Let's say 200 years ago it was discovered that gold has the atomic number 79.

My claim: possibility is conceivability without known defeaters.

201 years ago it was possible (because it was conceivable) that gold had any atomic number except for the ones which they'd already mapped (hydrogen=1, helium=2 were known defeaters against gold being 1 or 2). Note: and they may have known of some upper bound of stability which it couldn't be bigger than. So it may have been possible for the atomic number to be greater than 2 but less that (e.g.) 250.

Today it is not possible that gold has any atomic number other than 79 because the notion of any other weight is defeated by the evidence we've collected. Today's child may think it is possible that gold could have any atomic weight, but the child's claim would be incorrect relative to what today's educated people know. Also, the well educated adult from 201 years ago's claim is incorrect relative to today's knowledge, but relative to their knowledge 201 year ago, it was correct.

Similarly, we can safely assume that many of our current claims about possible states of affairs will be false relative to some future level of knowledge. However, today they are perfectly assertable/true/acceptable. Frankly, I'd think in this view, possibility claims are only ever assertable, never true (although 'true' is an easy shorthand, just like how scientific theories and the existence of their entities are 'true').

Robust views of possibility are not needed to explain why we make statements about possibility. Possibility statements are certainly useful, given our lack of knowledge, need for personal accountability, and sometimes just unwillingness to perform the required calculation, so I wouldn't suggest we stop using them.

Possibility is thus either epistemic or instrumental and therefore isn't required in our metaphysics.

Kripke's claim: no.

Explain.

- Jason Christie

## Thursday, September 6, 2007

## Wednesday, September 5, 2007

### Wormy Lumps

Two of Lewis' Five Objections:

Trans-World Glue

(found on pages 49-50 in Weatherson's “Stages, Worms, Slices and Lumps”, and 218 of Lewis'

“The general worry here is determining which stages are part of a particular worm and/or lump. That is, the worry is about how the worm or lump is held together. There are two ways in which this seems more problematic for lumps than for worm. First, the various parts of a worm are held together by a causal dependence of some parts on others. But since worlds are causally isolated, this cannot be the way that worms are held together. Secondly, to the extent that we need a similarity relation on top of the causal relation for worms, it is the similarity of one part to the nearby parts. Because there is no one-dimensional ordering of modal space matching the ordering of temporal space, the relevant similarities will have to be “a matter of direct similarity between stages.” (Lewis 1986a: 218).” (49)

Run-down of the above argument in two pieces:

A.)

(1)If the worm / lump theory is true, then the various parts of a worm are held together by a causal dependence of some parts on others.

(2)If the various parts of a worm are held together by a causal dependence of some parts on others, then worlds must not be causally isolated.

(3)So, if the worm / lump theory is true, then worlds must not be causally isolated.

(4)Worlds are causally isolated.

(5)Therefore, the worm / lump theory is not true.

(1) p → q premise (1)

(2) q → ~r premise (2)

(3) p → ~r sub-conclusion (1)-(2) HS

(4) ~~r → ~p (3) CONTRA

(5) r → ~p (4) DN

(6) r premise (4)

(7) ~p conclusion (5)-(6) MP

(N.B. Yes, I know there are two extra steps in the inference argument when reconstructed. I put in contraposition and double negation in there as extra steps to make everything explicit.)

B.)

(1)If the worm / lump theory is true, then the various parts of a worm are held together by both a causal dependence of some parts on others and a direct similarity relation between the stages or worms.

(2)If the various parts of a worm are held together by both a causal dependence of some parts on others and a direct similarity relation between the stages or worms, then there is a one-dimensional ordering of modal space matching the ordering of temporal space.

(3)So, if the worm / lump theory is true, then there is a one-dimensional ordering of modal space matching the ordering of temporal space.

(4)There is no one-dimensional ordering of modal space matching the ordering of temporal space.

(5)So, the worm / lump theory is not true.

