Let’s turn our attention now to the B Theory, in particular, to what I’ve called the Mixed B Theory. We can suppose that all actual substances and accidents, past, present, and future, are combined eternally with acts of existence, distinguishing them from merely possible substances and accidents. Is such a model compatible with Aristotelianism?
See my paper, “Aristotelians and the A/B Theory Debate,” delivered at the ACPA in 2019.
I think the central problem for the Aristotelian B Theorist is this (section 6): how can agents be said to actualize states in their patients at a point in time, if the patient’s states are eternally actual? In the paper, I consider the example of a banana whose ripeness is actualized by the sun at noon. To make sense of this, the Aristotelian B Theorist must be able to say that the banana’s ripeness was only potential before noon and actual afterward. There is no real difficulty in saying that the banana was potentially ripe. We could reasonably say that the banana was potentially ripe throughout its existence, if we use ‘potential’ in way that includes actuality. So, the problem for the Aristotelian B Theorist is to explain what it means to say that the ripeness of the banana was not actual before noon, given that it is eternally actual.
There are two ways of doing this. First, we could rely on a space/time analogy, thinking of substances and accidents as being located in “regions” of time, much as they are located in regions of space. Not actual before noon would consist in such a case in the ripeness of the banana’s not being actually located in the before-noon period. Second, we could suppose that there are two fundamental kinds of actuality: absolute actuality and actuality relative to an event. The banana’s ripeness is both absolutely actual and actual relative to the sun’s action at noon. Its absolute actuality is grounded by its actuality relative to the sun’s action taken together with the actuality of the sun’s action.
I don’t much care for the first alternative. Its inconsistent with Aristotle’s very non-spatial, non-locative account of time in the Physics. Time is the measure of change, and change is the actualization of a potential as such. To speak of being ‘located at a time’ is merely a metaphor, and an unhelpful one at that.
However, the second approach seems promising. We could suppose that things are absolutely actual in one of two ways: either by being necessarily actual (as in the case of God), or by being actual relative to some event that is absolutely actual. In other words, we can define absolute actuality recursively.
Let’s turn now to the question of the logic and semantics of time, given this metaphysical model. We should reject the standard Block Universe model of the B Theory as both too static and as lacking any representation of potentiality. The right semantic model is some variant of what is known as Ockhamite temporal logic. Such logic involves tree-like structures. Times are partially ordered (irreflexive, asymmetric, and transitive) and they are linear in the past (if two times are both earlier than a third time, then they are either identical or one is earlier than the other). Trees can branch into the future, representing the potentiality for contrary future conditions. So far, this model is identical to Prior’s Peircean model. What’s new is that we introduce a privileged linear pathway through the tree, a “Thin Red Line” (to use Nuel Belnap’s vivid language), that represents the actual future. Metaphysically, all of the events along the Thin Red Line are combined with acts of existence.
There is one important feature of Prior’s Peircean logic that has to be sacrificed. In the Peircean model, every proposition that is true in the past is necessarily true forever after. In the Ockhamite model, this is no longer the case. It is possible to change the past: there are possible actions with the potential of changing the past truth-values of certain propositions (or, to be more careful, at the very least to determine those truth-values to be different from what they were in fact in the past). Ockhamites must distinguish between hard and soft facts about the past. Hard facts are necessary and unchangeable/unaffectable, while soft facts can vary across future branches.
The language of Ockhamist logics has three basic operators: F, P, and <>. F and P function in a way that presupposes a linear time ordering. The diamond represents historical necessity, satisfying an S4 logic. We can treat Prior’s Peircean logic as a sub-theory, by defining the Peircean ‘F’ as the Ockhamite ‘[]F’, the Peircean ‘G’ as ‘[]G’, and the Peircean ‘P’ as ‘P’ (or equivalently ‘[]P’).
