2.1.Humean Chances and the Big Bad Bug

Following ^{Hall (2004)}, we lay down a few assumptions about chance. First, chances are probabilities defined over propositions. Second, chance is both time and world-dependent thus, we speak of the chance of a proposition being true at a time t and at a world w. We represent the chance distribution at t at w by the probability function P _t,w (·). Third, chances are defined for arbitrary Boolean combinations of propositions, provided they are defined for the propositions so combined. Fourth, what’s past is no longer chancy that is, for any proposition A entirely about the past (relative to a time t), either P _t,w (A) = 1 or P _t,w (A) = 0, depending on whether A is true or false at t. Fifth, the chance distribution P _t’,w (·) at a time t'>t may be obtained from the chance distribution at t by conditionalizing on the complete history of w from t up to and including t', i.e., P _t,w (·) = P _t,w (·|It,t'), where I _t,t is a proposition completely describing the history of w between t and t'. In general, we can represent the chance distribution at a world at any given time in terms of an ur-chance function urch _w and the complete history of w up to that time, so that, for any t, P _t,w (·) = urch _w (·|H _t,w ), where H _t,w is a proposition completely describing the history of w up to t. Finally, chances are law-governed, i.e., facts about chance at a world w at any given time are entailed by the laws of nature that hold at w and the complete history of w up to that time.

^{Lewis (1980)} formulated a very plausible credence-chance principle, thus giving a precise expression to the idea that it is rational for us to let chance guide credence:

The Principal Principle (PP). Let C be any reasonable initial credence function, ch _t (A) = x the proposition that the chance of A being true at time t equals x, and E a proposition compatible with ch _t (A) = x that is admissible at t. Then:

C(A|cht (A) = x & E) = x,

where E is the total evidence available to the agent at a given time. Lewis’ assumptions concerning admissibility allowed him to derive a special case of (PP):

The Principal Principle (special case) (PP). Let C be any reasonable initial credence function, P _t,w (·) the chance distribution at w and t, H _t,w the complete history of w up to t, and T _w the complete theory of chance for w. Then:

C(A|Ht,w  &  Tw) = Pt,w (A),

where T _w is a conjunction of all the members of the maximally consistent set of history-to-chance conditionals that hold at w (that is, strong conditionals whose antecedents are propositions of the form Ht,w (for some t in the history of w) and whose consequents specify facts about chance at w at t)¹. In this setting, chances are law-governed in the sense that, for any t, T _w , together with H _t,w , entail all the facts about chance at t at w.

Now, the Principal Principle conflicts with Lewis’ own reductive account of chance, namely Humean Supervenience. According to this view, the fundamental features of reality are spatio-temporal relations and local qualities of spatio-temporal points or point-sized occupants of those points (cf. Lewis 1986b, p. ix). All else supervenes on those qualities and those relations. In particular, chance supervenes on the Humean base in an indirect way. As we saw, facts about chance at a time t depend both on facts about the history of the world w up to t and the laws of nature that hold at w (probabilistic or not). So, chances supervene upon the Humean base via the supervenience of both history and the laws of nature. There is no peculiar philosophical problem as to how history supervenes upon categorical facts. But there is a philosophical problem concerning how laws and, in particular, probabilistic laws, may depend on purely categorical properties. Lewis’ answer is the best system account of lawhood: the laws at a world w are the theorems of the system that achieves the best balance between simplicity, strength, and fit. Even though they are not precise notions, we have enough grasp on the idea of simplicity and strength. However, the idea of fit is a tricky one. Lewis takes a two-step approach: first, discard the theories that make false statements about the history of w, and then look for the theory that assigns the highest probability to the actual course of events throughout the whole history of w this is the one that best fits the whole history of w.

Here is where the big bad bug bites. For suppose that the complete theory of chance for a world w does indeed supervene on the global distribution of local, categorical properties throughout all of w’s history. Let G _w be one such global distribution for w. By the supervenience thesis, there is a complete theory of chance entailed by G _w call it T _w . Now consider a time t well before G _w is complete. At t, G _w has a chance of obtaining, but also has a chance of not obtaining, so that P _t,w (G _w ) < 1. Moreover, if t is adequately selected, there will be a chance of a very different global distribution G'w obtaining, different enough from G _w so that T _w will no longer be the best fit for it. Rather, to it will correspond a different complete theory of chance T'w incompatible with T _w . It follows from this incompatibility that C(G'w|Ht,w  &  Tw) = 0. However, for this particular t, we have that P _t,w ( G'w) > 0. The Principal Principle turns this into a full-fledged inconsistency:

Lewis’ Result. (PP) and Humean supervenience are incompatible.

