So, under a J-analysis, WnRWmonly if tn ≠tm. If this seems confusing, keep in mind that the possibility I want R to characterize is understood as physical compatibility between worlds-at-times joined in causal relations, as parts of causal “paths.” To aid comprehension, our R can be understood and defined in terms of two other relations, which I’ll call the mother-relation and the daughter-relation. If WnRAm, if Wn is accessible from Am, we say that Wn is either a mother or a daughter of Am. Wn is a mother of Am iff WnRAmand Wn is temporally prior to Am. “Wn is a mother of Am” means basically that, where m>n, where n is “earlier” than m, the physical situation that obtains in Wn (W-at-time-n) and the causally efficacious events and states of affairs that obtain in Wn, together with the physical laws that govern the set of physically possible indices, are not physically incompatible with the physical situation that obtains in world-at-time index Am(intuitively, if it is physically possible that W-at-n could “give rise” to A-at-m). The daughter-relation is simply the reciprocal image of the mother-relation: if Wn is a possible mother of Am, Am is a possible daughter of Wn.
To take an example of how these relations work, suppose that W-at-t2—W2—includes as a feature a nuclear explosion at Amherst College, and that A-at-t3—A3—includes as a feature there being fewer than 20 rads of radiation at Amherst. In this case, A3 is not a possible daughter of W2, and so W2 is not a possible mother of A3: the situation and events in W2 are not physically compatible with, could not have given rise to, the situation in A3; A3 is not a possible causal consequence of W2. These two relations, then, are comprised by and together define our R-relation: if Wx RAy, then either Wx is the mother and Ay is the daughter, or Ay is the mother and Wx the daughter, depending only on which temporally distinct world-situation obtains first. Thus we can at this stage intuitively understand physical possibility as follows: if, in the actual situation that obtains now, I assert that it is physically possible now that p-now, my assertion will be correct if and only if the actual situation that obtained some temporal units ago35and gave rise to the actual situation that obtains now is not physically incompatible with, stands as a possible mother of, a situation now in which p obtains. Thus, even though the relation R is here defined as exclusively diachronic, it is easy to see that if p is here and now true, ◊p is here and now true (since the immediately-past situation that actually gave rise to the actual situation now in which p obtains is quite clearly not incompatible with the situation-now in which p obtains).
I’m gradually going to construct a visual apparatus in which I can represent the diachronic relation R between indices, the relation that will determine the interpretations and truth-values of tensed physical-modal propositions. Let’s look first at a tiny subset of the set of physically possible worlds-at-t2, the set {W2,A2}
We can say that W2, the actual world-at-t2, stands in an R-relation as the possible mother of all those possible worlds-at-t3 whose physical situations are not causally, physically incompatible with the situation and events in W2, all those 3-worlds to which W2 could possibly have given rise:
In the above diagram, A3, W3 and B3 are the daughters of W2: the physical situation and causally efficacious events in W2, together with general physical laws, are not incompatible with, and could have given rise to, A3, or W3, or B3. Here W3, the actual world-at-t3, is the “actual” daughter of W2: W2 did indeed give rise to W3.
Just as on this account a possible world-situation can have more than one possible daughter (although an actual world-situation can obviously have only one actual daughter), so too a world-situation can here have more than one possible mother (although an actual world-situation can have only one actual mother):
Here W3 has an actual mother and a different possible mother: one that could have given rise to W3 had it obtained and one that did obtain and did give rise to W3. B3 has two possible mothers. A2 has four possible daughters. W2 has one actual daughter and three possible daughters. And so on.
Note here that if we intuitively divide the R-relation into the mother-relation and the daughter-relation, the one vital feature of each of the two relations is that it is transitive: if A2 is a mother of D3 and D3 is a mother of C4, then A2 must be regarded as a sort of mother of C4, and C4 regarded as a sort of daughter of A2. This is because I want the device to be able to operate successfully with an ideally infinitely dense time-axis, in which between any two “successive” worlds-at-times there exists another world-at-time (a fact obviously not representable in a diagram).
