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!

1 comment:

LauLuna said...

Grim's second argument bans the concept of truth itself, for the rejected property T could just be 'being true'.

Is there a bijection from all truths to all truths? If yes, you can repeat Grim's second argument

From a set theoretic point of view a bijection, like any binary relation, is a set of ordered pairs. The obvious non existence of the set of all truths suggests there is no corresponding set of ordered pairs.

Inconsistent multiplicities are best dealt with by means of the concept of indefinite extensibility. I recommend the paper 'All Things Indefinitely Extensible' by Shapiro and Wright, in 'Absolute Generality', Rayo and Uzquiano eds. OUP, 2006.

In that book is also discussed the impossibility of unrestricted universal quantification. The thesis that we cannot speak of absolutely everything seems impossible to state, similarly to the thesis 'there is no proposition about all propositions not about themselves'.

I see a way out in the 'intensional' version of those claims:

1. The concept of universe of discourse implies the feature of being indefinitely extensible.

2. The concept of proposition about propositions not about themselves implies the feature of referring to an indefinitely extensible universe of discourse.

Regards