1. The Factivity Challenge

According to Moss, credences and SPJs are full beliefs in complex probabilistic contents (as opposed to complex attitudes with simple propositional contents).Footnote ¹ Probabilistic contents, in turn, are conceived as sets of probability spaces. Formally speaking, a probability space is an ordered triple, $s =\langle \Omega _s, \;\;{\cal F}_s, \;\;\mu _s\rangle$, containing a domain of possibilities, Ω_s, constituted by a non-empty set of possible worlds, an event space, ${\cal F}_s$, which is a set of subsets of Ω_s (and thereby a set of propositions), and a probability function, μ _s, that assigns each proposition in ${\cal F}_s$ a value between 0 and 1.Footnote ² To provide a quick example, the content that Jones probably smokes is representable as a set of probability spaces over a domain of possibilities containing two worlds: one where Jones smokes and one where she does not. The set contains each probability space that assigns a value above 0.5 (including 1) to the proposition that Jones smokes. All other possible probability spaces are excluded from the set.Footnote ³

Conceiving of probabilistic beliefs as full beliefs in probabilistic contents leads Moss (Reference Moss2018b: 86) to conclude that for a probabilistic belief to constitute knowledge means “just the same as whatever it traditionally means to say that beliefs constitute knowledge.” While it seems unproblematic to hold that credences and SPJs can be rational and/or justified, the truth-aptness of probabilistic beliefs is often questioned.Footnote ⁴ Knowledge is factive. If we have knowledge of any content ψ, it follows that ψ is true. A proponent of Mossean probabilistic knowledge has to show that probabilistic contents are truth-apt and, therefore, provide a theory of probabilistic truth. The factivity challenge for probabilistic knowledge consists in the obstacle of providing a plausible account of probabilistic truth.

In Probabilistic Knowledge Moss introduces a deflationary solution to the factivity challenge according to which a probabilistic content 〈ψ〉 is true iff ψ.Footnote ⁵ While deflationism formally solves the factivity challenge, it is frequently criticized for leaving probabilistic truth, and therefore probabilistic knowledge, a concept way too opaque to comfortably operate with.Footnote ⁶ I hold that this opaqueness is not the most pressing issue for the deflationary approach. Above all, Moss's deflationism leaves her ill-equipped to properly handle probabilistic AIs and thereby unable to defend the concept of probabilistic knowledge against skepticism.

2. The Challenge from Probabilistic Arguments from Ignorance

Without further ado consider the following standard lottery case:

Intuitively, we would like to judge that Thilo knows that his ticket has probably lost.Footnote ⁷ Let us assume, for the moment, that it is indeed the case that his ticket has probably lost. This suffices to satisfy Moss's deflationary definition of probabilistic truth. However, the claim that Thilo knows that his ticket probably lost is challenged by the following probabilistic argument from ignorance:

Let us consider each premise individually. The epistemic principle of K-closure underlying AI1 is prima facie plausible.Footnote ⁸ It seems that if I know that p, and that p entails q, I am, at least, in a position to know q. The content of ‘Thilo has probably lost' (ψ) and the content of ‘he has certainly won' (φ) are inconsistent. If Thilo has certainly won, it is false that he has probably lost.Footnote ⁹ Hence, if Thilo knows that he has probably lost (and recognizes the above inconsistency) he should be able to know that it is false that he has certainly won.

This takes us directly to AI2 according to which Thilo cannot not know that it is not the case that his ticket has certainly won. Following Moss, I call the content that Thilo has certainly won a nominally probabilistic content. The content of ‘Thilo has certainly won' is a set of probability spaces where each space has only worlds in its domain in which Thilo actually wins. The content of ‘Thilo probably lost' is different. It is thoroughly probabilistic, for it is represented by a set of probability spaces some of which do have worlds in their domain in which Thilo actually wins.Footnote ¹⁰ On Moss's semantics nominally probabilistic contents are type-shifted versions of simple propositional contents. This semantic approach is motivated by the fact that we often infer full beliefs from credences and vice versa. While Moss's account captures the logical connections between simple and probabilistic contents, disjunctive theories fail to account for the validity of arguments featuring both kinds of contents.Footnote ¹¹ This motivates a one-to-one correspondence between propositions and nominally probabilistic contents. Believing that p is then, strictly speaking, nothing more than having the nominally probabilistic belief that certainly p.Footnote ¹²

With this in mind we should be able to see why AI2 is hard to deny. It is standardly assumed that in lottery cases we are not in a position to know that it is not the case that we have (certainly) won, even though the chances of (certainly) winning are extremely low. It is widely accepted that having merely statistical evidence for p does not suffice for us to come to know p.Footnote ¹³ This explains why people regularly participate in lotteries with low chances of winning. After all, you never know.