Proof. By T _w and H _t,w we have that (1) P _t,w (G'w) ≠ 0. By the incompatibility between G'w and Tw, we have that (2) C(G'w|Ht,w  &  Tw) = 0. By the admissibility of Ht,w  &  Tw,C(G'w|cht (G'w) = x  &   Ht,w  &  Tw) = x is a derivable instance of (PP). Since by hypothesis ch _t (G'w) = x holds at w for some x, Ht,w  &  Tw entails ch _t (G'w) = x. Whence we get C(G'w|Ht,w&Tw) = x. Since P _t,w (·) is defined on G'w, P _t,w (G'w) = x, we get C(G'w|Ht,w  &  Tw) = P _t,w (G'w). Then, by (1), C(G'w|Ht,w  &  Tw) ≠ 0. This contradicts (2).

This problem actually affects a broad class of reductionist accounts, for the culprit is the assumption that chance is reducible to the complete arrangement of categorical facts throughout the entire history of the world, regardless of the particulars of the reductive mechanism and the nature of the base for reduction. This has come to be known as Lewis’ big bad bug: an analysis along reductionist lines is committed to the existence of undermining futures, that is, futures that (i) have a chance at t of occurring but (ii) whose occurrence would undermine the actual chances at t; and the preceding argument shows that the existence of undermining futures is incompatible with an unrestricted acceptance of the Principal Principle. Obviously, something has got to give.

2.2. Alternative Credence-Chance Principles

The way out for the reductionist is to find a credence-chance principle compatible with the existence of undermining futures, and to provide a motivation for it, so that the move to the new principle doesn’t involve an excessive cost². The most promising alternatives are ^{Hall’s (1994)} New Principle and ^{Ismael’s (2008)} pair of principles, the Unconditional Principal Principle and the General Recipe for reasoning under uncertainty concerning chance.³

^{Hall (1994}, ²⁰⁰⁴⁾ proposed a replacement for (PP):

The New Principle (NP). Let C be any reasonable initial credence function, E the agent’s total evidence, P _t,w (·) the chance distribution at w and t, H _t,w the complete history of w up to t, and T _w the complete theory of chance for w. Then:

C(A|Ht,w  &  Tw & E) = P _t,w (A|Ht,w  &  Tw  &  E)⁴

Hall derives (NP) from a deference principle that considers chance to be an analyst-expert (i.e., someone to whom we are willing to defer in virtue of considering her to be considerably better than us in evaluating the relevance of the information she’s given), thus providing an independent conceptual motivation for (NP): it is (NP), rather than (PP), that adequately captures our understanding of chance, for it is grounded upon the epistemic role chance would actually have for us. The problem with (PP) would be, then, that it treats chance as the wrong kind of expert (namely, as what Hall calls a database-expert, i.e., someone to which we are willing to defer in virtue of considering her to be substantially more informed than us).⁵

(NP) has some advantages over (PP). The most obvious one is that it is consistent with the existence of undermining futures, whereas (PP) is not, for even under the assumption that P _t,w (G'w) ≠ 0, by (NP) C(G'w|Ht,w  &  Tw) = P _t,w (G'w|Ht,w  &  Tw) = 0. Also, not only is it equivalent to (PP_S) under the assumption that there are no undermining futures, but it yields (PP) as a good approximation in garden-variety cases under the assumption that there are such futures, for in those circumstances P _t,w (A|T _w )≈ P _t,w (A)⁶. So (NP) seems to be, in a sense, a deeper principle than (PP_S), for it corrects it, it yields (PP_S) as a good approximation in ordinary cases, and it explains why it fails in the extraordinary ones. Moreover, it doesn’t depend, for its formulation, upon the notion of admissibility, which is not easy to characterize. And finally, Hall argues that (NP) correctly captures the purely epistemic role of chance.

