Saturday, December 20, 2008

Assertibility and Meaning (Take 2)

As I note in a comment on my previous post, I have a strong intuition that there should be some sort of systematic connection between assertibility-conditions and truth-conditions. The source of that intuition, I think, stems from the Principle of Charity, which states that, so long as it is contextually plausible to do so, one should try to interpret the statements a person makes in such a way that what they say would be rationally assertible for them. This is, I think, an essential heuristic principle of interpretation, one that connects the meaning that we impute to people's utterances to assertibility-conditions.

But it's not so easy to come up with a generalized statement of how assertibility-conditions bear on truth-conditions. There are several traps to avoid. Here are some:
  • We don't want assertibility to be a necessary condition for truth, since there can be truths that are not rationally assertible for anyone (except maybe God, but let's leave him out of this). For example, there is, presumably, a truth about how many times the last emperor of China sneezed, but I doubt that anyone is in a position to assert that the number of sneezes is equivalent to any particular number.
  • We don't want truth to be a necessary condition for assertibility, since there can be rationally assertible propositions that are false. For example, prior to Copernicus and Galileo, heliocentrism was rationally assertible, despite the fact that it turned out to be false.
  • We don't want to epistemicize truth or to equate it with warranted assertibility, at least not if we want to take a realist approach to metaphysics.
In my previous post, I tried this proposal:

(1) Whatever anyone must believe in order rationally to assert a proposition p is part of the meaning of p.

But as Rafal helpfully pointed out to me, this won't do. The most glaring problem, perhaps, is that there seem to be propositions that simply cannot be rationally denied, whether because they are obvious a priori truths (e.g., 1+1=2) or because denying them would land us in a performative contradiction (e.g., there are true sentences). Consequently, (1) yields the result that these propositions are part of the meaning of every proposition whatsoever. And that's just not plausible. So I went back to the drawing board and came up with a new proposal:

(2) Where q is a proposition that someone could rationally deny, if anyone must believe q in order rationally to assert p, then q is part of the meaning of p.

This gets around the problem by restricting the claim to propositions that someone could rationally deny. Rafal, however, raises two further problems. One concerns cases in which p is a proposition that cannot be rationally affirmed (e.g., 4+1=193). Given the standard semantics for counterfactuals, (2) would seem to imply that every proposition whatsoever is part of the meaning of 4+1=193. Fortunately, there is a simple revision of (2) that blocks these sorts of cases:

(3) Where q is a proposition that someone could rationally deny and p is a proposition that someone could rationally assert, if anyone must believe q in order rationally to assert p, then q is part of the meaning of p.

We're making progress toward a plausible principle, but we're not out of the woods yet, for Rafal has another counterexample, one that might rule out (3). Let p be "2+2=4". This is rationally assertible. Let q be "There are true mathematical statements". This is rationally deniable (or so mathematical fictionalists would have us think). It seems that one cannot rationally assert p without asserting q. Hence, by (3), we should conclude that q is part of the meaning of p. Yet, arguably, this is not so (at least, mathematical fictionalists would argue the point).

Now, I'm not quite sure what to make of this counterexample. I'm sufficiently skeptical of mathematical fictionalism, that I'm tempted to appeal to (3) as a reason for rejecting mathematical fictionalism. On the other hand, I don't know enough about the philosophy of mathematics to feel comfortable being so cavalier. Nor is it clear to me right now how to revise (3) in order to block this and similar counterexamples.

For the moment, then, I'm stuck. I'm not prepared yet to give up (3), but neither do I feel very certain that it's right.

Wednesday, December 17, 2008

Assertibility and Meaning

Thesis: Whatever anyone must believe in order rationally to assert a proposition p is part of the meaning of p.

Obvious case: One cannot rationally assert p unless one believes that p.

If the thesis is right, then it provides a test for whether a given proposition q is part of the meaning of p. In other words, it gives us a test for determining whether p semantically implies q.

Test: If you want to know whether q is part of the meaning of p, suppose that someone S does not believe q (or even believes not-q), and consider whether S could rationally assert p.

If "no", then q is part of the meaning of p.
If "yes", then q is not part of the meaning of p.

Example: Let's test whether the Peircean semantics for the future tense is correct. According to the Peircean, the future tense is intrinsically modal, and to assert that an event E "will" happen implies that the world is now tending strongly (probability > 0.5) toward E's happening.