AI1 is plausible on its own merits and AI2 is hard to deny on Moss's semantics. Moss indeed accepts both premises. If we do so, AI3 follows directly. Since probabilistic arguments from ignorance can be construed for a wide range of our credences and SPJs, namely against all probabilistic beliefs that are held on the grounds of merely statistical evidence, they call into question the possibility of probabilistic knowledge in a wide variety of cases. This seems false. If probabilistic knowledge, as Moss presents it, is supposed to be a plausible and fruitful concept for epistemologists, her account should allow for probabilistic knowledge in cases where we expect it the most: lotteries, coin-tosses, and, more broadly, in cases where our evidence does not suffice to justify full beliefs, but arguably credences.

Moss indeed takes the challenge from probabilistic AIs seriously and proposes a solution that seemingly allows her to salvage probabilistic knowledge without having to deny any of the premises A1–A3. Adopting relevant alternatives contextualism (RAC) for (probabilistic) knowledge, she argues that probabilistic AIs are such that they create contexts in which challenging nominally probabilistic contents become relevant. Since the RAC theorist holds that a subject S knows p iff S truly believes p on the basis of evidence that eliminates all relevant alternatives to p Footnote ¹⁴, in the context of the probabilistic AI Thilo will indeed lack probabilistic knowledge. However, this does not imply that Thilo could never know that his ticket probably lost. Moss holds that at most ordinary contexts nominally probabilistic alternatives are not relevant and therefore need not be ruled out in order to arrive at probabilistic knowledge.Footnote ¹⁵

However, issues are lurking. Moss seems to be convinced that by raising a nominally probabilistic alternative against a thoroughly probabilistic belief we create contexts of a similar kind as by raising brain-in-a-vat-like (BIV-like) alternatives to ordinary full beliefs. But, intuitively, nominally probabilistic contents – like the one that Thilo's ticket has certainly won – are not as aberrant as the skeptical alternatives raised in standard AIs.

Proponents of RAC provide different principles for determining whether an alternative counts as relevant at a given context. Moss explicitly accepts the following criterion by Lewis (Reference Lewis1996: 559):

ATT states that we cannot properly ignore an alternative if we actively consider the alternative. Only because a subject is ignorant of a possibility it does not follow that she is in a position to properly ignore it. In addition to ATT we are, therefore, in need of further criteria to determine when alternatives are relevant. Consider two further rules given by Lewis (Reference Lewis1996: 554, 556):

ACT and RES in combination yield that if, at context C, a true (and thereby relevant) alternative saliently resembles another possibility, this possibility will be relevant no matter whether it is ignored or false at C. Of course these rules only give us a vague guidance as to when alternatives are relevant. Nonetheless, together with the above observation that the nominally probabilistic alternatives in probabilistic AIs are way less aberrant than the alternatives raised by standard skeptical AIs they already suffice to uncover some pressing issues for Moss's RAC approach.

Let us focus on ATT first. It seems that while a proponent of RAC for simple knowledge claims can properly claim that in most ordinary contexts (outside the philosophy classroom) I do not pay attention to the possibility of being a BIV, the same does not obviously hold for nominally probabilistic alternatives that are raised against thoroughly probabilistic beliefs in probabilistic AIs. Consider again Lottery. Thilo knows how lotteries work. He is aware that winning is a possibility. Hence, he also is likely to be aware that it is possible that his ticket has (certainly) won instead of probably lost. This indicates that nominally probabilistic alternatives of the above kind are ordinarily and frequently relevant and, therefore, very different from other skeptical possibilities. If it is true that we ordinarily do pay attention to nominally probabilistic alternatives, we will frequently fail to have knowledge of the most ordinary probabilistic contents.Footnote ¹⁶