But (NP) is not the only possible replacement. ^{Ismael (2008)} distinguishes between reasoning within a theory of chance and reasoning about which theory of chance is correct. When we reason in the first way, the rational thing to do is to equate our credence to the chances dictated by the theory. Thus, the correct deference principle for this case is a simplified version of the Principal Principle:

The Unconditional Principal Principle (UPP). Let C _t be the credence function of an agent at time t, P _t,w (·) the chance distribution at time t and world w, and H _t,w the history of w up to t. Then:

C _t (A|H _t,w ) = P _t,w (A)

We may tidy up this formulation a bit by dropping the time subscript for C and adopting talk of initial credence functions. Basically, what we get, restricting the evidence to H _t,w , is C(A|H _t,w ) = P _t,w (A), where C is any reasonable initial credence function and H _t,w is the agent’s total evidence. If we adopt the ur-chance notation, we may bring to the surface the parallelism between credence and chance, for (UPP) amounts to the claim that C(A|H _t,w ) = urch _w (A|H _t,w ). Importantly, Ismael’s contention is that (UPP), rather than (NP) or (PP), is the principle that adequately captures the epistemic role of chance: if we are taking a theory of chance T _w as an epistemic policy rather than reasoning about its correctness, then our credence simply should follow chance. Now, given that we don’t know which of the possible theories of chance is the correct one, our epistemic situation is not one of certainty concerning chance, and (UPP) is not much of a guide for us. Then, the deference principle we should use in attempting to adequate credence to chance, given our epistemic situation, is:

The General Recipe (GR). Let C be the credence function of an agent, CH tPt,w the proposition that the chances (at t at w) are given by P _t,w , and P _t,w (·) any possible chance distribution at t at w. Then:

C(A) = ∑Pt,w C(CH tPt,w)P _t,w (A)⁷

(UPP) and (GR) also present advantages with respect to (PP). Neither of them entails a contradiction under the assumption that there are undermining futures. The former entails that C(G'w| H _t,w ) ≠ 0, given that Pt,w (G'w) ≠ 0, which is alright for certainly C(G'w| H _t,w ) ought to be greater than zero. The latter entails that C(G'w) ≠ 0, which is also alright for we are not sure which theory of chance is true, and as long as we give some positive credence to a theory of chance that assigns positive chance to G'w (and we should, for G'w is entailed by a consistent theory of chance given H _t,w ), C(G'w) ought to be greater than zero. (UPP) and (PP) are not, in general, equivalent. However, if we are certain with respect to the correct theory of chance, then (PP) yields (UPP), for C(A|Tw  &  Ht,w) simplifies to C(A|H _t,w ). Moreover, (PP) entails (GR), even though the converse entailment doesn’t hold: (GR) is just (PP)’s recommendation for setting the unconditional degree of belief in A under uncertainty concerning chances (slightly re-written).

As it turns out, (NP) is incompatible with (UPP) and (GR), if we allow for undermining futures, for they entail different ways of determining unconditional credence. According to (NP), under certainty concerning the correct theory of chance, the credence in A should be P _t,w (A|CH tPt,w), whereas both (UPP) and (GR) yield P _t,w (A); and under uncertainty concerning which theory of chance is the correct one, the degree of unconditional credence in A should be ∑Pt,w C(CH tPt,w)P _t,w (A|CH tPt,w) according to (NP), whereas, according to (GR), once we introduce uncertainty concerning history, it should be ∑Pt,w C(CH tPt,w)P _t,w (A) and, if we admit self-undermining chances, P _t,w (A|CHtPt,w) and P _t,w (A) yield, in general, different values. Hence, the reductionist has more than one recommendation to deal with, in facing the motivation problem. (If we do not allow undermining, then the recommendations are the same, for P _t,w (A|CH tPt,w) = P _t,w (A). So, the non-reductionist has only one recommendation to deal with.)

