8. If some sets have truth values and others do not, then there is an explanation of why this is the case.
9. So if some sets are propositions, then there is an explanation of why some sets have truth values and others do not. (7,8)
10. There is no explanation of why some sets have truth values and others do not.
11. So it's not the case that some sets are propositions. (7,10)

I think someone like Lewis can resist (10) with some plausibility. Here's a view that Lewis and some of his opponents, like Salmon and Soames, both seem to hold:

Propositions are pieces of information semantically encoded by well-formed declarative sentences. They are truth-apt objects of cognitive attitudes (like belief, etc).

This characterization serves to specify the role of propositions. Something is a proposition iff it's the best candidate for that role. If that turns out to be shoes or fish or whatever, then propositions may be identified with shoes, fish, whatever. Now Lewis holds that certain sets occupy this role. (For what it's worth, Salmon and Soames give their theories of propositions in set-theoretic terms but are not explicit about whether the set-theoretic entities are supposed to be propositions or if they merely represent them.) If he's right about that, then it seems he has a not implausible explanation of why (e.g.) some sets are true and others have no truth value. There's more that can be said about this objection, but I'll leave it at that for now.

King's second objection is a version of the Benacerraf problem. (Link requires JSTOR access.) It requires a bit of set-up. Consider sentence *:

* Brendan loves Adam

Suppose we held that propositions were ordered n-tuples. Consider the following ordered triples:

(i) bLa
(ii) aLb
(iii) Lab
(iv) Lba
(v) abL
(vi) baL

Furthermore, there are many ways to construct ordered n-tuples. For each way, there is a non-equivalent set that corresponds to each of (i)-(vi). Let's suppose there are only seven ways. Then there are 42 sets: each of (i)-(vi) constructed in each of the seven ways. Here's the objection:

12. If propositions are sets, then there is a unique most eligible candidate among the sets for being the proposition expressed by * in English.
13. There are (at least) 42 sets that are equally eligible candidates for being the proposition expressed by * in English.
14. If there are (at least) 42 sets that are equally eligible candidates for being the proposition expressed by * in English, then there is no unique most eligible candidate among the sets for being the proposition expressed by * in English.
15. So there is no unique most eligible candidate among the sets for being the proposition expressed by * in English. (13,14)
16. So it's not the case that propositions are sets. (12,15)

Carl offered a reason for denying (1): given a multiplicity of equally eligible candidates, a proponent of the "propositions are sets" view could hold that it's indeterminate which of the 42 sets is the proposition that Brendan loves Adam. One could add that picking any of the 42 to represent the information that Brendan loves Adam is harmless as long as one makes the appropriately uniform choices for representing other propositions. One could also hold that it's appropriate to talk about the proposition that Brendan loves Adam iff according to any legitimate way of eliminating the indeterminacy, there is only one candidate for the proposition.

Some thoughts:

A. The sharpenings would have to be done with care. Suppose one pursued the same tack for numbers. There may be admissible sharpenings for propositions according to which S is a proposition and admissible sharpenings for numbers according to which S is a number; 0, for instance. Then 0 would have a truth value and it would be possible to believe 0. That's no good. But it seems like it could be prevented by adding the relevant constraints on admissible sharpenings.

B. I worry on the indeterminacy proposal that it would be true that, were we to have decided on a different sharpening, then the proposition expressed by 'Brendan loves Adam' would have been the proposition expressed by 'Adam loves Brendan' (while all the facts about the English sentences 'Brendan loves Adam' and 'Adam loves Brendan' remain fixed). The counterfactual strikes me as false. There are probably ways around this too: there are similar views about vagueness according to which there are several admissible sharpenings for 'red' and 'orange' and some things that are red under one sharpening are orange on another, but under no sharpening are some things both red and orange. But note a lack of parallel: all of the set-theoretic candidates for being the proposition that Brendan loves Adam are the set-theoretic candidates for the proposition that Adam loves Brendan. In spite of the disanalogy, I confess that the objection does not strike me as especially serious.

C. It would be self-refuting for me to believe that there are no beliefs. On one usage of 'belief', the word refers to the objects of belief. On this understanding, 'I believe there are no beliefs' expresses a proposition that entails that I bear a relation to the proposition that there are no propositions. Contrast this with my (pretend) belief that there are no sets. This does not seem similarly self-refuting. But it would be on the "propositions are sets" view. This is the basis for a Leibniz's Law objection, but I think it's better than Plantinga's because it does not rest on the contention that it's just obvious that sets don't have truth-values. Divers will cry "hyperintensionality" here, but I don't buy it. The two beliefs really strike me as different in the way described.

