Dynamic Accessibilities

Various notions of accessibility appear in the literature. Accessibility is given a par-ticular understanding in this essay, in keeping with the tradition of branching time logics.

Roughly, a nodefor clarity, call it the target nodeis accessible from another nodecall it the source nodejust in case the target is realizable or actualizable from the source; that the source might transition into the target. For instance, there is a genuine transition or potential transition between the states depicted in Figure 2.3.1 on page 30. This notion of accessibility will be called the dynamic account of accessibility. The dynamic account is relatively intuitive and has been the prevalent take on accessibility in branching time logics, used heavily throughout this essay. Such focus, especially in the eld of temporal logic, is no doubt inuenced by Prior, to whom Peter Geach suggested this understanding of accessibility for temporal logics.¹¹

There is no pretension that the brief discussion here proves that the dynamic notion of accessibility is the best. Ocially, the dynamic account is taken as a background assumption.

It is nevertheless worthwhile to clarify Priorian accessibility, comment on its generality, and explain why it is relevant to this project.

In ordinary language, a thing is accessible to someone if and only if the person can get to

11[Copeland(1996)]

it. If and only if Jones can reach a book on a shelf, the book is accessible to her. Provided that the book is inaccessible to her, then she cannot get to it, perhaps because it is locked in a vault. One might say that a target node is accessible from a source node if and only if the source can get to the target. This portrayal makes it look as if nodes are like people in space, some of whom can touch some others. Another analogy might convey nodes as places, like Detroit and Chicago. Detroit and Chicago are connected by the interstate highway. They are accessible from one another; that is, people can go from one to the other. Maybe a target node is accessible from a source node if and only if one can go from the source to the target.

These analogies provide an intuitive start, but they appear circular if mistaken for expli-cations. Accessibility is itself a modal notion, indicating that a (grammatical) subject can access an object, and was just described using modal terms; in this case, can. The course can get to the target, one can go from the source to the target, and so forth. Modality, in turn, is explicated using accessibility; hence the apparent circularity.

Seeking to avoid circularity, many philosophers nowadays use consistency as the foun-dation of accessibility.¹² Propositional modality plays a signicant role here. The weakest form of accessibility is just logical accessibility. Every node is accessible to every other node.

For propositional modalities in general, a source node accesses a target node just in case the source satises the laws of the modality, given that consistency is not violated. This deni-tion makes it so that accessibility depends on what is true at nodes, with modal statements playing an especially important role in limiting accessibility; even though modal operators are characterized in terms of accessibility.

There is no circularity, at least no vicious circularity, in interdening accessibility relations and modal operators using the above method; but such denitions do not capture very much in the way of content. That is exactly what one would expect for accessibility and modality.

Propositional modalities specify accessibility structure in connection with true propositions.

12This is not to say that those scholars think that there is no more to modality than consistency. One must know what the laws are, which laws are relevant to genuine modality (whatever that means), how a formal system must be developed to represent modal statements involving individuals, and so forth. Such issues are tangential to this discussion.

There is no more content to be found when accounting for propositional modality because those modalities are so diverse. For instance, physical and deontic modality may not have much in common besides the fact that both lend themselves to this type of characterization in terms of propositional modality. Additional content, accounts of accessibility relations and modal operators, is only to be found when addressing those rather dierent modalities individually, not in a more general account including both. The sources of content for physical and deontic modality, if not independent, are at least disparate.

The account of accessibility indicated so far is static. The static notion is based primarily on consistency, which generates no ow or movement, not even possible ow or movement.

The ordinary notion of accessibility, to the contrary, involves accessing.

It has been argued that the dynamic content associated with nodes accessing each other is only metaphorical.¹³ There are, however, reasons to include dynamic content. Many philosophers, especially those who use branching time systems, hold that time ows.¹⁴ In branching time logics, the present moves along accessibility paths. The debate about whether or not time ows has a very long history, dating back at least to Heraclitus and Parmenides in the Western canon, and remains too contentious to address in satisfactory detail here.

The dynamic view of time is taken as a background assumption for most of this essay.

The problems of logical and theological fatalism are in part about what might become the case or, put in terms of agents, what is within an agent's power to bring about. If it is within Jones' power to drink coee tomorrow, then she can make her coee-drinking real or actual tomorrow. Put in terms of moments, there is some moment in which Jones drinks coee tomorrow and it is in her power to make that moment real. Agents aside, if some event is future-contingent, then there is a moment representing its occurrence that might become real. A static account of accessibility is insucient to explicate libertarian freedom and future contingency, which have dynamic content. Freedom and contingency are not just

13See, for instance, [Smart(1949),Nerlich(1998), Sider(2003)].