4.1. A Formal Jusatification of Credence-Chance Principles

^{Pettigrew (2012)} seems to provide the beginning of a solution to this problem. The general strategy is to provide a formal justification of any of the credence-chance principles proposed, depending on the particular assumptions we make. The central idea is that, for each possible world w ∈ 𝑊, there is a credence function v _w that is vindicated at w (i.e., that is perfect or exactly correct at w). It’s possible to measure the distance of any credence function b∈ B (with B the set of all credence functions definable over a given algebra of propositions F) from being vindicated at w, for any w, in terms of its inaccuracy measure, that is, in terms of its distance at w with respect to v _w . Define V as {v _w : w ∈ W}, and define the convex hull of V as the smallest set V ⁺ which contains every vindicated credence function in V and, for any two credence functions which belong to V ⁺ , all of their mixtures (where the mixtures of two credence functions b and b’ are all the credence functions λ b + (1 - λ) b’, for every λ ∈ [0,1]). Then, provided the inaccuracy measure satisfies certain constraints, it is possible to establish the following result:

Joyce’s Theorem. Let B be the set of all probability functions definable over a (finite) algebra of propositions F, and I: B x W → R an inaccuracy measure. Then, for any b ∈ B:

1. b ∈ B & b ∉ V ⁺ → ∃b'[b' ∈ V+  &  ∀w (I(b',w) < I(b,w))]

2. b ∈ V ⁺ → ¬∃b'[b' ∈ B  &  ∀w (I(b',w) ≤ I(b,w))]

That is, an agent whose credence function is not in V ⁺ will do better (epistemically speaking), no matter how the world turns out to be, if she moves to a credence function in V ⁺ (i.e., her credence function will be closer to vindication in every world), and an agent whose credence function is in V ⁺ will do worse (epistemically speaking) in at least one world if she moves to another credence function (i.e., her credence function will be further from vindication in at least one world).

This theorem provides a formal justification for the following norm schema:

(GN)An agent ought to have a credence function in V⁺.

However, (GN) won’t be of any use until the identity of V ⁺ is fixed, and this won’t happen until the identity of v _w is fixed. There are a few natural candidates for being the vindicated credence function at a world. On the one hand, we may suppose that the vindicated credence function at a world w is the “truth” credence function for w, that is, the credence function that assigns 1 to all propositions true at w, and 0 to all propositions false at w. On the other hand, we may consider the credence function vindicated at w to be the ur-chance function for w, or something close to it. Then, it is possible to prove the following results:

These results yield four specifications of (GN):

(What about satisfying (Prob) in (PP-N) - (GR-N)? Since we’ve assumed that chance is a probability function, any credence function satisfying (PP), (NP) or (GR) will also satisfy (Prob).)¹⁰ If the non-reductionist can show that v _w (·) = urch _w (·), then she can show that it is rational to follow the credence-chance principle she favors. (Note that she need not decide between urch _w (·) and urch _w (|CHurchw): since, for the non-reductionist, there need not be self-undermining chances, she may take it that urch _w (·) = urch _w (·| CHurchw).) More precisely, if she assumes (as she most certainly will) that there are no self-undermining chances and she identifies v _w (·) with urch _w (·), then we have a formal justification of (PP): it is rational for an agent to follow (PP) because, if she does otherwise, she will risk being in a worse epistemic situation. Moreover, since under the assumption that there are no undermining futures, (PP), (NP) and (GR) are equivalent, this amounts to a formal justification of all the credence-chance principles so far proposed. Note that if we allow self-undermining chances, we lose the possibility of justifying (PP) (as it should be), and which principle we end up justifying depends on whether we consider urch _w (·) or urch _w ( ·| CHurchw) to be the credence function vindicated at w. So, this strategy is open to the reductionist also (we’ll come back to this in 4.3).

4.2 Vindicating Chance

To solve the explanation problem, the non-reductionist needs to show that urch _w (·) is the credence function vindicated at w. So, the question for her is: Why not truth? That is, why not v _w (·) = truth _w (·)? Let’s start by noticing that we have two different conceptions of indeterminism. One is epistemic, and consists in the impossibility of knowing all the truths about the future (relative to a time t) of a world w from the complete knowledge of the laws of nature that hold at w and the complete knowledge about the entire history of w up to t. The other is metaphysical, and considers all possible futures (relative to a time t) as “real” or “objective” alternatives (^{Belnap et al. 2001}, chap. 6), rather than as epistemic possibilities arising from the unknowability of future history. There seems to be little point, if any, in discussing which of these conceptions reflects “true” indeterminism, or which of them better captures our ordinary conception of an indeterministic world. What’s important for our purposes is the constraints each of these conceptions place on the structure of history.