Here are examples of propositions about the past that aren’t necessary (let p refer to a sea battle tomorrow):
(20) HFp
(21) P(x)F(x)p
For Ockhamites, supposing that the sea battle will take place tomorrow, it was always true that it would take place on that date. But this truth is itself contingent: in the possible future situation in which the sea battle doesn’t take place, it won’t be true that it was always the case that it would take take place.
Technically speaking, there are at least two distinct ways of carrying out this program. First, there are the simple Thin Red Line models, in which we simply add to the Peircean models a function that assigns a linear “history” to every time in the branching structure. Every time belongs to its own assigned history. We can further designate one of those histories as the actual one. The actual times are the ones assigned the actual history. A formula is evaluated at an ordered pair consisting of a time and a linear history to which the time belongs.
M, t |= Fp iff (Et’)(t < t’ & t’ <epsilon> H(t) & M, t’ |= p)
M, t |= Pp iff (Et’)(t’ < t & M, t’ |= p)
M, t |= <>p iff (Et’)(t < t’ & M, t |= p)
Second, there are the bundled tree models, developed by Hans Kamp, John Burgess, Mark A. Reynolds, and Alberto Zanardo. These are also sometimes called Leibnizian models. There are several equivalent ways of carrying out this idea. I will describe two briefly. First, we could build a model with a set of times and a set of worlds and define truth at world-time ordered pairs. We will then need two accessibility relations—one between times (representing linear time priority) and one between worlds (representing the overlap between worlds). Second, we could add to the simple Ockhamite models a set of histories (the model’s bundle B), with the requirement that every time belongs to at least one history in the bundle. Here are the bundle-theoretic definitions:
M, t |= Fp iff (Et’)(t < t’ & M, t’ |= p)
M, t |= Pp iff (Et’)(t’ < t & M, t’ |= p)
M, t |= <>p iff (Eh)(Et’)(t <epsilon> h & t’ <epsilon> h & & h <epsilon> B & t < t’ & M, t |= p)
Let’s go back to the Thin Red Line. The main drawback is the fact that the TRL model gives us the power to make something false even if at every point in the past it was going to happen. The following formulas are not valid in this approach:
(22) p -> PFp [not valid in TLR]
(23) p -> P(x)F(x)p [not valid in TLR]
Future-tensed propositions are possibly ineffective.
The Leibnizian or bundle approach eliminates this problem, but it doesn’t adequately capture the reality of branching potentialities. Each time belongs to a single world, with a linear temporal order, and potentiality is captured only by an accessibility relation to other worlds. No two worlds truly overlap mereologically.
More importantly, the Leibnizian version of Ockham’s solution requires that we have (in some sense) the power to change the past. Let’s consider again Aristotle’s future sea battle, to bring out the dilemma facing the Aristotelian B Theorist. According to the Thin Red Line model, it is possible for the battle to have been always going to happen and yet not happen (recall that the operator ‘G’ represents it has always been the case that):
(24) Possibly [G(the sea battle will happen) & the sea battle doesn’t happen]
(25) Possibly [(n days ago)(the sea battle will happen in n days) & the sea battle doesn’t happen]
Neither (24) or (25) are satisfiable in the Leibnizian model. Instead, we get (26) and (27) as satisfiable:
(26) G(the battle will happen) & Possibly[G(the battle will not happen) & the battle doesn’t happen]
(27) (n days ago)(the sea battle will happen in n days) & Possibly[(n days ago)(the sea battle will not happen in n days) & the sea battle doesn’t happen]
Remember, possibly here means, not abstract, logical possibility or conceivability, but the condition of something’s having the causal power to bring about the relevant situation. The Leibnizian-Ockhamite is imagining that I (the Greek admiral) now have the causal power to make it the case that something that has always been true (viz., that a sea battle would take place tomorrow) be false. Does this mean the power to change the past? Yes, and no. No, in the sense that if I were to cancel the sea battle, the resulting situation would be one in which it had always been the case that the sea battle would not happen. But, Yes, in the sense that the result of my action would involve some past-truth-values being different from what they actually are. On any reasonable conception of ‘power’ and ‘change’, this is the power to change the past.