14This idea is prevalent in Prior. See, for instance, [Prior(1957),Prior(1967)]. McCall also advocates time flow [McCall(1998)].

a matter of consistency, but of change or potential change.

A good candidate source of accessibility's dynamic character is the so-called ow of time.

In branching time models, present moments move along the tree as if it were a road.¹⁵ The notion of time ow works well with the ordinary idea of accessibility and has been used in formal settings involving both branching time logics and relativistic branching space-time logics.

A common question (or criticism) is, how fast does time ow? The standard answer is, one second per second.¹⁶ Consider an analogy. One might watch a lm at various speeds

fast, slow, even backwards. The characters do the same things, although faster, slower, or backwards. Similarly, one can imagine present moments moving through a branching time model at dierent speedsagain faster, slower, or even backwards. That movement is faster, slower, and backwards is only in comparison between the time scale of the lm or the model to another, relatively external time scale of the viewer. From the perspective of the

lm characters or someone in the model, there is no dierence in time owit ows at one second per second. From the perspective of the viewer, time also ows at one second per second, although the viewer apprehends that the lm's time scale can dier. This analogy illustrates two points. First, no matter how the viewer plays the lm or the logician imagines the model, it makes no dierence to those in the lm or model. Second, no matter which frame of reference one is in, time ows at the same rateone second per second, although the rate at other frames may appear dierent.

A more rigorous account of dynamic accessibility is called for. Such an explication might be given by just tacking on the property of being dynamic to the static account; but that approach is misleading, if not backwards. The static account may be taken to start with logical accessibility, using it to build stronger types. This direction of construction is not appropriate for the dynamic account because it is not evident how movement is supposed to

15The same may be said of branching space-time models, although past, present, and future are frame specific. See, for instance, [Belnap(1992),McCall(1976),McCall(1994),McCall(1998)].

16See [Prior(1967),McCall(1998)]. Cf. [Smart(1949),Nerlich(1998)].

come from logical accessibility.

Whatever accessibility is involved in future contingency and libertarian freedom might be called all-things-considered (ATC) accessibility. It was indicated above that ATC acces-sibility has a dynamic character that goes beyond mere consistency. Future contingency is about what might become the case; libertarian freedom is about what an agent can bring about. Time ow is part ofATC accessibility. SoATC accessibility, not logical accessibility, is the starting point of the dynamic account. Other propositional modalities are derived fromATC modality, and the former inherit the latter's dynamic character.

Suppose that ATC accessibility is propositional. In fact, physical accessibility is a good candidate forATCaccessibility, although the issue is ocially left open here. Given thatATC accessibility is propositional, its structure can be represented in terms of consistency just as in the static account described above. The structure of ATC accessibility is represented using consistency, but ATC accessibility is not derived from logical accessibility per se. It follows that ATC accessibility is free to retain its dynamic character. ATC accessibility is more than its structure.

Let P be the set of laws characterizing ATC accessibility. P represents all of the rules to be considered in all-things-considered accessibility. Thus, if S is a nonempty subset of P , then S generates a some-things-considered accessibility. If S is the empty set, then S leads to a no-things-considered accessibility, which is just logical accessibility. If S is a proper superset of P , then S yields an extra-things-considered accessibility. Et cetera.

Here are some examples. It is not ATC-possible to drop a stone from atop a building without it falling. However, ignoring physical principles, it is possible to drop the stone without it falling. Considering only some principles or no principles at all, the type of accessibility can be weakened. Suppose that it is ATC-possible for Jones to punch her boss in the face without getting red. Given extra considerations, like the rules of her workplace, Jones will inevitably get red for punching her boss. So it is not possible in this stronger sense for Jones to punch her boss without getting red.

ATC accessibility can be weakened, strengthened, or some combination thereof to con-struct any propositional modality. The dynamic character of ATC accessibility is inherited by those accessibility relations derived from it. Thus, propositional modalities in general are dynamic under this interpretation of accessibility. It does not follow that all formal modalities are dynamic, nor that propositional modalities are dynamic if they are given a dierent interpretation, one by which ATC modality is not taken as primary. The dynamic interpretation is nevertheless the most appropriate for considering future contingency and libertarian freedom.