Storrs McCall’s Falling Branches model (1976) is the most attractive version of the A Theory of time for Aristotelians, since it relies on the changing modal status of possible events. It is also easy to combine with the idea that all possibilities branch off from the actual world.
We can suppose that every event or possible categorical fact in the past and present has the absolute attribute of actuality. Alternatively, we could limit actuality to the present moment. As time passes, actuality moves forward, and events that were possible cease to be possible, as the world passes by the relevant branch points. This will require a modality that can model historical necessity, with every categorical fact in the present and past necessary, and every possible event located either in the past or present or on a branch diverging from the world in the present or future. This sort of modality will validate the S4 logic. Things that are possibly possible are possible, and (equivalently) things that are necessary are necessarily necessary. However, it will not give us S5, since things that are possible (now) are not necessarily possible (since they might cease to be possible later).
The Inegalitarian A Theory differs from Presentism in its account of quantification. Inegalitarians include merely past and merely future beings in the domain of quantification. Both actuality and presentness are somehow distinguished from merely possible and merely past or future situations. The Moving Spotlight model, in which actuality is a peculiar kind of quality, would be one of way of realizing this. However, Aristotelian Thomists have much better option: the present moment of the actual world is distinguished by the presence of acts of existence (actus essendi). The actuality of present states and facts is not some peculiar quality they have, but something that gives them an enhanced ontological status, one upon which all merely possible and merely past and future facts depend.
There is an important point of common ground between Presentists and Inegalitarian Thomists: both treat ‘past’ and ‘future’ as non-factive, alienating modalities. Inegalitarians of this kind will have to distinguish between properties that entail actual existence and those that do not. If F is a non-actual-existence-entailing property, and something was F in the past and is no longer F, then it is simply not F. Past and future times are like merely possible worlds. Things in those worlds lack all qualities and quantities, they lack spatiotemporal location, and they lack causally power (all of which are actual-existence-entailing properties). They do have* some of these properties, where x has* property P just in case x would be P if x were actual. My past pet Schnauzer and my future sixth grandchild’s pet do have* caninity, since my pet was a dog when it existed, and my grandchild’s future pet will be a dog when it comes into existence. But neither is in fact a dog.
What sort of properties are not actual-existence-entailing? Many relational properties, like quantitative or qualitative comparisons (using the having* or belonging* relation). For example, having* the same height as Julius Caesar has*. Also, intentional relations, like being known, thought of, or admired. Mereological relations and relations of numerical identity and distinctness. Some causal-explanatory relations. Also modal and temporal properties, like being possible or being such as to exist in the future.
Arthur Prior was a pioneer in developing a tense logic, and I will follow some of his nomenclature. We can let ‘P’ represent truth at some point in the past, ‘F’ to represent truth in some point in the future in all branches, ‘H’ for truth at every point in the past (“has always been the case heretofore”), and ‘G’ for truth at every point in the future in all branches (“will always be the case from now on”). Given the linearity of the past, the H operator is definable as the dual of the P operator: Hp =df ~P~p. In contrast, the G operator and the F operator are not interdefinable.
A branching future model like that of Prior or McCall (which Prior called ‘Peircean tense logic’, after C S Peirce) will not give us the validity of the future-tensed LEM:
- F(p) v F(~p). [not valid]
This should be distinguished from the relevant instance of the simple LEM:
- F(p) v ~F(p), [valid]
which is valid. We do get some version of the necessity of the past:
- Pp -> GPp [valid]
The corresponding schema involving futurity is not valid:
- Fp -> HFp [not valid]
Present facts are necessary, but need not have been necessary in the past:
- p -> GPp [valid]
- p -> HFp [not valid]
But they must have been at least possible throughout the past:
- p -> H~F~p [valid]
Transitivity gives us these axioms:
- Hp -> HHp [valid]
- Gp -> GGp [valid]
- F(p v Fp) -> Fp [valid]
We don’t have certain counterparts to axioms B or 5 of modal logic, since the temporal order is certainly not symmetric:
- p -> GFp [not valid]
- p -> HPp [not valid]
- ~Fp -> G~Fp [Not valid]
- Pp -> HPp [not valid]
(For a complete axiomatization, see Burgess 1980 or Zanardo 1990.)
It might be more illuminating to introduce a measure of time, with two new operators, P(x) and F(x), for ‘x units of time ago’ and ‘x units of time in the future’. We can then define Prior’s operators:
Fp = (Ex) F(x)p
Gp = (x) F(x)p
Pp = (Ex) P(x)p
Hp = (x) P(x)p = ~(Ex) P(x)~p
Here are some relevant proposition schemata:
- P(x)p v P(x)~p [valid]
- F(x)p v F(x)~p [not valid]
- p -> P(x)F(x) p [not valid]
- p -> F(x)P(x)p [valid]
- p -> P(x)~F(x)~p [valid]
This would seem to be a complete theory of time for Aristotelian A Theorists. The great drawback to this approach is that we cannot even express propositions about the actual future. We can use ‘Fp’ to express that p must be true at some future time, and ‘~Fp’ to express that p might be true at some future time, but there is no way to express simply that p will in fact be true in the future. This lack of expressibility corresponds with the fact that in the Aristotelian A-Theoretic metaphysics, there could be no truthmaker for a contingent actual truth about the future.