(1) p → (q & r) premise (1)

(2) (q & r) → s premise (2)

(3) p → s (1)-(2) HS

(4) ~s premise (4)

(5) ~p (3)-(4) MT

Now here is the Lewis version of the argument from

“(1) The temporal parts of an ordinary thing that perdures through time are united as much by relations of causal dependence as by qualitative similarity. In fact, both work together: the reason the thing changes only gradually, for the most part, is that the way it is at any time depends causally on the way it was at the time just before, and this dependence is by and large conservative. However, there can be no trans-world causation to unite counterparts. Their unification into trans-world individual can only be by similarity.

(2) To the extent that unification by similarity does enter into perdurance through time, what matters is not so much the long-range similarity between separated stages, but rather the linkage of separated stages by many steps of short-range similarity between close stages in a one-dimensional ordering. Change is mostly gradual, but not much limited overall. There is no such one-dimensional ordering given in the modal case. So any path is as good as any other; and what's more, in logical space anything that can happen does. So linkage by a chain of short steps is too easy: it will take us more or less from anywhere to anywhere. Therefore it must be disregarded; the unification of trans-world individuals must be a matter of direct similarity between stages.”( L 218)

Here's my difficulty. After the first couple of readings of Weatherson it seemed that his rendition of Lewis' argument was reasonably faithful to Lewis. But, now after reading it over again a few times, I'm not so sure it is. The reason being is that the argument presented causes problems for any wormy theory, which is Lewis' theory as well. Which means Lewis has a hard time denying any of the premises above, as they are writen. Bad. Lewis forwards these as arguments that cause greater difficulties for lumps, then for worm theories such as his own. Something needs to be added that will capture the difference between Weatherson's lumps and Lewis' worm. Hopefully, I'm not completely wrong here, but the something should be that Lewis can chose between the counterpart or mereological components of his view. Whereas Weatherson doesn't accept counterparts (it's just the lump, even if the lump includes what Lewis would refer to as a counterpart of

Anyhow tell me what ya' think. And yeah, tell me what needs to be clarified and further explicated.

Trans-World Glue

(found on pages 49-50 in Weatherson's “Stages, Worms, Slices and Lumps”, and 218 of Lewis'

*Possible Worlds*)“The general worry here is determining which stages are part of a particular worm and/or lump. That is, the worry is about how the worm or lump is held together. There are two ways in which this seems more problematic for lumps than for worm. First, the various parts of a worm are held together by a causal dependence of some parts on others. But since worlds are causally isolated, this cannot be the way that worms are held together. Secondly, to the extent that we need a similarity relation on top of the causal relation for worms, it is the similarity of one part to the nearby parts. Because there is no one-dimensional ordering of modal space matching the ordering of temporal space, the relevant similarities will have to be “a matter of direct similarity between stages.” (Lewis 1986a: 218).” (49)

Run-down of the above argument in two pieces:

A.)

(1)If the worm / lump theory is true, then the various parts of a worm are held together by a causal dependence of some parts on others.

(2)If the various parts of a worm are held together by a causal dependence of some parts on others, then worlds must not be causally isolated.

(3)So, if the worm / lump theory is true, then worlds must not be causally isolated.

(4)Worlds are causally isolated.

(5)Therefore, the worm / lump theory is not true.

(1) p → q premise (1)

(2) q → ~r premise (2)

(3) p → ~r sub-conclusion (1)-(2) HS

(4) ~~r → ~p (3) CONTRA

(5) r → ~p (4) DN

(6) r premise (4)

(7) ~p conclusion (5)-(6) MP

(N.B. Yes, I know there are two extra steps in the inference argument when reconstructed. I put in contraposition and double negation in there as extra steps to make everything explicit.)

B.)

(1)If the worm / lump theory is true, then the various parts of a worm are held together by both a causal dependence of some parts on others and a direct similarity relation between the stages or worms.

(2)If the various parts of a worm are held together by both a causal dependence of some parts on others and a direct similarity relation between the stages or worms, then there is a one-dimensional ordering of modal space matching the ordering of temporal space.

