Wednesday, January 11, 2006

Peircean vs. Ockhamist Tense Logic

So far as I am aware, all philosophers agree that if it rains on a particular Tuesday then it will be the case every day thereafter that it rained on that particular Tuesday. If we let P stand for "it was the case that ...", F stand for "it will be the case that ...", N stand for "it is now the case that ...", and p stand for the proposition "It is raining on Tuesday" then we can express this symbolically as

(1) N(p) --> F(P(p)).

(1) claims that if it is now the case that it is raining on Tuesday then it will be the case hereafter that it was the case that it is raining on Tuesday. In other words, once something has happened and is past, it cannot change.

But not all philosophers agree that the following is true:

(2) N(p) --> P(F(p)).

(2) claims that if it is now the case that it is raining on Tuesday then it always was the case that it will be the case that it is raining on Tuesday. In other words, from the beginning of time it was already a settled truth that it would rain on that particular Tuesday. Indeed, (2) implies that there is already a settled truth about what will happen tomorrow and the next day and the next ... and so on, until the very end of time.

It is probably true that most philosophers would accept (2). Following Arthur Prior, we'll call them 'Ockhamists' after the medieval logician William of Ockham. Again following Prior, we'll call those who reject (2) 'Peirceans', after the nineteenth century logician Charles Sanders Peirce.

Those in the Ockhamist camp are apt to regard (2) as obviously correct, a truism, a platitude. Peirceans, however, regard (2) as a gross non sequitur, an invalid inference. Who is right? It all depends on the interpretation of "will" in

(3) "It will be the case that p" uttered at time T.

For the Ockhamist, to say that something "will" happen implies absolutely nothing about its chances of happening, except to say that its chances are not zero. So if someone tosses a coin and, while the coin is still in the air, another person predicts "the coin will land heads", an Ockhamist would not construe that to be implying that the coin was likely to land heads. Instead, he would construe the prediction as saying nothing more than "in fact the coin does land heads subsequent to T". Consequently, the mere fact that the coin does land heads is sufficient to make it true beforehand that it "will" land heads. In summary, then, the Ockhamist takes (3) to mean

(4) It is the case that p at some time T' subsequent to T.

If this is the right way to read statements like (3), then (2) is correct.

The Peircean, however, construes "will" differently. For the Peircean, to predict that something "will" happen is to say that the chances of its happening are 100% (or very close). In other words, they hold that to say that something "will" happen implies that the future is fixed, not by what does happen in the future (as the Ockhamist supposes) but by its being a causally necessary consequence of what is the case right now. If we follow the Peircean and construe "will" in this sense, then it should be pretty obvious that (2) is false. After all, says the Peircean, the mere fact that a tossed coin lands heads only proves that it was antecedently possible that it land heads, not that it was causally necessary that it land heads. In summary, then, the Peircean takes (3) to mean

(5) It is causally necessary that p occur at some time T' subsequent to T.

If this is the right way to read statements like (3), then (2) is false.

2 Comments:

At 1/11/2006 5:48 PM, Blogger Tom said...

Congrats on the new blog, Alan. Best wishes.

Can you tell me where in Pierce's writings can one find his own discussion of this issue?

Tom

 
At 1/11/2006 6:01 PM, Blogger Alan Rhoda said...

Hi Tom,

I would start with the references in Prior's footnotes. Probably the clearest passage is this one (from Peirce's Collected Papers, vol. 6, para. 638). I've highlighted the key sentence:

"Nominalists uniformly speak of Aristotle’s view of future contingents as really absurd. It may be so; but it is certainly the only doctrine which their principles leave room for. A certain event either will happen or it will not. There is nothing now in existence to constitute the truth of its being about to happen, or of its being about not to happen, unless it be certain circumstances to which only a law or uniformity can lend efficacy. But that law or uniformity, the nominalists say, has no real being; it is only a mental representation. If so, neither the being about to happen nor the being about not to happen has any reality at present; and the most that we can say is that the disjunction is true, but neither of the alternatives. If, however, we admit that the law has a real being, not of the mode of being of an individual, but even more real, then the future necessary consequent of a present state of things is a real and true as that present state of things itself."

 

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