The Ockhamite distinction between hard and soft facts is supposed to make this consequence acceptable. We can change (in this sense) soft facts about the past (e.g., semantical facts involving the future) but not hard facts (categorical facts intrinsic to the past). But is power over even soft facts plausible? The very idea of a past fact seems to involve a kind of determinacy and settledness. Moreover, the Ockhamite view would entail that it is impossible for anyone to have any knowledge of a future contingent fact (assuming that facts about what is known must be hard facts).
Is there another alternative for the Aristotelian B Theorist?
A B-Theoretic Three-Valued solution?
The B Theory does not deny the possibility of change. Consequently, there is no reason why we couldn’t suppose that the class of propositions and facts are subject to change, coming into existence or gaining definite truth values as time passes.
We could suppose either that future contingent propositions don’t exist, or they exist but have a third truth-value (as per Łukasiewicz). B propositions can’t change from true to false or vice versa, but is there any reason that they can’t go from indeterminate to true or false?
But on this view, what are we to say about future-tensed statements and thoughts about future contingencies? I think we have to say that in those cases we simply fail to express a proposition—something we might have to say about the Liar (‘this sentence is false’) or the Truth-Teller (‘this sentence is true’). Future-tensed sentences would fail to express a proposition accidentally rather than essential, like contingent Liars (e.g., ‘the 18th sentence on the 18th page of my essay is false’). The content of what we express (rigidified by the use of a Kampian ‘Now’ operator) would become true as time passes. In the meantime, semanticist can assign them ersatz propositions, lacking either of the classical truth-values. Since all of these propositions will eventually have a classical truth-value, we can perhaps use these ersatz propositions in formulating the probability calculus, with ‘Prob(p)’ measuring the likelihood of the eventual truth of whatever proposition will eventually replace the ersatz proposition.
The Thomistic theory of acts of existence might also be of help here. Suppose we conjecture that all propositions that exist at time t have domains of quantification that are limited to acts of existence that exist at or before t. Recall that I insist on an actualist semantics for quantification over acts of existence: all such acts are actual. We could similarly employ a hybrid semantics here: Presentist (present-plus-past-ist) for acts of existence, and Eternalist for all other entities.
I’ll have to say that there will be acts of existence that don’t exist now, but I can do that in the same way that Presentists can—by placing the quantifier within the scope of a future-tense operator, and refusing to export it in this cases, since we are quantifying over acts of existence.
Is this still a version of the B Theory? I think so. It doesn’t privilege any particular moment of time, from a God’s eye point of view. The domain of quantification at every moment of time is limited in exactly the same way. We don’t have to designate one moment as absolutely present. We are limited in what acts of existence we can quantify over now, but only in the same way that people are at every moment. And, there is no suggestion of a real passage of time. There is no meta- or hyper-time involved. If one insists, one could perhaps classify this as a hybrid A-B theory: A Theoretic about acts of existence and B Theoretic about everything else.
So, it’s clearly a B Theory, since we can recognize the existence of a God’s eye perspective, from which all moments of time are on a par, with a truly universal domain of quantification (something no A Theorist can recognize). The point is simply that our being located in time entails certain limitations on the propositions that we can grasp or express.
What we can do is to introduce a new temporal-modal operator, E for ‘in eternity’. I will insist on presentist-actualist semantics for this operator, invalidating the Barcan formula. So, this will be invalid:
(EBC). If in eternity there is a proposition x such that F(x), then there is a proposition x such that in eternity F(x).
Now, we can say that every rigidified future-tense sentence, like ‘It is now the case that a sea battle will take place tomorrow’, expresses in eternity either a true or a false proposition, and that God knows the true one. Thus, we can escape open theism without falling into logical fatalism, and without the Ockhamite distinction between hard and soft facts.
The Analytic Thomist 