(3)So, if the worm / lump theory is true, then there is a one-dimensional ordering of modal space matching the ordering of temporal space.

(4)There is no one-dimensional ordering of modal space matching the ordering of temporal space.

(5)So, the worm / lump theory is not true.

(1) p → (q & r) premise (1)

(2) (q & r) → s premise (2)

(3) p → s (1)-(2) HS

(4) ~s premise (4)

(5) ~p (3)-(4) MT

Now here is the Lewis version of the argument from

*Possible Worlds*:“(1) The temporal parts of an ordinary thing that perdures through time are united as much by relations of causal dependence as by qualitative similarity. In fact, both work together: the reason the thing changes only gradually, for the most part, is that the way it is at any time depends causally on the way it was at the time just before, and this dependence is by and large conservative. However, there can be no trans-world causation to unite counterparts. Their unification into trans-world individual can only be by similarity.

(2) To the extent that unification by similarity does enter into perdurance through time, what matters is not so much the long-range similarity between separated stages, but rather the linkage of separated stages by many steps of short-range similarity between close stages in a one-dimensional ordering. Change is mostly gradual, but not much limited overall. There is no such one-dimensional ordering given in the modal case. So any path is as good as any other; and what's more, in logical space anything that can happen does. So linkage by a chain of short steps is too easy: it will take us more or less from anywhere to anywhere. Therefore it must be disregarded; the unification of trans-world individuals must be a matter of direct similarity between stages.”( L 218)

Here's my difficulty. After the first couple of readings of Weatherson it seemed that his rendition of Lewis' argument was reasonably faithful to Lewis. But, now after reading it over again a few times, I'm not so sure it is. The reason being is that the argument presented causes problems for any wormy theory, which is Lewis' theory as well. Which means Lewis has a hard time denying any of the premises above, as they are writen. Bad. Lewis forwards these as arguments that cause greater difficulties for lumps, then for worm theories such as his own. Something needs to be added that will capture the difference between Weatherson's lumps and Lewis' worm. Hopefully, I'm not completely wrong here, but the something should be that Lewis can chose between the counterpart or mereological components of his view. Whereas Weatherson doesn't accept counterparts (it's just the lump, even if the lump includes what Lewis would refer to as a counterpart of

*x*), so he cannot divide as finely as Lewis can in his response. Which as far as I can tell is what is needed for Lewis to show that there are greater difficulties for the lump view as opposed to the worm view.Anyhow tell me what ya' think. And yeah, tell me what needs to be clarified and further explicated.

## Sunday, August 26, 2007

### Kit Fine, "The Problem of Possibilia"

I'd like to try and standardize Fine's argument against "proxy reduction."

Fine is concerned with certain sorts of (actualist) attempts to make sense of possibilist discourse. One option for the actualist is to make sense of possibilist discourse by employing “proxies,” in something like the following manner:

“With each possible x is associated another entity x’, acceptable to the actualist, and any statement Φ(a,b,…) about the possibles a,b,…is then understood in terms of a corresponding statement Φ’(a’,b’,…) about the associated entities a’,b’,…”

According to Fine the most natural way of thinking about the relationship between the entities in the first set and those in the second is in terms of the identity relation. And if that is the assumption, he argues, then the following argument can be presented against any form of actualism employing this form of proxy reduction:

Where Mx: x has the (modal) property of possibly-being-the-world

Rxy: x ‘goes proxy’ for y

w: some possible world w

r: any actualistically acceptable proxy: i.e: a maximal consistent set of propositions or states of affairs; a maximal structural property; or a ‘way a world could be.’