D. There are cardinality problems for the view that propositions are sets. There are several ways to state these. Here's one. The proposition that absolutely everything is self-identical is (logically) true. But there is no ordered pair with absolutely everything as one member and the property of being self-identical as the other. That is because there is no set that has as a proper subset absolutely everything. That is because if sets are things, there are too many things for all of them to be a subset (even an improper subset) of a set. (Given any set, the set of all of its subsets has a strictly greater cardinality. So any candidate for being a set that has absolutely everything as a subset is such that there's a "bigger" set: the set of all of its subsets.) Furthermore, if there were a proposition that absolutely everything is self-identical, and it was a set, then it would be a proper subset of itself (since it, too, is one of absolutely everything). This violates standard axioms of set theory ("well-foundedness"). Upshot: if the "propositions are sets" view is true, then there is no proposition that absolutely everything is self-identical. So if propositions are sets, then some logical truth is not true.

I take the last sort of problem to be the most serious. But note it will not do to rest with the claim that propositions are not sets. A positive theory is needed. And part of the burden of the proponent of the positive theory is to show that propositions don't run into cardinality problems anyway. More work is called for.

At any rate, my main purpose in posting this was to recap some of the discussion and elicit further thoughts on the thesis that all propositions are sets.

Reducing Modality

For "Reduction is Good" see Sider's "Reductive Theories of Modality". (For his in-progress attempt at a non-Lewisian reductive theory see here. See also his "The Ersatz Pluriverse" for a detailed formulation of the sort of AR view I was sketching last time.)

For "Reduction is Bad" see Plantinga's "Two Concepts of Modality: Modal Realism and Modal Reductionism" (JSTOR access required.)

Objection from Alien Properties

Def’n:
“An alien natural property is a natural property that is not instantiated by any individual in @, and is not analyzable as a conjunctive or structural property built up from constituents that are all instantiated by parts of @”
Divers p.115 quoting Lewis (1986a: 91)
Let’s assume that Lewis can give a good account of natural properties, since if he can’t that is an objection in and of itself. The objection can proceed as follows:
1) It’s possible for alien natural properties to be instantiated
2) Postulates O1 – O12 exhaustively and exclusively represent reality.
3) If (2) then (4)
4) No individuals (speaking un-restrictedly) instantiate alien natural properties
5) If (4) then (6)
6) ~(1)
7) (1)&(6)
8) ~(2)

In support of (1), we intuitively think it’s possible that there are ways things could have been that are different than ways things are on a fundamental level. That is, things could have been so different that Humean recombination cannot account for that difference. For instance, things could have been massless, colourless, shapeless, but still had properties we can’t conceive of. Alternately, the universe could have been Newtonian.
In support of (3), postulates O1 – O12 postulate the actual world, and us. It then uses postulates O8-O10 and O12 to generate the rest of the worlds. Any individuals generated in this manner can’t be fundamentally different from actual individuals in the way previously described.
In support of (5), according to GR P is possible iff P at some world. If no individual at any world instantiates natural property A, then A is impossible by the lights of GR.
The most assailable principle is (2). A proponent of GR may agree with the conclusion, and promise a suitable addition to the postulates O13 - On that may account for alien natural properties. Here’s a possible O13 that may account for alien properties:

O13: For any world w and for any number n, there exists a world w* such that w* has n number of instantiated natural properties that are alien to w AND w* has parts that instantiate every natural property in w.

O13 will guarantee that there are a plenitude of alien natural properties that are instantiated in other worlds. It’s also more defensible in the face of the argument in divers on p 118. That objection tries to set up a model which satisfies (GR) + (OAN), and then set up another model that also satisfies (GR) + (OAN) but fails to include all of the alien natural properties of the first model (thus saying GR + OAN is apt to lose out on some). That is more difficult with (GR) + (O13). Should you generate a w* with respect of @, such that it excludes some number of alien natural properties, O13 posits another world w** with properties alien to both w* and @ that may include them after all. O13 will posit a world that includes n alien natural properties as well as all the natural properties of @. It will in turn posit a world with all of those natural properties as well as other natural properties not yet accounted for, and iterate. Recombination (O12) guarantees that there are worlds with only alien properties, and different combinations of those properties. Note also that alien properties that only come in pairs (or triplets, or quadruplets etc.) can be generated using O13. While it’s still true that no particular alien property is guaranteed to be accounted for, none are explicitly un-accounted for. The drawback is that by O13, it’s impossible for a world to instantiate all of the natural properties. O12 would like there to be such a world, but the qualification “if there is a spacetime big enough to hold them all” would have to be used to say that this is an exception.
Should that fail, the next most assailable principle is (3). One would need a good account of natural properties to ensure that O1-O12 don’t in fact capture them all. In other words, this objection may be a little premature.