probabilistic stit

our indeterminist world

we can represent the branching histories model as a partially ordered rooted tree; the root represents the single first past moment. each node of the tree is a branch point, at which two possible futures diverge. some branch points are choice points; that is, a choice point is a branch point for which some agent chooses which of several possible future histories to make actual.

agents at choice points cannot choose a specific history, however. rather, the histories through the choice point are partitioned into choice sets, and the agent chooses one of those sets. which history within that set becomes actual will depend on the outcome of future branch points.

in Belnap's presentation of the branching histories model, the terms moment and instant are distinct technical terms. an instant is a collection of moments, which are best thought of as the same wallclock time, but along different distinguishable histories.

for any moment, there is one history flowing through for each possible future of the world, but of course these different histories cannot be distinguished. if they could be, then they would amount to different moments; that is, a moment is a single state of affairs, snapshotted at a particular moment of time.

a proposition is said to be settled at a particular moment if its truth has become certain—which is to say, if it has the same truth value on all histories through that moment.

for example, let the proposition p be at 2pm Alice purchases a cup of coffee. at 1pm, p is not settled. at 3pm, p is now settled (whether true or false)—all the histories through any single moment at 3pm will have the same truth value for p. but at 1pm, p is not settled, since there are histories through such a moment at 1pm in which Alice does purchase the coffee, and others on which she does not. since we must always evaluate all propositions at moment/history pairs, and since the different histories through any 1pm moment have different values for p, the proposition is not settled at that time.

finally, note that branch points come in two flavors: some are the decisions of agents, while others simply happen. the ones that simply happen can be characterized as decisions of a special agent known as nature. this essay is concerned with taking the branch points at which nature operates far more seriously than most stit discussions consider.

sees to it that

in this paper, i will be concerned with the achievement stit. at a given moment/history pair we say that [α stit: p] iff there is some past choice moment m for α, such that on every history through the option α chose at m, p is true, and there is some history through that m in which p is not true. the second condition, called a counter, is part of the definition in order to block the claims of king Canute to bring in the tides.

nature is radically indeterminist

for example, as i write this i depend on the presence of oxgyen in the air i breathe. but the laws of statistical mechanics, which govern the way gases behave, assign a determinate probability to the chance that all the oxygen atoms in this room should collect in a corner, leaving me to suffocate. to be sure, the probability of this unfortunate occurrence is exceedingly small—nothing to worry about—but it is not zero.

we can model this possibility in a branching histories structure with little difficulty. at each instant, there is a branch point, at which nature chooses either that the gas in the room remains mixed in the usual manner, or one of the various chances it could behave in bizarre and unusual ways. we know that the history representing the usual manner of behavior is vastly more likely than any of the others, but the branching histories model does not mention probabilities.

and so, if we try to evaluate a stit expression, where the result depends on whether i can continue to breathe, we find that these unusual possibilities radically affect the result. for example, in a choice diagram like the following, we might be inclined to say that [α stit: s]. but inclusion of the actual branching points that statistical mechanics notes, gives the following diagram. because there is always a vanishingly small chance (but not strictly zero) that my choice will fail, we see that the success condition for a stit sentence will never be satisfied for any ordinary real-world situations. and, similarly, there will always be a counter, even if a very unlikely one.

in normal life, we ignore possibilities which have vanishingly small probabilities, but the stit semantics and the branching world model have no way to account for them. one could adopt a model of ignoring what we know must be ignored, but it is the goal of this paper to offer a more formal and principled account of where and how we should go about this important task.

probabilistic stit

the second condition is the requirement that there be a counter to the agent's achievement of the complement. here we want to require that the counter not be of vanishingly small likelihood.

first we need to augment the branching histories model to include mention of the probabilities of natural events. we do this by adding probabilities to the various choices made by nature at natural branch points, as seen in figures 2 and 3. to avoid unnecessary complication, we assume that the space of histories at nature's branch points is discrete; that is, that there are no possible events with probability zero.

we do not want to fix ahead of time what the probability thresholds should be, so we are defining in fact a family of connectives, stit_p,q, where p is to be the threshold on the success condition, and q the threshold on the counter condition.

we want our probabilistic stit to hook up with normal stit, so it must be the case that:

α stit _p,q: s ⇒ α stit _p,r: s (whenever q < r)
α stit _p,q: s ⇒ α stit _r,q: s (whenever p < r)
α stit _1,1: s ⇔ &alpha stit: s.