So we’re now in a position to use these visual features of system J to analyze our two relevant problems in the scope of tense- and physical-modal operators. Let’s first take another look at the terrorist case. I specify for our purposes three times: t2, t3, and t4. t2 = the day before yesterday; t3 = yesterday (alias P1); t4 =today. Here, then, is a simple theoretical physical-possibility structure, Φ, that begins its representation with the actual world-situation that obtained at t2 and ends with all physically-possible-worlds-at-t4. W’s are still actual worlds:
Say that at index W2—the actual world the day before yesterday—the actual situation that obtains includes such features as the nuclear weapon being brought onto the Amherst campus, its being rendered fully functional, the terrorists discussing what to do, the head terrorist limbering up his trigger-finger, etc. Features of W3, the actual world-situation-at-t3, include the head terrorist sitting with his finger resting on the trigger, the trigger and weapon being perfectly operational, the head terrorist being completely unconstrained, etc. Features of the actual world-situation-at-t4, W4, include there being fewer than 20 rads of radiation on the Amherst campus (represented in the diagram). I hold that, in the terrorist case, with t4 being today, the proposition ~◊P1E is today true, while the proposition P1~◊E is today false. The above physical-possibility-structure diagram allows us to justify this claim in something more than an intuitive way.
To demonstrate the truth today of ~◊P1E (or the exactly equivalent and better-formed t4~◊t3E) via the visual device, we go first to the actual world-situation that obtains at t4. This is the index W4. We look “back” from W4 at that set of all possible-worlds-at-t3 that stand in the accessibility relation R to W4. This is the set of all possible mothers of W4, the set of world-situations-at-t3 which, given prevailing physical laws, could have “given rise” to W4 had they obtained. This is the set {C3, W3, D3, E3}. We look now to see if E (Explosion) is a feature of any of these possible mothers of W4. As I’ve constructed the diagram, it is not. This means that, given the actual world-situation-at-t4, it is not today, at t4, possible that an explosion did occur at t3 (alias P1), that is, that ~◊P 1E.
Now I’ll try to demonstrate the falsity here of the stronger, fatalist proposition P1~◊E (or t3~◊t3E). Note first that to evaluate this proposition we must look, not to W4, but to the set of all physically-possible-worlds-at-t 3. This is nothing other than the set of all worlds-at-t 3 that stand in the R-relation to W2, the set of all possible daughters of the actual-world-at-t2. This is the set {A3, B3, C3, W3, D3, E3}. It is of course important to see that this set is not identical to the set of all possible mothers of W4. P1~◊E (and so t3~◊t3E) would be true iff E were not a feature of any of the physically-possible-worlds-at-t3, any of the possible daughters of W2. Referring to the diagram, we can see that E is presented as a feature of these possible-worlds-at-t3: A3 and B3. We can thus see that at t3, relative to the situations obtaining at t2 and t3, it was here physically possible for the terrorist to push the trigger at t3, and so it was at t3 physically possible for a nuclear explosion to occur at t3, and so that, in precisely the same structure with which we demonstrated the truth here of ~◊P 1E, we can demonstrate the manifest falsehood here of P1~◊E. (For formal reasons that will be explained, I will ultimately want to eliminate reference to t3 in all analyses of what is physically-possible-at-t3, and make exclusive reference to the actual situation that obtained at t2, but the results, and the implications for the Taylor argument, will rem
ain exactly the same.)
Under this sort of informal use of system J, its structure Φ, and the analyses of the terrorist case they afford us, then, (IV-5) and (IV-6) are not equivalent: (~◊P1E) ≢ (P1~◊E). Note that under a Φanalysis, (P1~◊E) implies (~◊P1E), but not vice-versa. It’s easy to see why this is so: the set of possible world-situations-at-t3 ranged over by the physical modal in (~◊P1E) is but a proper subset of the set of possible world-situations-at-t3 ranged over by the physical modal in (P1~◊E).