Suppose, then, that S does not believe that the world is strongly tending toward E's happening. Indeed, suppose that S believes that E's happening is highly improbable. Could S believe that and, at the same time, rationally assert "E will happen"?

It seems clear to me that the correct answer is "no". Hence, it follows that the Peircean semantics for the future tense is correct.

To resist this argument, a proponent of an Ockhamist semantics for the future tense must reject my thesis. The Ockhamist believes that the future tense is not intrinsically modal. On his view, to say that E "will" happen implies nothing about its probability, save that it is non-zero. Rather, to say that E "will" happen is simply to say "E does happen subsequently", nothing more. Since I think my thesis is pretty plausible, I think it gives us a good reason for rejecting Ockhamism.

I wonder if anyone out there has an plausible counterexamples for my thesis.

Thursday, December 11, 2008

A Cantorian Argument for Open Theism?

I'v just read an interesting paper on Enigman's website entitled "Omniscience and the Odyssey Theodicy". At one point in the paper, he employs Patrick Grim's well-known Cantorian argument against omniscience to argue for open theism over against an essentially epistemically static (EES) deity (my term, not his). The argument is intriguing.

According to set theory, anything can be a member of a set, including other sets. Hence, there can be no such thing as the set of all sets, for reasons pointed out by Cantor. Take any non-empty set (e.g., {1,2}) and form the set of all subsets (the power set) of the original (e.g., {{∅}, {1}, {2}, {1,2}}). That set will always have more members than the original set. And since this operation can be carried out for any set, there can be no such thing as the set of all sets.

This creates a problem for omniscience if that notion is defined set-theoretically, e.g., believing every member of the set of all truths. One can argue along Cantorian lines that there can be no such thing as the set of all truths. That is, for every set of truths, one can construct new truths that are not already members of the set.

As Enigman points out, if this is right, then it follows that an essentially epistemically static (EES) God - i.e., a God who cannot either acquire or lose beliefs - cannot be omniscient (in a set-theoretical sense). Such a God cannot know all truths. Moreover, there would have to exist truths that are forever outside the ken of an EES God. In contrast, according to open theism God is not epistemically static. He can acquire new beliefs. Indeed, his knowledge is, as Enigman puts it, "indefinitely extensible". As a result, there are no higher-order set-theoretical truths that God cannot eventually come to know. And this constitutes an advantage for open theism. On neither account can God know all truths, but, unlike EES theism, open theism is compatible with the idea that there are no truths that God cannot come to know.

Now, I think this is a very interesting argument for open theism, one that I have not heard of or considered before. I'm not sure that it's sound in its present formulation, though, because I'm not sure that omniscience is best understood in a set-theoretical fashion.

I would suggest that omniscience, in its primary sense, is best construed as a kind of knowledge by acquaintance. On this view, "omniscient" means being fully acquainted with all that is, where "all that is" need not be conceived as a set of discrete constituents, but rather as a continuous field. Here's an analogy. Consider a continuous plane surface. On that surface one may analytically isolate or pick out individual points and lines. In so doing, one brings those points or lines to the foreground, so to speak, but only over against a background, the continuous surface. Since the surface is continuous, no analysis of it can bring all of it into the foreground. There are always more points one could identify, more lines one could draw, etc. If this is right, then an omniscient God's knowledge doesn't come in presliced propositional packets; rather, it exists as a plenum of intelligibility. (I take the term "plenum" from Richard Creel's book Divine Impassibility.) Any part of that plenum can be analyzed out of it as a proposition, but no set of propositions can exhaust the plenum.

Now, even if that way of thinking about omniscience is on the right track, it may still be possible for an Enigman-style argument in favor of open theism to get off the ground. For if the plenum cannot be exhaustively analyzed in propositional terms, then no deity can have exhaustive propositional knowledge. However, an EES deity, on the one hand, is permanently stuck with whatever propositional knowledge he starts out with, whereas an open theist deity's propositional knowledge is indefinitely extensible. There is no point at which he can bring the entirety of the plenum into the foreground, but at the same time there is no part of the plenum that cannot be brought into the foreground. Hence, even on this analysis of omniscience an open theist God turns out to have a higher-quality sort of omniscience than an EES God.

By the way, while I don't endorse Enigman's "Odyssey theodicy" I think that's a really cool name. Kudos to Enigman for a stimulating paper.