the success condition on α stit _p,q: s is easy to state. the agent's choice must have made s to have a likelihood of at least p.

the counter condition, however, is a bit trickier. we cannot meaningfully ask what the probability of ¬ s is given that α does not make the choice she actually did, since those probabilities are assigned for each of nature's branch points, but not for discriminating between the particular choices that α might have made. it seems that the correct condition is that there is some choice that α could have made, under which the probability of ¬ s is appreciable. but we cannot simply insist that there is a choice for α such that the probability of ¬ s is at least q. in diagram 4, we want to say that [α stit _p:s] for some high p, (having chosen to try and write) but there is no counter with a sufficiently high probability if we require that the counter hav a probability of at least p.

we find the right condition by looking to our motivating considerations. we want the result to be that which we would get if we were to ignore low probability events. so a counter is an outcome from some choice such that the outcome did not have a vanishingly small probability. the criteria are then:

[&alpha stit _p,q: s] iff there is a witness w such that
the probability of s given the choice α made at w is ≥ p, and
there is some choice α could have made for which the probability of ¬ s was > 1 − q.

it is clear that this definition passes the minimal criteria above. we will abbreviate stit _p,q as stit _p, and write stit ₁ as simply stit, without loss of precision.

suppose a very unlikely event actually does happen. the conditions above could still have it that [α stit _n: s] on a history in which s is settled false! so the conditions above are not yet sufficient; we must add an additional success condition: m/h models [α stit _p,q: s] only if m/h models s.

higher probability events

consider a person who plays russian roulette with his friend: he puts one bullet in the revolver, spins it, and fires at his friend. from the diagram it is clear that [&alpha stit _1/10: β is killed]. this is often enough for responsibility. but if β was going to have the same risk of death either way, then the sense in which α has responsibility would seem to be changed. (as is usual in discussions of stit, i am not here concerned with whatever subjective or internal conditions might apply to responsibility.) and in such a case, we would find that there was no applicable counter.

modeling of other agents

when we allow for probabilistic stit, however, it is clear that we can have [α stit _p: [β stit _q: s]], provided p < q. the success condition for [α stit _p: [β stit _q: s]] allows for ignoring events with a small chance of happening, but if p < q, then those chances might still be big enough to serve as a counter for [β stit _q: s].

more importantly, from the perspective of a single agent, other agents might appropriately be conceived of as probabilistic choosers in the manner nature is. and from that perspective, probabilistic considerations similar to those of the immediately preceding section are also operative. such modeling of other agents' psychology is a strikingly absent element in conventional stit analysis, and probabilistic stit provides a way to begin getting a handle on the problem.

an alternative way to account for negligibly probable histories

however, this strategy is ill conceived. what is it for an event to be probable at a particular moment/history pair? if we are speaking of future events which either happen or not, the probabilities will always be either zero or one.

first, it does not correspond to the branching histories model. look back at figure 2. note that on the right-hand history, p(oxygen collects in corner of room) = 1. indeed, once a proposition is settled at some moment, isn't its probability always unity?

similarly, it does not give us a clean solution to the counter condition. is a counter to be any history on which the probability p(s) is not greater than some threshold? but this is so if there is any history on which ¬ s, which is to say, we have failed to exclude negligible events.

we can address the first problem by carrying forward probabilities from nature's branch points to the resultant states. this can indeed be done rigorously; the probabilities to be carried forward are those between the moment at which we evaluate the stit and the witness for it. so we need a notion of probability relative to a witness.

similarly we can solve the second problem by considering countering choices rather than just countering histories. however, how is this to be written in stit semantics, such that [α stat: s] continues to have the same truth conditions as before, except when s is a probabilistic statement, and then it has the new choice-sensitive notion of a counter?

it is not clear that these adjustments could be succesfully made (and preserve a univalent set of truth conditions for stit). but more to the point, making them already concedes the inability of conventional stit to handle the negligible-probability situations. if one needs a more flexible connective than ordinary stit, why not simply choose a straightforward generalization such as stit _p,q?

indeed, this generalization also more naturally maps to the way we think about negligible probability situations. we do not say i made it very likely that i would finish the paper, we say i saw to it that i finished the paper. our practice is to ignore the very low probability events, and the stit _p,q connective explicitly considers this process of ignoring, rather than shifting it onto the complement.

conclusion