(1) □∀x ∀y [(x = y)-->(Fx-->Fy)]

(2) □∀x ∀y [~(Fx-->Fy)-->~(x=y)] (1) CONTRA

(3) □∀x ∀y [~(~Fx v Fy)-->~(x=y)] (2) IMPL

(4) □∀x ∀y [(Fx & ~Fy)--> ~ (x=y)] (3) DEM & DN

(5) ∀x ∀y [~(x = y)--> ~Rxy]

(6) Mw

(7) ~Mr

(8) Mw & ~Mr (5&6)

(9) ~(w=r) (4,7)

(10)~(r=w) (9)

(11) ~Rrw (5,10)

This seems to be faithful to the English argument given by Fine. I *believe* it is valid as presented. If anybody would comment and let me know if and where I am going off-course with this, it would be appreciated.

Fine is concerned with certain sorts of (actualist) attempts to make sense of possibilist discourse. One option for the actualist is to make sense of possibilist discourse by employing “proxies,” in something like the following manner:

“With each possible x is associated another entity x’, acceptable to the actualist, and any statement Φ(a,b,…) about the possibles a,b,…is then understood in terms of a corresponding statement Φ’(a’,b’,…) about the associated entities a’,b’,…”

According to Fine the most natural way of thinking about the relationship between the entities in the first set and those in the second is in terms of the identity relation. And if that is the assumption, he argues, then the following argument can be presented against any form of actualism employing this form of proxy reduction:

Where Mx: x has the (modal) property of possibly-being-the-world

Rxy: x ‘goes proxy’ for y

w: some possible world w

r: any actualistically acceptable proxy: i.e: a maximal consistent set of propositions or states of affairs; a maximal structural property; or a ‘way a world could be.’

(1) □∀x ∀y [(x = y)-->(Fx-->Fy)]

(2) □∀x ∀y [~(Fx-->Fy)-->~(x=y)] (1) CONTRA

(3) □∀x ∀y [~(~Fx v Fy)-->~(x=y)] (2) IMPL

(4) □∀x ∀y [(Fx & ~Fy)--> ~ (x=y)] (3) DEM & DN

(5) ∀x ∀y [~(x = y)--> ~Rxy]

(6) Mw

(7) ~Mr

(8) Mw & ~Mr (5&6)

(9) ~(w=r) (4,7)

(10)~(r=w) (9)

(11) ~Rrw (5,10)

This seems to be faithful to the English argument given by Fine. I *believe* it is valid as presented. If anybody would comment and let me know if and where I am going off-course with this, it would be appreciated.

## Wednesday, August 15, 2007

### the end of last reading group

Soams gives an objection to himself starting at the end of page 22.

Consider an agent A, the actual world @, and two worlds W and W*.

Suppose further that A has certain beliefs at W about W* (that is to say, were W instantiated then A would have beliefs about W*). So, W must have some defining propositions that result in belief ascriptions to A. The problem gets tricky when we try and apply knowledge ascriptions. I'll give the important sentences right here (with our modifications).

12) p is true at w*

13) A believes truely that p is true at w*

15) if (conditions for A's knowledge of S) then A knows that S

14) A knows that p is true at w*

The worry is that there may be no necessary&sufficient conditions for knowledge. If that's right, then if A is able to know S at W, then "A knows S" must be a defining proposition of W. If it were not, "A knows S" would be derivable from the defining propositions of W, and that would require conditions for knowledge. But, if "A knows S" must be a defining proposition, there are problems. Consider:

8c) A knows that: ~Saul philosophizes & it is true at @ that Saul philosophizes

If (8c) is true at W, we can infer (8b)

8b) ~ Saul philosophizes & actually Saul philosophizes

But we know apriori the truth of (8a)

8a) Saul philosophizes iff actually Saul philosophizes

So, we can know a-priori that W isn't instantiated. This is bad. Note that Soames's original strategy won't work, since if (8c) is a defining proposition of W we can take it by the indexical mode of presentation if we like. For simplicity, consider W a tiny world with only A and non-philosophical Saul, and all A does is know that (8c), and Saul does nothing.

Here's a reminder of EP1 and EP2.