As ^{Thomason (1970)}, ^{Belnap et al. (2001)}, and ^{MacFarlane (2003)}, among others, have variously argued, a metaphysically indeterministic conception of history calls for a forward-branching time structure without a designated history. The picture is well known, but it is worth reviewing. Time in a metaphysically indeterministic world may be represented as a tree:

where the graph is oriented from left to right. In the graph, t ₀ represents the present; the absence of branching to the left of t ₀ encodes the assumption that what’s past is no longer chancy; h ₁ , h ₂ and h ₂ are possible histories with respect to t ₀ , that is, possible ways the history after t ₀ might go; the absence of a designated history represents the view that no history is “ontologically privileged”, that none is “actually going to take place”.

The non-reductionist may endorse a metaphysical understanding of indeterminacy, and I think she should, for two reasons. First, for the possibility of justifying the assumption that v _w (·) = urch _w (·), as we’ll shortly see. Second, because it is more congenial with primitivism about chance. This will become clearer when we move to considering how reductionism may address the explanation problem. For the moment, just notice that the non-reductionist doesn’t rely on the existence of a metaphysically privileged future to make sense of her conception of chance, so she’s not constrained to posit such a thing. For her, the global pattern of outcomes and frequencies need not be given before chances can be given her direction of explanation is the other way around: first she recognizes chances as primitive features of the world, and then she considers any pattern of outcomes and frequencies that may eventuate to be a consequence of a real, objective openness of future history.

From this non-reductionist point of view, truth may strike us as the vindicated credence function for a world only if we adopt an external perspective on that world: we imagine the world as if it were in front of us, in its entirety, so to say, and from that vantage point we ask which credence function is the correct one for that world. And we adopt a “wait and see” attitude: the correct credence function is the one that assigns 1 to any proposition that turns out true at w, and 0 to any proposition that turns out false at w. But things look importantly different when we adopt the internal perspective. From this perspective, the metaphysically indeterministic nature of future history gives rise to semantically defective discourse: unless a given aspect of the future is settled at t, that is, unless it is determined at t that that aspect will come about, any proposition describing that aspect of the future lacks truth value at t. Consequently, at that time there is no fact of the matter as to which credence function “gets it right” in terms of truth. True: we can always, in making retrospective assessments, claim that one credence function “got it right”, as it were. We can even say that the credence function that corresponds to truth at w is the one that gets everything right throughout the whole history of w. But this doesn’t mean that it was the credence function that was right at that moment. It is this possibility of making retrospective assessments what strikes me as generating the impression that truth is the vindicated credence function in a metaphysically indeterministic world. But once we look closer at what happens with truth values in a metaphysically indeterministic world w, it turns out that the chance function for w is better suited to play the role of the credence function vindicated at w.

How does retrospective assessment work? Suppose S ₁ asserts at t ₀ a certain proposition A (say, that there will be a sea-battle within the day, that the toss of the coin will come up heads), whereas S ₂ asserts, also at t ₀ , the negation of A. Basically, we ascertain the correctness or the incorrectness of the assertions by waiting and seeing how history actually goes: if history continues in a way that makes A true (say, h ₁ ), then we say that S ₁ was right after all, that what she said was true. We may even say things such as that S ₁ was correct, or spoke truly, because she said of history that it was going to continue the way it actually did. If history continues in a way that makes A false (say, h ₂ or h ₃ ), then we say that S ₁ was mistaken, or that what she said was false. Again, we may even say that S ₁ spoke falsely because she said of history that it was going to continue in a way it actually didn’t. (The same goes, mutatis mutandis, for S ₂ ’s assertion.) That is, in order to assess an assertion about the indeterministic future, we stand from a time after the assertion is made, a time at which the events that yield a definite truth value for the asserted proposition either definitely occurred or definitely did not, and then, from that vantage point, we determine whether the assertion was true or false.

The potential non sequitur comes when we slide from these formulations to saying that S ₁ was right, or spoke truly, because she said of history that it was going to continue in the way it was actually going to continue. Note here the change in the verb for there is a natural yet inadmissible move from talk of the (possible) future that actually came to pass, to talk of the (possible) future that, at the time of the assertion, was actually going to take place. It is this last formulation that is inadmissible: if the world is metaphysically indeterministic, then it is not settled, at t ₀ , which of all the possible continuations of history is going to take place. Hence, at t ₀ , there is no unique way in which the history after t ₀ is actually going to develop. So, even though we may say, from the vantage point of the way history actually developed, that S ₁ was correct or incorrect, this doesn’t imply that there was, at t ₀ , a fact of the matter as to which prediction was the correct one.

