Monday, December 14, 2009
Wednesday, December 09, 2009
Iterative Conception of Propositions
(Cross Posted at joshuaspencer.net)
The iterative conception of propositions works like this. First, we start out with some propositions at the bottom level. None of these propositions contain as a constituent the property of being true. These propositions are combined in a Boolean kind of way to get conjunctions, disjunctions, conditionals etc. Then, for each proposition at the base level and for each proposition constructed in a Boolean kind of way, there is the proposition that that proposition is true. That is, for any P such that P is a proposition at the base level or P is a proposition formed by Boolean operations on propositions at the base level, there is the proposition that P is true. This is the second level of propositions. Now, we combine these propositions and the ones at the base level and the ones formed by Boolean operations in Boolean kind of way. We get even more propositions and we do the trick over again ad infinitum. There are no propositions other than the ones formed by this series of operations.
This iterative conception of propositions will look a lot like typed propositions but will not involve a hierarchy of truth predicates. Moreover, I believe this conception of propositions will avoid certain kinds of paradoxes. There will be no liar proposition. Moreover, for any proposition P, the proposition that P is true is not a conjunct of P. Hence, there will be no Russell proposition that has itself as a conjunct iff it does not. This iterative conception of propositions is rather attractive given that it avoids these paradoxes.
We might also add to our iterative conception of propositions a kind of anti-deflationist principle. That is, for any proposition P, the proposition that P is true is not identical to P. This anti-deflationist principle will be attractive to all those who believe that there really is a property of truth.
Here is a principle that seems to follow from the iterative conception of propositions combined with the anti-deflationist principle:
Particularized Principle of Sufficient Reason (PPSR):
For any true proposition, P, there is a particular sufficient reason, S, such that (i) S is identical to the proposition that P is true, (ii) S is true, (ii) necessarily: S only if P, (iii) S is not identical to P or to any contingent conjunct of P.
However, in An Essay on Free Will, Peter van Inwagen presented a strong argument against the principle of sufficient reason (which is implied by the particularized principle above). Hence, there is a strong argument against PPSR. Very briefly, I will set out the assumptions that underlie van Inwagen’s argument and present the argument against PPSR. My formulation will follow Hudson’s formulation of van Inwagen’s argument (“Brute Facts” AJP March 1997).
Here are the five assumptions that underlie the argument:
A1. There are contingently true propositions.
A2. Any conjunction of contingently true propositions is itself contingently true.
A3. Any true proposition is either contingently true or necessarily true.
A4. For any P and Q, if both Necessarily: P and Necessarily: P only if Q, then Necessarily: Q
A5. If there are contingently true propositions, then there is a conjunction of all contingently true ptopositions.
The argument against PPSR is rather straightforward.
1) P is a conjunction of all contingently true propositions (A1, A5)
2) Hence, P is contingently true (1, A2)
3) Hence, there is a sufficient reason for P, S, such that S is true and S is identical to the proposition that P is true. (2, PPSRi, PPSRii)
4) Hence S is either contingently true or necessarily true. (3, A3)
5) S is not necessarily true (proof below)
6) Hence S is contingently true. (4, 5)
7) But, S is not contingently true. (proof below)
The proof of 5 is as follows:
a) S is necessarily true (reductio assumption)
b) Hence, necessarily: S only if P (3, PPSRii)
c) Hence, necessarily: P (b, A4)
d) It is not the case that P is necessary (2)
e) Hence S is not necessarily true (from a-d)
The proof of 7 is as follows:
f) S is contingently true (reductio assumption)
g) Hence, S is a contingent conjunct of P (e, 1)
h) S is not a contingent conjunct of P (2, 3, PPSRiii)
i) Hence, S is not contingently true (from f-g)
Since PPSR followed from the iterative conception of propositions and the anti-deflationary principle, we can conclude that A1-A5, the iterative conception of propositions and the anti-deflationary principle are jointly inconsistent. I take it that A1-A4 are all very strong. So, that leaves us with a choice. We must decide on whether we are going to give up on the iterative conception of propositions, the anti-deflationary principle, or the assumption that if there are contingent truths then there is a conjunction of all such truths. Those who are attracted to the iterative conception of propositions because it sidesteps paradox must decide between being a deflationist about truth (that is accepting that for any proposition P, the proposition that P is identical to the proposition that P is true) or denying a conjunction of all contingent truths.