EP1: A world state w is epistemically possible iff w is a way the world can coherently be conceived to be, which it cannot be known apriori not to be

EP2: A world-state w is epistemically possible iff w is a way the world can coherently be conceived to be, and one cannot know apriori that w is not a way the world could be (or have been)

The objection depends upon two principles:

1) ~E sufficient and necessary conditions for knowledge

2) EP1

If we adopt EP2 then W is still epistemically possible.

Consider an agent A, the actual world @, and two worlds W and W*.

Suppose further that A has certain beliefs at W about W* (that is to say, were W instantiated then A would have beliefs about W*). So, W must have some defining propositions that result in belief ascriptions to A. The problem gets tricky when we try and apply knowledge ascriptions. I'll give the important sentences right here (with our modifications).

12) p is true at w*

13) A believes truely that p is true at w*

15) if (conditions for A's knowledge of S) then A knows that S

14) A knows that p is true at w*

The worry is that there may be no necessary&sufficient conditions for knowledge. If that's right, then if A is able to know S at W, then "A knows S" must be a defining proposition of W. If it were not, "A knows S" would be derivable from the defining propositions of W, and that would require conditions for knowledge. But, if "A knows S" must be a defining proposition, there are problems. Consider:

8c) A knows that: ~Saul philosophizes & it is true at @ that Saul philosophizes

If (8c) is true at W, we can infer (8b)

8b) ~ Saul philosophizes & actually Saul philosophizes

But we know apriori the truth of (8a)

8a) Saul philosophizes iff actually Saul philosophizes

So, we can know a-priori that W isn't instantiated. This is bad. Note that Soames's original strategy won't work, since if (8c) is a defining proposition of W we can take it by the indexical mode of presentation if we like. For simplicity, consider W a tiny world with only A and non-philosophical Saul, and all A does is know that (8c), and Saul does nothing.

Here's a reminder of EP1 and EP2.

EP1: A world state w is epistemically possible iff w is a way the world can coherently be conceived to be, which it cannot be known apriori not to be

EP2: A world-state w is epistemically possible iff w is a way the world can coherently be conceived to be, and one cannot know apriori that w is not a way the world could be (or have been)

The objection depends upon two principles:

1) ~E sufficient and necessary conditions for knowledge

2) EP1

If we adopt EP2 then W is still epistemically possible.

## Wednesday, August 8, 2007

### Grim & Plantinga on Cantorian arguments

There's something on the back of Chelsey's folder that says we can't quantify over all the truths... that seems bad. Plantinga & Grim have a pretty lengthy discussion of it, which I'll try and outline here. First, the arguments. I'll give 3 forms of the cantorian argument, one involving sets, one involving properties, and one involving propositions. I'll give what I take to be the most brutal version of each.

sets:

1) E a set T of all truths (assume for reductio)

2) If (1) then (3)

3) T has a power set PT

4) For each member p* in PT there is a truth

5) If (4) then (6)

6) There are more truths than members of T

7) If (6) then (5)

8) T is not the set of all truths

9) If (8) then (10)

10) ~(1)

Fairly innocuous. One merely has to say that even if the truths do not form a set we can still quantify over them. Grim says that the only notion of quantification we have is in terms of sets, but assents that it's not a crucial point. There are other possible ways of quantifying. Quick support for (4): for any given t1, either "t1 is in p*" or "t1 is not in p*" is a truth.

Properties:

to avoid controversial use of "mappings" (which derive from set theory) I'll give Grim's mappings via relations:

"A relation R gives us a mapping from those things that are P1 that is one-to-one and onto those things that are P2 just in case (here we merely add a conjunct):

AxAy[[P1x & P1y & Ez(P2z & Rxz & Ryz)] -> x = y]

& Ax[P1x -> EyAz(P2z & Rxz <-> z = y)]

& Ay[P2y -> Ex(P1x & Rxy)].(3)" -Grim

I'll suplement that with an ONTO relation:

A relation R' gives a mapping from things that are P1 ONTO things that are P2 iff