Now, to drive the point home, consider all super-asserters for a world w at a given time t. A super-asserter at t for w asserts two very complex propositions: the proposition H _t,w , and a proposition F _t,w that completely describes a possible continuation of w’s history after t, that is, a proposition completely describing some possible future of w. (We leave aside assertions of propositions not about the history of w, even though they are part of what is true or false at a world, for the complication adds nothing of importance.) Let’s suppose, for simplicity, that, for each complete proposition about the future of w after t, there is exactly one super-asserter that asserts it, and let’s stipulate that, if two propositions F _t,w are necessarily equivalent, they have the same super-asserter. Then, super-asserters and possible futures are in one-to-one correspondence. Obviously, super-asserters represent all the possible truth-value assignments to propositions about the entire history of w at a given time. For entirely historical propositions, there is only one such assignment. For propositions about the future, each possible distribution of truth values is represented by one super-asserter.

We may see the unfolding of w’s history as a way of discarding super-asserters. Each time a new event takes place, one or more super-asserters are discarded, namely the ones that made assertions incompatible with the way history actually continued. The super-asserters are reduced to one only when the history of w gets closed, that is, only when there are no more chancy events in the history of w. This may happen because the history of w comes to its end, or because it reaches a time t _c at which all the events after t _c are already necessitated by the state of the world at t _c plus the laws that hold at w. It doesn’t matter for our purposes which one it is. All that matters is that we can say which super-asserter “got everything right” only from the vantage point of this “end of history”, not a moment before. Crucially then, we cannot say, at any given time before the history of w gets closed, that there is a super-asserter that got it right already, but we do not and cannot know which: this would be endorsing some kind of “thin red line” view on the future that is incompatible with a metaphysically indeterministic understanding of the world. At any given time before history gets closed, there is no actual way history is going to develop. Consequently, there is no super-asserter that actually got it right.

Obviously, super-asserters are just literary devices for representing all the candidate “truth” credence functions for w at any given time t. They are useful devices because what happens to super-asserters happens to the candidate credence functions for truth in a metaphysically indeterministic world: it is not settled which credence function is the “truth” credence function for w until the history of w gets closed, that is, until there are no more chancy events. This is so because there is no actual way history is going to develop at any time prior to the time at which history gets closed. It’s not that propositions about the indeterministic future have a determinate truth value that is inaccessible to us; it’s not that they have a truth value, but it is indeterminate which one it is: rather, they have no truth value at all their truth or falsity is not determined yet.¹¹ Hence, there is no fact of the matter as to which candidate for being the “truth” credence function for w is the correct one. In a sense, then, no one is: reality lacks the determinateness that is required for it to select one of them as the credence function that reflects truth for w. But this is so, crucially, because what is truth for w itself is underdetermined by reality.

This suggests that the vindicated credence function at w cannot be the credence function that corresponds to truth at w, for there is no such unique credence function rather, we have quite a good deal of credence functions that compete for being the truth credence function for w, none of which enjoys any metaphysical or epistemological advantage over the others. The chance function for w at t (for any t), on the other hand, is perfectly well defined at t itself (provided the chance distribution at t is well defined). This makes it the perfect candidate for being the credence function vindicated at w for, at any given time, there is a perfectly determinate sense in which the chance function “gets it right”: it is exactly correct in what it says about chances at w.

4.3.Unvindicating Reductionism

So, non-reductionists are able to meet their share of the explanation problem. What about the reductionist? Suppose she successfully answers the problems identified by ^{Briggs (2009)} and summarized in section 3, or suppose she accepts them as a cost she has to live with. Can she successfully meet her share of the explanation problem? The strategy outlined in 4.1 is open to her, provided she can show that either urch _w (·) or urch _w (·| CHurchw) is the vindicated credence function at w, rather than truth _w (·). So, if she can show that truth _w (·) is not the vindicated credence function at w, she’ll have a formal justification of the credence-chance norm she favors (provided she takes the extra step of adjudicating vindication between urch _w (·) and urch _w (·| CHurchw)). I want to suggest that she won’t fare well at this task.

