Time, metaphysics of

5. The A-theory and McTaggart’s paradox

In The Nature of Existence, (McTaggart 1927) McTaggart offered an argument against the existence of time itself (see McTaggart, J.M.E. §2). His strategy was to argue that time essentially involves the A-series, but that the A-series is inherently self-contradictory. Since the existence of time requires the existence of the A-series, but the A-series cannot exist, it follows that time cannot exist either. Opponents of the A-theory have taken up that component of McTaggart’s argument directed against the existence of the A-series.

McTaggart’s argument (often called McTaggart’s paradox) that the notion of the A-series involves a contradiction goes as follows. The first step is to note that the A-characteristics of pastness, presentness and futurity are incompatible with each other. Nothing can exemplify more than one of these characteristics. The next step is to consider the effect on that fact of the other aspect of an A-theoretic conception of time, namely, the fact that time is dynamic, so that what is future becomes present and then becomes past. When the flow of time is added to the picture, we find that, since events are continually changing their A-series locations, each one exemplifies every A-characteristic. But this is in direct conflict with the first step, that nothing can exemplify more than one A-characteristic.

The natural response at this point is to note that nothing exemplifies more than one A-characteristic at the same time, so there is really no contradiction between these two aspects of an A-theoretic conception of time. However, McTaggart anticipates this move. He argues that in order fully to cash out this response, we must specify when an event exemplifies each of the incompatible A-characteristics, and furthermore, we must do so in A-series terms. If, instead, we used B-series terms to specify when an event exemplified each of the incompatible A-series characteristics, our attempt to characterise the A-series would collapse into a characterisation of the B-series. If, for example, we said that an event, e, is future on Sunday, present on Monday, and past on Tuesday, this description would be a permanent, unchanging one, equivalent to the B-theoretic description that e occurs on Monday, is later than Sunday and earlier than Tuesday. But this describes unchanging, B-series facts. So, instead, we would have to say that e is now present, will be past, and has been future. But this move, McTaggart argues, introduces compound tenses, and when we consider the entire range of compound tenses (there are nine of them), we can see that they are just as incompatible with each other as the three simple tenses. Furthermore, given the flow of time, every event exemplifies every compound tense. So, the contradiction has not been avoided by switching our attention from the three simple tenses to the nine compound tenses. Again, the reply could be made that no event exemplifies any of the incompatible compound tenses at the same time. But again, McTaggart can reply that specifying in A-series terms exactly when an event exemplifies the incompatible compound tenses generates a third level of even more complex tenses, and a similar incompatibility exists among them. Given the flow of time, every event exemplifies every third-level compound tense, so once more the contradiction has not been avoided. The regress that we have thus embarked upon is vicious. At no level is the contradiction removed.

In response to McTaggart’s paradox, A-theorists attempt to identify the faulty move in his argument (Smith 1993; Craig 2000). There are also some B-theorists who agree that the argument fails to establish the incoherence of the A-series (Savitt 2000). But a significant cohort of B-theorists argue that it succeeds in establishing the unreality of tense (Oaklander 1984; Horwich 1987; Le Poidevin 1991; Mellor 1998; Dyke 2002).