A temporal framework, plus

Motivation: I have been reading John Horty’s (2019) paper integrating deontic and epistemic logic with a framework of branching time. As a preliminary to exploring his examples and problem cases I want to outline one way to understand indeterminism and time, and a simple way in which such a framework can be given ‘attachments’ to accommodate modalities. Like Horty, I follow the main ideas introduced by Thomason (1970), and developed by Belnap et al. (2001).

The terms ‘branching time’ and ‘indeterminist time’ are not apt: it is the world, not time, that is indeterministic, and the branching tree diagram depicts possible histories of the world. I call a proposition historical if its truth or falsity in a world depends solely on the history of that world. At present I will focus solely on historical propositions, and so worlds will not be separately represented in the framework I will display here.

We distinguish what will actually happen from what it is settled now about what will happen. To cite Aristotle’s example: on a certain day it is unsettled, whether or not there will be a sea-battle tomorrow. However, what is settled does not rule out that there will be a sea-battle, and this too can be expressed in the language: some things may or can happen and others cannot.

Point of view: The world is indeterministic, in this view, with the past entirely settled (at any given moment) but the future largely unsettled. Whatever constraints there are on how things may come to be must derive from what has been the case so far, and similarly for whatever basis there is for our knowledge and opinion about the future. Therefore (?), our possible futures are the future histories of worlds whose history agrees with ours up to and through now.

Among the possible futures we have one that is actual, it is what will actually happen. This has been a subject of controversy; how could the following be true:

there will actually be a sea battle tomorrow, but it is possible that there will not be a sea battle tomorrow?

It can be true if ‘possible’ means ‘not yet settled that not’. (See Appendix for connection with Medieval puzzles about God’s fore-knowledge.)

Representation: A temporal framework is a triple T = <H, R, W>, where H is a non=empty set (the state-space), R is a set of real numbers (the calendar), W is a set of functions that map R into H (the trajectories, or histories). Elements of H are called the states, elements of R the times.

(Note: this framework can be amended, for example by restrictions on what R must be like, or having the set of attributes restricted to a privileged set of subsets of H, forming a lattice or algebra of sets, and so forth.)

Here is a typical picture to help the imagination. Note, though, that it may give the wrong impression. In an indeterministic world, possible futures may intersect or overlap.

If h is in W and t in R then h(t) is the state of h at time t. Since many histories may intersect at time t, it is convenient to use an auxiliary notion: a moment is a pair <h, t> such that h(t) is the state of h at t.

An attribute is a subset of H, a proposition is a subset of W. For tense logic, what is more interesting is tensed propositions, which is to say, proposition-valued functions of time.

Basic propositions: if R is a region in the state-space H, the proposition R^(t) = {h in W: h(t) is in R} is true in history h at time t exactly if h(t) is in R. It is natural to read R^(t) as “it is R now”. If R is the attribute of being rainy then R^(t) would thus be read as “It is raining”.

I will let ‘A(t)’ stand for any proposition-valued function of time; the above example in which R is a region in H, is a special case. For any particular value of t, of course, A(t) is just a proposition, it is the function A(…) that is the tensed proposition. The family of basic propositions can be extended in many ways; first of all by allowing the Boolean set operations: A.B(t) = A(t) ∩ B(t), and so forth. We will look at more ways as we go.

Definitions:

• worlds h and k agree through t (briefly h =/t k) exactly if h(t’) = k(t’) for all t’ ≤ t.
• H(h, t)= {k in W: h =/ t k} is the t-cone of h, or the future cone of h at t, or the future cone of moment <h, t>.
• SA(t)= {h in W: H(h, t) ⊆ A(t)}, the proposition that it is settled at t that A(t)

The term “future cone” is not quite apt since H(h, t) includes the entire past of h, which is common to all members of H(h, t). But the cone-like part of the diagram is the set of possible futures at for h at t.

Thus S, “it is settled that”, is an operator on tensed propositions. For example, if R is a region in the state-space then SR^(t) is true in h at t exactly if R has in it all histories in the t-cone of h. Logically, S is a sort of tensed S5-necessity operator. In Aristotle’s sea-battle example, nothing is settled on a certain evening, but early the next morning, as the fleets approach each other, it is settled that there will be a sea-battle.

There are two important notions related to settled-ness: a tensed proposition A(t) is backward-looking iff membership in A(t) depends solely on the world’s history up to and including t. That is equivalent to: A(t) is part of SA(t), and hence that A(t) = SA(t). If A is a region in H then A^(t) is backward-looking iff each future cone is either entirely inside A, or else entirely disjoint from A.

Similarly, A is sedate if h being in A(t) guarantees that h is in A(t’) for all t’ later than t (that world has, so to say, settled down into being such that A is true). Note well that a backward- looking proposition may be “about the future”, because in some respects the future may be determined by the past. Examples of sentences expressing such propositions:

“it has rained” is both backward-looking and sedate, “it will have rained” is sedate but not backward looking, and “it will rain” is neither.

Tense-modal operators can be introduced in the familiar way: “it will be A”, “it was A”, and so forth express obvious tensed propositions, e.g. FA(t) = {h in W: H(h,t’) ⊆ A for some t’> t}. More precise reckoning can also be introduced. For example if the numbers in the calendar represent days, then “it will be A tomorrow” expresses the tensed proposition TomA(t) = {h in W: h(t+1) is in A}.