Reductionists seem to be committed to an epistemic understanding of indeterminism. After all, the way reductionists explain chance requires that there be an ontologically privileged history, a unique way history is actually going to develop, for there to be a unique global pattern of outcomes and frequencies. The direction of explanation is the inverse with respect to non-reductionism: first the pattern, then the facts about chance. That is, the pattern has to be already given, so that chance may supervene upon it. So it seems that the epistemic construal of indeterminacy is the view she has to endorse.

Let’s start by noticing that epistemic indeterminism is compatible with two different time structures: a linearly ordered history and a forward-branching time structure with a designated history (informally, the history that will actually take place)¹². These structures allow for a perfectly well-defined truth distribution at any given point of history. In the case of a linearly-ordered time structure, sentences (propositions) are evaluated with respect to the only possible continuation of history, thus being true or false at any time within that history. In the case of a forward-branching time structure with a designated history, matters are a bit more complex but essentially the same: at any given time t, sentences (propositions) not involving operators such as “It is settled that” and “It is possible that” will be evaluated with respect to the designated history, while sentences (propositions) involving those operators will be evaluated with respect to every history passing through t, to prevent collapse of operators.¹³ In any event, the alternative, non-designated histories in this framework have the status of epistemic possibilities, which are not ontologically equivalent to the designated one. So, in either case, facts about chance are settled by the actual history of the world, be it because there is only one possible history, be it because there is an ontologically privileged one.

Hence, at any given time t within the history of w there is a fact of the matter as to which, of all the truth distributions compatible with the history of w up to t, is the truth distribution that matches the complete history of w. Now, if the complete truth distribution for w is well-defined, then it seems that truth _w (·), and not urch _w (·) or urch _w (|CHurchw), is the credence function vindicated at w. For truth _w (·) contains all the information chance has, and more besides with respect to the past, relative to any time t, it agrees entirely with chance; with respect to facts about chance at t, it states all of them by assigning a truth value to statements reflecting those facts; and with respect to outcomes concerning the future relative to t, it gives more information than chance does. So, if one is exactly correct with respect to the complete history of w, it seems to be truth.

This invites the question: Can the reductionist adopt a forward-branching time structure without a designated history as her conception of time, so that truth be undefined? If at all compatible with her understanding of indeterminacy and chance, the cost of doing so would be excessive. For one thing, absent any privileged history, there is no unique, privileged global distribution of categorical properties for chance to supervene on. Also, as we saw in 4.2, such a structure gives rise to semantically defective discourse: certain propositions will lack truth value at certain points in history. The problem is that not only statements about outcomes will be semantically defective in this sense, but also statements about the chances of outcomes will turn out to be so. This is, at least, one lesson to draw from the existence of undermining futures in a forward-branching framework without a designated history. For statements about chances will receive their truth values in the same way other statements do: by supervaluating over possible histories. If, following the reductionist, we posit that chances are relative to complete histories, and we allow histories that assign different chances to the same events, then, relative to different histories, statements of the general form ch _t (A) = x will have different truth values. This means that these statements will, in general, turn out to be undefined also.¹⁴

There are a couple of further moves the reductionist may attempt. First, she might remark that, under uncertainty concerning which outcome will come about, it is only reasonable to guide oneself by chance, not by truth. Second, she may reinforce this by saying that, since truth is in principle inaccessible to us (for complete knowledge of the laws of nature and complete knowledge of the entire history up to any given time do not yield complete knowledge about the future with respect to that time), it cannot be the credence function vindicated at w hence, it must be chance (or something close to it). She should resist the temptation of giving “reasonable” a prudential reading: it should not be claimed that it is reasonable in a practical sense to do so, for the reasonableness or rationality involved in the explanation problem is epistemic. Her best hope seems to be to argue that, due to our less-than-ideal epistemic situation, agreement with chance is all we get, and that this is enough for the vindication of chance. However, such considerations seem somewhat extraneous to the concept of vindication, and we are owed a fuller justification. So, until such a justification is offered, the dialectical advantage seems to be on the non-reductionist’s side.