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.