Ax(P2x->Ey(P1y&yR'x))

Note that if P1 does not map ONTO P2, then P2 must have a strictly larger extension then P1

the argument:

1) E a property T which applies to all and only truths (call the things in T's extension t's)

2) If (1) then (3)

3) E a one-one relation R from t's to truths

4) E a property S0 which applies to nothing and a series of properties S1,S2,... which apply to one or more t's

5) There are at least as many truths as there are S properties (each S property has a corresponding truth)

6) (3)&(5)

7) If (6) then (8)

8) There are at least as many t's as there are S properties

9) if (8) then (10)

10) There is a relation from the t's ONTO the S properties

11) for any relation R' from the t's onto S properties, E an S property (D) which is the property of being a t such that tR'S* and ~S*t

12) if (11) then (13)

13) ~(10)

14) (10)&~(10)

15) ~(1)

propositions:

Consider a proposition P which is about all propositions. Call the propositions which P is about p's.

1) E a proposition P which is about all propositions.

2) if (1) then (3)

3) E a one-one relations between the p's and all the propositions

4) E a proposition not about any proposition, and some propositions S1,S2,... about one or more p's

5) if (4)&(3) then (6)

6) there are at least as many p's as S's

7) if (6) then (8)

8) there is a relation from p's ONTO the S's

9) for any relation R' from p's ONTO S's, there is an S that is about those p's such that pR'x -> x is not about p

10) if (9) then (11)

11) ~(8)

12) (8)&~(8)

13) ~(1)

That's 3 cantorian arguments. The first is to the effect that there is no set of all truths. The second concluded that there is no property had by all truths. The third, that there is no proposition which is about all propositions. In the second two arguments, all set-theoretic notions have been put in terms of properties and relations. Thus, showing that even if the problems lie in set theory, they carry over to common philosophical notions that we'd want to keep around. That in itself busts Plantinga's first rebuttle, but fear not, he doesn't fail us.

Plantinga's first strategy is to deny the "diagonal premise" in each argument (except the first, which lacks one). The diagonal premises are these:

in the first argument:

11) for any relation R' from the t's onto S properties, E an S property (D) which is the property of being a t such that tR'S* -> ~S*t

in the second argument:

9) for any relation R' from p's ONTO S's, there is an S that is about those p's such that pR'x -> x is not about p

These are perhaps the easies to deny on face value. After all, they involve a rather cryptic property (or proposition), and when we deny that there is such it seems like we're denying a rather obtuse philosophical entity. Not so says Grim. Grim asserts that each diagonal premise is built from fundamental principles, and to deny the diagonal entities one would have to deny one of the principles that allowed them to be built. For instance, from what I can tell, all that's needed for the diagonal premise in the property case would be a principle like (P1):

P1) For any set of conditions, there is a property whose extension is those entities which satisfy those conditions

I doubt one would want to give up (P1). Granted lots of properties generated by such a principle will have an empty extension, but that's allowed for in the argument.

Plantinga notes that each premise is less disasterous to reject than the conclusion. Therefore, reject a premise, any premise... we NEED to! To bolster this point, he notes that the conclusion is impossible to state. Take the conclusion of the proposition argument:

CP) there is no proposition about all propositions

That's about all propositions!!!! We don't merely get a bad conclusion out of these arguments, we get lunacy! Therefore, reject a premise for the love of Bhudda! In addition, many of the premises are themselves illegally universal by the conclusion. So if we accept the conclusion we have to reject premises ANYWAY. So just reject the premises, keep the conclusion, and count our losses.

I agree with Plantinga here. We can either toss out everything, or toss out (P1) and it's mates. That's contradiction baby. But perhaps something can be salvaged if we have a suitable replacement for (P1). I suggest something like (P2):

(P2) For any set of conditions which is not illegally self-referential, there is a property whose extension is those entities which satisfy those conditions

I know not what is involved in legal self-reference, but it does seem like (D) is self-referential in a strange way. Consider this property:

Q) Ax, Qx iff ~Qx

Is that a genuine property? Or is it illegally self-refferential? This is the liar paradox all over again. So if we declare liar paradoxes illegal for generating properties, would (D) be illegal? How would this work for the diagonal premise in the proposition argument?