Under the above analysis, the terrorist case, with the designated sufficiency-relation between nuclear explosions and high radiation, and the low radiation today, does yield the conclusion ~◊P1E, but does not yield the stronger, fatalist conclusion P1~◊E. This should obviously resonate pleasantly with most people’s modal intuitions about the case. And why only the non-fatalist modal proposition is derivable from the case accords with people’s reasonable intuitions, too: the non-fatalist proposition derives from our evaluating a modality, asserted to hold at t3, in the context of the physical situation obtaining at t4, and it is precisely this situation-at-a-time, with its low radiation, that is presented in the case itself as having the modally limiting force. The stronger, Taylor-ish proposition could derive only from our evaluating a modality, asserted to hold at t3, in the context of the physical situations obtaining at and before t3; the above evaluation of the terrorist case does not yield physical-impossibility-at-t3 precisely because the modally limiting situation in the case (low radiation at t4) does not obtain at these times; there is absolutely nothing in the physical situations obtaining at and before t3 that renders E-at-t3 physically impossible, nothing in these situations with which E-at-t3 would be physically incompatible.
An identical analysis, using the intuitive Φ-structural features of the system J, can be performed on the original Taylor problem itself. Let W4 be the actual world-situation that obtains today, with the absence of a sea-battle being a prominent feature. Let W3 be the actual world-situation yesterday, when I, the admiral, was in a position to give orders. Let W2 be the actual world-situation that obtained the day before yesterday. To show that (t4~◊t3O) is true, we look to W4, the actual battle-free world-at-t4, and examine the set of its possible mothers for the presence of battle-order O. Given that, as Taylor constructs the case, a battle-order in a t3 mother-situation would have physically-necessarily yielded a battle in W4, we can conclude that O is not a feature of any of the possible world-situations-at-t3 that stand in the relation R to W4, that is, that (t4~◊t3O), given the character W4.
In order to show that a battle-order was physically-impossible-at-t3 in the context of the physical situations obtaining at and before t3, however, we are required to examine, not the set of possible mothers of W4, but rather the set of possible daughters of W2; that is, we examine the set of all physically-possible world-situations-at-t3. Now, since Taylor gives absolutely no evidence for any physical circumstances limiting the admiral’s freedom or ability or power to give a battle-order being features of all the members of the set of possible daughters of W2, it is easy to conceive of order O being a feature of at least one possible world-at-t3, given the physical situations obtaining at t2 and t3. Taylor is able legitimately to conclude only that, at t4, the order cannot have been given by the admiral at t3, not that, at t3, the admiral could not give it.
Thus in his own fabricated example Taylor has really shown only that the physical modalities that are asserted to obtain at a time-and-situation are sensitive to the contexts created by the mother-and daughter-relations enjoyed by that situation-at-a-time. I hold that, regardless of the time-interval the modality is asserted to range over, the evaluation of the modality must, if features of other situations-at-times are alleged to bear on that modality, be carried out in the context of those other situations-at-times, not in the context of the situation-at-a-time the modality is asserted to range over. If this point still seems opaque, simply remember that what is physically possible to have occurred at t3 relative to the situation that actually obtains at t4 is not necessarily the same as what is physically possible to occur at t3 relative to the situations at t3, t2, t1, etc., that, under a proper understanding of physical possibility, whatever else is closed, the future remains very much “open.” This is so simply because physical modalities are understood here as sensitive to time and sensitive to world-situations causally joined in mother- and daughter-relationships, as parts of causal paths. And this understanding of physical modality seems to point to a way to solve the Taylor problem, to show that even under the most generous acceptance of his premises and reading of his argument, the fatalistic conclusion he wants to “force” upon us simply does not validly follow.
