Monday, October 09, 2006

(Actually p) iff p

Here's an interesting schema from the point of view of thinking about the relationship between conceptual truth and necessary truth:

A: (Acutally p) iff p

For true but contingent propositions p, this is possibly false: worlds where not-p are still worlds where actually p. Do people reckon it's conceivable that A is false for such p?

Even if it *is* conceivably false, you might think it's the sort of thing you can tell is true just by thinking about the concepts it involves. This might lead you to suspect there are two grades of conceptual truth:

1. Things you can tell are true just by thinking about the concepts involved,
2. Things such that it is not conceivable that they are false.

The second grade looks strictly stronger. (Although I'd be interested to hear if people think they have counterexamples to this claim.)

(Thanks to Daniel for making me think about this!)

3 comments:

djc said...

Well, some would hold that 'Hesperus is Phosphorus' and the like satisfy 2 but not 1. At least, it looks like there is a sense of 'conceivable' in which A is conceivably false and 'H=P' is not conceivably false. This is roughly Kripke's sense, where conceivability can itself depend on a posteriori matters, and tracks metaphysical possibility. This sense, applied to your 2, yields a notion of "conceptual truth" that isn't strictly stronger than your sense 1 (at least if sense 1 of conceptual truth is understood so that it roughly tracks cases that are typically held to be a priori).

On the other hand, there seems to be another sense of 'conceivable' in which both A and 'H=P' are conceivably false. One might gloss this sense by saying that P is conceivable iff it it is not a priori that it is necessary that P. Combining this with your 2 yields a third notion of "conceptual truth" that is plausibly strictly stronger than your 1, at least given these glosses. (If it's a priori that it's necessary that P, then it's a priori that P.) I suspect that the "strictly stronger" version of (2) that you have in mind is grounded in a notion of conceivability that's closely related to this sense.

(Of course there's room for different notions in the vicinity of each of these three senses, depending on e.g. one's views about the a priori, whether one casts the notions in terms of imaginability or in terms of non-excludability, and so on.)

I think these three senses can be naturally modeled two-dimensionally: they correspond at least roughly to versions of primary conceivability, secondary conceivability, and "1-2-conceivability", in the terms of "Does Conceivability Entail Possibility".

Brit Brogaard said...

Hi Carry,

"For true but contingent propositions p, this is possibly false: worlds where not-p are still worlds where actually p"

Is it possibly false? Well, doesn't that depend on the semantic value of the operator 'actually'. If 'actually' is indexical (just like 'it is the case now that'), then the actual world will be in the proposition expressed by 'actually p' (this is the standard view). In that case there will be worlds at which the bi-conditional is false (not-p worlds if p is true in the actual world). But if 'actually' is not an indexical but functions more like 'possibly' and 'necessarily' (which are not indexical once we have fixed the accessibility relation), then there is no way that the bi-conditional could be false.

An analogy: the "tensed" proposition that there will be people inhabiting the moon will be true at any (world, time) pair just in case the embedded proposition is true at a future-shifted (world, time) pair. The truth-value of the "tensed" proposition does not depend on the world of utterance.

Likewise, I am suggesting that there may be a reading of 'actual' where the extension of a sentence containing it does not depend on the world of utterance but only on the world of evaluation. In fact, I think most standard uses of 'actual' are like this.

Compare:

#If John said "I am hungry now" and he was right, then he is hungry now.

If John had said "I am actually hungry" and he were right, then he would actually have been hungry.

The second sounds better to me.

Carrie Jenkins said...

Thanks Dave and Brit.

Dave - you're right, I had in mind the kind of conceivability which is closely related to a prioricity.

Brit - I was intending the (philosophically) standard indexical use of 'acutally', but I agree with you that there's room for dispute about how well this fits the everyday usage of the word.