I need to mull this over now, but hopefully with this explication of the debate everyone can share in the frustration!

sets:

1) E a set T of all truths (assume for reductio)

2) If (1) then (3)

3) T has a power set PT

4) For each member p* in PT there is a truth

5) If (4) then (6)

6) There are more truths than members of T

7) If (6) then (5)

8) T is not the set of all truths

9) If (8) then (10)

10) ~(1)

Fairly innocuous. One merely has to say that even if the truths do not form a set we can still quantify over them. Grim says that the only notion of quantification we have is in terms of sets, but assents that it's not a crucial point. There are other possible ways of quantifying. Quick support for (4): for any given t1, either "t1 is in p*" or "t1 is not in p*" is a truth.

Properties:

to avoid controversial use of "mappings" (which derive from set theory) I'll give Grim's mappings via relations:

"A relation R gives us a mapping from those things that are P1 that is one-to-one and onto those things that are P2 just in case (here we merely add a conjunct):

AxAy[[P1x & P1y & Ez(P2z & Rxz & Ryz)] -> x = y]

& Ax[P1x -> EyAz(P2z & Rxz <-> z = y)]

& Ay[P2y -> Ex(P1x & Rxy)].(3)" -Grim

I'll suplement that with an ONTO relation:

A relation R' gives a mapping from things that are P1 ONTO things that are P2 iff

Ax(P2x->Ey(P1y&yR'x))

Note that if P1 does not map ONTO P2, then P2 must have a strictly larger extension then P1

the argument:

1) E a property T which applies to all and only truths (call the things in T's extension t's)

2) If (1) then (3)

3) E a one-one relation R from t's to truths

4) E a property S0 which applies to nothing and a series of properties S1,S2,... which apply to one or more t's

5) There are at least as many truths as there are S properties (each S property has a corresponding truth)

6) (3)&(5)

7) If (6) then (8)

8) There are at least as many t's as there are S properties

9) if (8) then (10)

10) There is a relation from the t's ONTO the S properties

11) for any relation R' from the t's onto S properties, E an S property (D) which is the property of being a t such that tR'S* and ~S*t

12) if (11) then (13)

13) ~(10)

14) (10)&~(10)

15) ~(1)

propositions:

Consider a proposition P which is about all propositions. Call the propositions which P is about p's.

1) E a proposition P which is about all propositions.

2) if (1) then (3)

3) E a one-one relations between the p's and all the propositions

4) E a proposition not about any proposition, and some propositions S1,S2,... about one or more p's

5) if (4)&(3) then (6)

6) there are at least as many p's as S's

7) if (6) then (8)

8) there is a relation from p's ONTO the S's

9) for any relation R' from p's ONTO S's, there is an S that is about those p's such that pR'x -> x is not about p

10) if (9) then (11)

11) ~(8)

12) (8)&~(8)

13) ~(1)

That's 3 cantorian arguments. The first is to the effect that there is no set of all truths. The second concluded that there is no property had by all truths. The third, that there is no proposition which is about all propositions. In the second two arguments, all set-theoretic notions have been put in terms of properties and relations. Thus, showing that even if the problems lie in set theory, they carry over to common philosophical notions that we'd want to keep around. That in itself busts Plantinga's first rebuttle, but fear not, he doesn't fail us.

