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.
Sunday, August 26, 2007
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.
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