This is problematic for at least three reasons. First, we can clearly express in natural language a proposition asserting such a future contingent truth. We do so, for example, whenever we make bets about the future. I’m not betting about whether p could happen or must happen, but about whether it will happen. Second, without future contingents we can’t have well-defined probabilities (either subjective or objective) about the future. Third, there are theological concerns. It seems that at least in some cases God knows what will happen contingently in the future, but this account can’t make any sense of such claims.
To overcome the first problem, the Aristotelian A-Theorists would have to abandon truthmaker theory, at least to some extent. They would have to suppose that there are certain propositions that necessarily lack both truthmakers and falsitymakers. This could be done in at least three ways. First, they might suppose that some propositions have a third value, in addition to truth and falsity. Future contingent propositions would all take this third value. Complex propositions would then be evaluated by some kind of three-valued semantics, like Strong Kleene. This was the approach suggested by Aristotle in De Interpretatione and developed fully by Jan Łukasiewicz. Second, one could adopt a supervaluationist approach, as Richmond Thomason did (1970).[1] On this view, future contingent propositions are neither true nor false, but some future-tensed propositions can be “super-true”, if they come out as true on all future branches. Third, we could follow John McFarlane’s more recent relativistic model (Phil Quarterly 2003), in which propositions are assigned truth-values relative to two contexts or indices: a context of utterance and a context of assessment. This also requires a truth-value gap. Like the supervaluationist approach, it can affirm the truth of the future LEM, and even of the law of semantic bivalence:
(FLEM) (Fp v F~p)
(FBIV) (True(Fp) v True(F~p))
The supervaluationist will say that both are super-true, since both disjunctions have one true disjunct on every branch. McFarlane’s relativist will similarly count both as true (in this case, simply true), since each will be true relative to any pair of contexts, since again it will have a true disjunct on every history. In fact, the necessitation of both laws will be super-true (for the supervaluationist) and absolutely true (for the relativist). In both cases, there are serious problems about being able to express coherently in its own terms. How can one affirm the existence of truth-value gaps while also affirming the necessity of the law of bivalence?
So, perhaps the three-valued approach of Łukasiewicz and Kleene makes more sense in the end. They can recapture some of MacFarlane’s intuitions by introducing Hans Kamp’s NOW operator (Kamp 1971) an operator that picks out the present time rigidly. The three-valuers can then assert F(NOW(Fp v F(~p))), while denying (Fp v F(~p)).[2] Similarly, they can assert F(NOW(True(Fp) v True(F~p))) while denying (True(Fp) v True(F~p)). This would enable them to make sense of the idea that the proposition I am now asserting, although it necessarily lacks truth or falsity when asserted, can eventually acquire truth.
How does this approach fare with the three problems I identified earlier? First, thanks to Kamp’s NOW operator, we can express a proposition which lacks a truth value now but will acquire one in the future. Of course, we can also do this by stating a future contingent B-proposition, one that relates events to other events (including the times defining a conventional dating system).
On the second point, the probability calculus, things are not so clear. We could take the probabilities involved to be probabilities of future truth, for some future date. This would enable us to assign reasonable probabilities to the propositions that will have truth-values on that date, but there will always be many more propositions that will still be value-less. There are some technical issues here that I’ve not looked into.
Finally, nothing in the new model will be much help with divine foreknowledge. God could choose to know certain facts about the future by resolving to make them inevitable now. Whenever He does so, He will be depriving in advance some creatures of their future freedom of action, so He could not do so on a wholesale fashion without destroying secondary causation altogether. We might wonder if we could add an omega point to our timescale—a point in time infinitely far in the future, with no later times, a point at which all tensed propositions have acquired definite truth-values. We could locate God’s omniscience at that omega point, in something like the way Boethius describes God’s omniscience in the Consolation of Philosophy, as a spectator able to observe the entire cavalcade of time. However, I don’t think this option is available for Aristotelians, since it would violate an important axiom about the finite connectedness of time—namely, that every actual instant must have an immediate predecessor (and an immediate successor) within the class of actual instants. (I will discuss this principle in more detail in two weeks.) In addition, this Boethian proposal would seem to violate causal finitism (not a problem for all Aristotelians, but a problem for Alex and me).
[1] The second and third options do not involve positing a third value (as Łukasiewicz and Kleene did) but instead involve the claim that some propositions lack either of the two values. The second and third options do not involve introducing a third value which figures compositionally in determining the value of compound propositions.
[2] I’m using ‘denial’ here as an act or attitude one can correctly take toward a proposition that is neither true nor false. It is not supposed to be equivalent to affirming or asserting the negation.
The Analytic Thomist 