Plantinga's first strategy is to deny the "diagonal premise" in each argument (except the first, which lacks one). The diagonal premises are these:

in the first argument:

11) for any relation R' from the t's onto S properties, E an S property (D) which is the property of being a t such that tR'S* -> ~S*t

in the second argument:

9) for any relation R' from p's ONTO S's, there is an S that is about those p's such that pR'x -> x is not about p

These are perhaps the easies to deny on face value. After all, they involve a rather cryptic property (or proposition), and when we deny that there is such it seems like we're denying a rather obtuse philosophical entity. Not so says Grim. Grim asserts that each diagonal premise is built from fundamental principles, and to deny the diagonal entities one would have to deny one of the principles that allowed them to be built. For instance, from what I can tell, all that's needed for the diagonal premise in the property case would be a principle like (P1):

P1) For any set of conditions, there is a property whose extension is those entities which satisfy those conditions

I doubt one would want to give up (P1). Granted lots of properties generated by such a principle will have an empty extension, but that's allowed for in the argument.

Plantinga notes that each premise is less disasterous to reject than the conclusion. Therefore, reject a premise, any premise... we NEED to! To bolster this point, he notes that the conclusion is impossible to state. Take the conclusion of the proposition argument:

CP) there is no proposition about all propositions

That's about all propositions!!!! We don't merely get a bad conclusion out of these arguments, we get lunacy! Therefore, reject a premise for the love of Bhudda! In addition, many of the premises are themselves illegally universal by the conclusion. So if we accept the conclusion we have to reject premises ANYWAY. So just reject the premises, keep the conclusion, and count our losses.

I agree with Plantinga here. We can either toss out everything, or toss out (P1) and it's mates. That's contradiction baby. But perhaps something can be salvaged if we have a suitable replacement for (P1). I suggest something like (P2):

(P2) For any set of conditions which is not illegally self-referential, there is a property whose extension is those entities which satisfy those conditions

I know not what is involved in legal self-reference, but it does seem like (D) is self-referential in a strange way. Consider this property:

Q) Ax, Qx iff ~Qx

Is that a genuine property? Or is it illegally self-refferential? This is the liar paradox all over again. So if we declare liar paradoxes illegal for generating properties, would (D) be illegal? How would this work for the diagonal premise in the proposition argument?

I need to mull this over now, but hopefully with this explication of the debate everyone can share in the frustration!

## Friday, August 3, 2007

### a time argument

Here's an argument that was recently presented to me...

Assume 4 dimensionalism. Consider the time-worm Dyck. Let A be the individual corresponding to Dyck from conception to death. Let B be the individual corresponding to Dyck from conception to 5 minutes before his death. Here's the argument:

1) []~(A=B)

2) Dyck could have died 5 minutes before he actually died

3) if (2) then (4)

4) <>(A and B share all their parts)

5) If (4) then (6)

6) <>(A=B)

7) ~[]~(A=B)

8) (1)&(7) contradiction

I'm not sure I understand the argument quite right, because it seems like one could easily deny (5) (if they're any sort of hecceatist). Also it seems like you could run a spatial argument in a similar way. Let A be Dyck, and B be the parts of Dyck vital to survival. Let (2) be "everything not vital to Dyck's survival could have been amputated". Anyway, I'm confused as to how this is supposed to work, I'm sure I don't have the best formulation of the argument.

Assume 4 dimensionalism. Consider the time-worm Dyck. Let A be the individual corresponding to Dyck from conception to death. Let B be the individual corresponding to Dyck from conception to 5 minutes before his death. Here's the argument:

1) []~(A=B)

2) Dyck could have died 5 minutes before he actually died

3) if (2) then (4)

4) <>(A and B share all their parts)

5) If (4) then (6)

6) <>(A=B)

7) ~[]~(A=B)

8) (1)&(7) contradiction

I'm not sure I understand the argument quite right, because it seems like one could easily deny (5) (if they're any sort of hecceatist). Also it seems like you could run a spatial argument in a similar way. Let A be Dyck, and B be the parts of Dyck vital to survival. Let (2) be "everything not vital to Dyck's survival could have been amputated". Anyway, I'm confused as to how this is supposed to work, I'm sure I don't have the best formulation of the argument.

## Tuesday, July 31, 2007

### 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

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.

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.

Subscribe to:
Posts (Atom)