Logic: A Very Short Introduction (Very Short Introductions) by Priest Graham

Author:Priest, Graham [Priest, Graham]
Language: eng
Format: epub
Publisher: Oxford University Press
Published: 2000-10-11T21:00:00+00:00

The first inference is valid, since if r is true in some situation, s0, then in any situation to the right of s0, say s1, Pr is true (since s0 is to its left). But then, FPr is true in s0, since s1 is to its right. We can depict things like this:

The second inference is valid, since if FHr is true in s0, then in some situation to the right of s0, say s2, Hr is true. But then in all situations to the left of s2, and so in particular s0, r is true:

Moreover, certain combinations of tenses are impossible, as one would expect. Thus, if h is a sentence that is true in just one situation, say s0, then Ph & Fh is false in every s. Both conjuncts are false in s0; the first conjunct is false to the left of s0; the second conjunct is false to the right. Similarly, e.g., PPh & FFh is false in every s. I leave you to check the details.

Now, how does all this bear on McTaggart’s argument? The upshot of McTaggart’s argument, recall, was that, given that h has every possible tense, it is never possible to avoid contradiction. Resolving contradictions in one level of complexity for compound tenses only creates them in another. The account of the tense operators that I have just given, shows this to be false. Suppose that h is true in just s0. Then any statement with a compound tense concerning h is true somewhere. For example, consider FPPFh. This is true in s-2, as the following diagram shows:

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