A clear grasp of the features and merits of system J’s physical possibility semantics requires that J be more formally characterized and presented via explicit rules for well-formedness in the system’s language, and some formal semantic rules for the evaluation within the system of tensed physical-modal propositions.36
Let Φ (a physical possibility structure) be a set of distinct but intersecting paths ji-jn, each of which is a set of functions, L’s, on ordered pairs 〈t, w〉 (〈time, world-situation〉), such that for any Ln, Lm in some ji, LnRLm, where R is a primitive accessibility relation corresponding to physical possibility understood in terms of diachronic physical compatibility. (That is, the structure is made up of distinct but not necessarily exclusive paths, j’s, each path representing an individual trans-temporal causal chain between world-situations, where for each world-situation-at-a-time in a single path ji, it stands in some mother- or daughter-relation to every other world-situation-at-a-time in ji.) Any well-formed formula in the language of J must begin with an operator specifying the index of temporal evaluation. When no such operator is indicated in a natural-language proposition, the proposition, in order to be well-formed in the language of J, must be translated into the language as bound by an operator designating the moment “now.” So, where a formula p contains no tense- or physical-modal operators, where tn and tm serve as any moment-designating temporal operators, where W and W′ designate world-situations, with W being actual, where ◊ = “It is physically possible that ... ,” and □ = “It is physically necessary that ... ,” P= “In the past,” and F= “In the future,” the following semantic rules of J can be used to evaluate the sorts of propositions with which we are concerned.
Rule 1) [[tnp]]w = 1iff [[p]]w, = 1.
Rule 2) [[tn◊tmp]] w = 1 iff ∃jx ∧ ∃ W′ such that 〈W,tn〉 ∈ jx ∨ 〈W′ ,tm〉 ∈ jx ∨ [[p]]w′, = 1.
Rule 3) [[tn□tmp]]w = 1iff[[tn∼◊∼tmp]]w = 1.
Rule 4) [[Fp]]w, = 1iff ∃jx ∨ ∃tm such that 〈W,tn〉 ∈ jx ∧ 〈W,tm〉 ∈ jx ∧ m > n ∧[[p]]w, = 1.
Rule 5) [[Pp]]w, = 1iff ∃jx ∧ ∃tm such that 〈W, tn〉 ∈ jx ∧ 〈W,tm〉 ∈ jx ∧ m
These rules allow us formally to demonstrate the inequivalence of the alternative tensed physical-modal propositions derived from the Taylor problem and the terrorist case. In the terrorist case, the inequivalence can be shown by proving the compatibility within the system of the following two propositions:(a) Yesterday it was possible for there to be an explosion.
and(b) It is not possible today that there was an explosion yesterday.
By rule (1), (a) is true iff(a′) It is possible that there is an explosion
was true yesterday. But (a′), in order for its translation to be well-formed in the logic of J, must be translated into an expression with two temporal operators. This is because of the stipulation that every wff in the language must appear under the scope of a temporal operator specifying an index of evaluation. “There is an explosion” is a formula (like proposition p in rule (1) above), and takes an operator, say tn. But “It is possible that tnE” is itself also a formula, and must also appear under the scope of a temporal operator—tm◊tnE. Recall further that, in J, physical possibility is defined in terms of a diachronic accessibility relation R;
the only way a state of affairs in one world-at-a-time can be “possible” is relative to another, distinct world-at-a-time. Thus the only plausible way to interpret (a′) under the rules of the system is as saying that at a time tn, where tn is designated the time yesterday:(a”) A few moments ago (tm) it was possible for there now (tn) to be an explosion.
Thus the translation of (a′) into the logic of J does indeed turn out to be:(a′”) t m ◊t n E (where n is a few moments later than m).
And (a′”) is true in the actual world today iff there is some path jx from a few moments before tn (yesterday) in the actual world to some world contemporary with yesterday in which there was an explosion, by rule (2).
Let’s now consider (b), which has, if to is designated today and tn is designated yesterday, the form:(b′) Today (to) it is not possible that yesterday (tn) there was an explosion,
Fate, Time, and Language Page 19