Let’s consider the second option. It seems that the second option is inconsistent with the iterative conception of propositions. After all, at any level of proposition construction, we are supposed to perform Boolean operations on all those propositions constructed up to that point. We should get a conjunction of all contingent propositions at the end of our (infinitely long) construction procedure. Remember that the construction procedures should not be thought of as an actual process. Rather, we should think of the iterative conception as a thesis that involves a base clause (about the existence of certain propositions) and a recursive clause (that tells us what propositions exist given the existence of those in our base clause). The recursive clause will have a quantifier over propositions. Hence, if we are to hold on to the iterative conception without accepting a conjunction of all contingently true propositions, we must say that the quantifier in the recursive clause is indefinitely extensible. If this is all correct, then the only way a person who believes in the iterative conception of propositions can avoid a conjunction of all contingently true propositions is by endorsing a rather radical thesis about quantification (namely that the quantifiers that range over propositions are indefinitely extensible).
Hence, it looks like the defender of the iterative conception of propositions is faced with a choice. Become a deflationist about truth. Say that for every proposition P, the proposition that P is true is identical to P. Or, alternatively, claim that quantifiers that range over propositions are indefinitely extensible. Neither option seems too attractive.
Monday, November 09, 2009
Truthmakers for Negative Existentials
- Atomic singular sentences of the form a is F encode atomic Russellian propositions.
- Atomic sentences that contain non-referring names encode atomic gappy propositions.
- Atomic sentences that contain a name for a property that does not exist encode atomic gappy propositions.
- Atomic sentences that include 'exists' or cognates encode the first-order property of existence.
- Non-singular existence statements encode a second-order property of existence.
- All atomic gappy propositions are false. All of their negations are true.
That a proposition is true or false is not a fundamental fact. Truthmaker theorists want to capture this nonfundamentality by holding that for every true proposition there is some state of affairs that makes it true. (I'm ignoring the trope option here. I'm also joining truthmakerists in ignoring the issue of falsemakers. After all, that something is false is no more fundamental than that something is true. But falsemakers never seem to come up. Maybe the assumption is that we could do falsemakers in terms of truthmakers if we could just nail truthmakers. I'm not sure this assumption is true, but I'll bracket it here.) True negative existentials are a major bugaboo for truthmaker theorists. Two main constraints on truthmakers is that they must necessitate the relevant truth and the truths must be about them. Another is that they are supposed to not be "suspicious". (This is related to the power of truthmakers to "catch cheaters".)
It is widely held that the best shot for being a truthmaker for a singular or non-singular true negative existential is something like the entire world plus the fact that there's nothing more. Opponents of truthmaker complain that this sort of truthmaker does bad on all counts. I wonder if there is not a better one to be had. (I don't know the truthmaker literature very well though.)
Here's the idea: the gappy proposition < __, smokes > is false and it represents nothing at all as smoking. This is to say that it represents the gappy state of affairs [ __, smoking ] as obtaining. Paradoxes aside, suppose a liberal account of states of affairs, including gappy ones. To be true is to represent a state of affairs that is among the states of affairs that obtain. To be false is to fail to do that.
So < NEG, < __, exists > > is true because [ __, exists ] is not among the states of affairs that obtain.
< NEG, < hobbits, exist > > is true because nothing has the property of being a hobbit. That is to say, [ __, is a hobbit ] is not among the states of affairs that obtain, and neither is any state of affairs [ o, is a hobbit ] for any o that exists.
Following Kripke, suppose 'is a unicorn' does not express a property. Then 'Unicorns don't exist' encodes something like < NEG, < __, exist > > where the existence property is second-order. This is true for reasons parallel to the first-order case.
This view seems to beat others in terms of necessitation and aboutness, though perhaps gappy states of affairs are suspicious. I don't know.
Perhaps the fact that the gappy states of affairs are not among those that obtain is not itself a fundamental fact. Maybe it holds in virtue of the totality of things plus the fact that there is nothing more. Maybe that eases suspicion while maintaining aboutness. But it does not merely collapse into the usual account. If I say that Joshua's life is longer than JonBenet Ramsey's, I am talking about their lives. But that is consistent with their lives being nonfundamental. So I think this is okay.
Perhaps one could object to the intrinsic weirdness of gappy states of affairs. But if one accepts gappy propositions, one should not have this problem with gappy states of affairs. Russellian propositions are so very states-of-affairs-like that I can't think of a principled reason for accepting one and not the other. The link is even tighter if we accept the identity theory of truth (I don't): the true propositions just are the states of affairs that obtain. Then gappy propositionists get gappy states of affairs "for free".
What's not to like?