Attachments

If T is a temporal framework then an attachment to T is any function that assigns new elements to any entities definable as belonging to T. The examples will make this clear.

Normal modal logic

Let T = <H, R, W> be a temporal framework and REL a function that assigns to W a binary relation on W. Define:

◊A^(t) = {h in W: for some k in W such that REL(h, k), k(t) is in A}

Read as the familiar ‘relative possibility’ relation in standard possible world semantics, a sentence expressing ◊A^(t) would be of the form “it is possible that it is raining”.

But such a modal logic has various instances. In addition to alethic modal logic, there is for example a basic epistemic logic where the models take this form. There, possibility is compatibility with the agent’s knowledge, ‘possible for all I know’. In that case a reading of ◊A^(t) would be “It is possible for all I know that it is raining”, or “I do not know that it is not raining”.

Deontic logic

While deontic logic began as a normal modal logic, it has now a number of forms. An important development occurred when Horty introduced the idea of reasons and imperatives as default rules in non-monotonic logic. There is still, however, a basic form that is common, which we can here attach to a temporal framework.

To each moment we attach a situation in which an agent is facing choices. What ought to be the case, or to be done, depends on what it is best for this agent to do. Horty has examples to show that this is not determined simply by an ordering of the possible outcomes, it has to be based on what is best among the choices. (The better-than ordering of the choices can be defined from a better-than ordering of the possible outcomes, as Horty does. But that is not the only option; it could be based for example on expectation values.)

Let T = <H, R, W> be a temporal framework and SIT a function that assigns to each moment m = <h, t> a situation, represented by a family Δ of disjoint subsets of the future cone of m, plus an ordering of the members of Δ. The cells of Δ are called choices: if X is in Δ then X represents the choice to see to it that the actual future will be in X. The included ordering ≤ of sets of histories may be constrained or generated in various ways, or made to depend on specific factors such as h or t. Call X in Δ optimal iff for all Y in Δ, if X ≤ Y then Y ≤ X. Then one way to explicate ‘Ought’ is this:

OA(t) = {h in W: for some optimal member X of Δ in SIT(<h, t>), X ⊆ A(t)}

This particular formulation allows for ‘moral dilemmas’, that is cases in which more than one cell of Δ is optimal and each induces an undefeated obligation. That is, there may be mutually disjoint tensed propositions A(t) and B(t) such that a given history h is both in OA(t) and in OB(t), presenting a moral dilemma.

An alternative formulation could base what ought to be only on the choice that is uniquely the best, and insure that there is always such a choice that is ‘best, all considered’.

Subjective probability

We imagine again an agent in a situation at each moment <h, t>, this time with opinion, represented by a probability function P<h,t> defined on the propositions. (If the state-space is ‘big’ the attributes must be restricted to a Boolean algebra (field) of subsets of the state-space, and thus similarly restrict the family of propositions.)

This induces an assignment of probabilities to tensed propositions: thus if R is a region in H, P(R^(t)) = r is true in h at t exactly if P<h, t>({h in W: h(t) is in R}) = r. Similarly, the probability FR^(t) is true in h at t, is P<h,t>({{h in W: h(t’) is in R for some t’> t}). So if R stands for the rainy region of possible states, this is the agent’s opinion, in moment <h,t>, that it will rain.

In view of the above remarks about the dependency of future on the past, the subjective probabilities will tend to be severely constrained. One natural constraint is that if h =/t h’ then P<h,t> = P<h’,t>.

In Horty’s (2019) examples (which I would like to discuss in a sequel) it is clear that the agent knows (or is certain about) which futures are possible. In that case, at each moment, the future cone of that moment has probability 1. For any proposition A(t), its probability at <h, t> equals the probability of A(t) ∩ H(h, t).

APPENDIX

I am not unsympathetic to the view that only what is settled is true. But the contrary is also reasonable, and simpler to represent. However, we face the puzzle that I noted above, about whether it makes sense to say that we have different possible futures, though one is actual, and future tense statements are true or false depending on what the actual future is.

In the Middle Ages this came up as the question of compatibility between God’s foreknowledge and free will. If God, being omniscient, knew already at Creation that Eve would eat the apple, and that Judas would betray Jesus, then it was already true then that they would do that. Doesn’t that imply that it wasn’t up to them, that they had no choice, that nothing they could think of will would alter the fact that they were going to do that?

No, it does not imply that. God knew that they would freely decide on what they would do, and also knew what they would do. If that is not clearly consistent to you — as I suppose it shouldn’t be! — I would prefer to refer you to the literature, e.g. Zagzebski 2017.

REFERENCES

(I adapted the diagrams from this website)

Belnap, Nuel; Michael Perloff, and Ming Xu (2001) Facing the Future; Agents and Choices in our Indeterministic World. New York: Oxford University Press.

Horty, John (2019) “Epistemic Oughts in Stit Semantics”. Ergo 6: 71-120.

Müller T. (2014) “Introduction: The Many Branches of Belnap’s Logic”. In: Müller T. (eds) Nuel Belnap on Indeterminism and Free Action. Outstanding Contributions to Logic, vol 2. Springer, Cham. https://doi.org/10.1007/978-3-319-01754-9_1

Thomason, R. H. (1970) “Indeterminist Time and Truth Value Gaps,” Theoria 36: 264-281. 

Zagzebski, Linda (2017) “Foreknowledge and Free Will“, The Stanford Encyclopedia of Philosophy (Summer 2017 Edition), Edward N. Zalta (ed.).