Despite the above worry, let us assume that there are contexts where Thilo is not aware of the possibility that his ticket has (certainly) won. Let us further assume that Thilo believes that his ticket has probably lost. According to Moss, Thilo will be able to know this. He is in a position to rule out other relevant thoroughly probabilistic alternatives (e.g. that it is exactly .4 likely that his ticket has lost) on his evidence and he does not need to rule out the possibility of (certainly) winning because it is not relevant at the context of his belief. But now imagine that Thilo transforms his initial probabilistic belief into a slightly stronger one, namely the belief that his ticket has certainly lost. We can imagine that nothing further changes. Thilo continues to be ignorant of the possibility of his ticket being the winner. It seems that we would have to conclude that Thilo knows that he has certainly lost. But this is obviously false. In fact, ACT in combination with RES explain this verdict. Suppose that ticket #37 wins. According to ACT the possibility of ticket #37 winning is thus relevant. Even if Thilo is not the holder of ticket #37, RES tells us that, since the possibility of Thilo's ticket winning saliently resembles the relevant alternative of ticket #37 winning, Thilo's ticket winning is a relevant alternative as well. Thilo cannot rule out that his ticket has (certainly) won on his evidence; therefore he cannot know that his ticket has (certainly) lost.Footnote ¹⁷

If the nominally probabilistic alternative is relevant for Thilo according to ACT and RES at the ordinary context of Thilo's nominally probabilistic belief, why should it not be relevant at the ordinary context of Thilo's previously held thoroughly probabilistic belief? This is what I call the asymmetry challenge. The proponent of RAC as a solution to probabilistic AIs has to give an explanation as to why switching between thoroughly and corresponding nominally probabilistic beliefs has an impact on context such that in the latter case nominally probabilistic alternatives are naturally relevant while in the former case they are not. As long as such an asymmetry cannot be made plausible, Moss faces a dilemma. Either she has to conclude that we are sometimes able to know on the basis of merely statistical evidence that our lottery ticket has (certainly) lost, or we have to admit that whenever it is impossible for us to know a proposition p on the basis of merely statistical evidence, it will also be impossible for us to have probabilistic knowledge of p (e.g. knowledge that probably p).

To be sure, I am not denying that changing the content of one's belief can invoke shifts in context that could make certain previously irrelevant alternatives relevant. Assume, for instance, that Thilo fully believes that he has hands at an ordinary context where he does not consider any skeptical hypothesis. Now imagine that Thilo moves from believing that he has hands to believing that he is not a BIV. Obviously, this change of belief also invokes a broader contextual shift. By believing that he is not a BIV Thilo considers the possibility of being a BIV. He does not hold this belief at an ordinary but rather at a skeptical context. However, this mechanism does not obviously apply for the change from believing a thoroughly to believing a corresponding nominally probabilistic content. Remember that in order for RAC to deliver a proper defense of probabilistic knowledge against skepticism, we have to assume that we can (at least sometimes) properly ignore challenging nominally probabilistic contents. It is not clear at all why Thilo should be able to properly ignore the possibility of his ticket winning while he believes that it has probably lost, but cease to be able to do so as soon as he starts believing that his ticket has (certainly) lost.

Though Moss does not directly discuss the asymmetry challenge she defends the following rule

We can charitably assume that if GAR is true, it will also turn out to be true that nominally probabilistic contents are more challenging for other nominally probabilistic contents than for thoroughly probabilistic ones. Anyways, I hold that Moss's arguments for GAR fail. Moss sketches two possible reasons to assume that GAR is correct. Let us consider each of them in turn.

The first reason Moss cites in favor of GAR rests upon the idea that certain nominally probabilistic contents are ‘routinely present’ when we hold thoroughly probabilistic beliefs. When Thilo believes that his ticket probably lost, he necessarily believes that his ticket might be the winning ticket. According to Moss, this makes the alternative that his ticket has (certainly) won routinely present for him and thereby less challenging for his probabilistic belief. Other thoroughly probabilistic alternatives are not routinely present for Thilo and appear more challenging for this reason. Extending this reasoning, we could argue that when Thilo believes that his ticket (certainly) lost he does not believe that his ticket might have won. His belief is thereby more easily challenged by the alternative of (certainly) winning.

I hold, however, that this approach is flawed. First of all, note that Moss's introduction of challenging alternatives is infelicitous. It masks what kind of asymmetry GAR really has to account for. As I see it, some content ψ is challenging for some other content φ iff ψ is inconsistent with φ and relevant at the context of φ's use. Furthermore, if ψ is inconsistent with φ, ψ, by definition, constitutes an alternative to φ. In this light, the possibilities introduced by probabilistic AIs are certainly alternatives. Whether they challenge the content in question does then wholly depend on whether they are ordinarily relevant. This is why the first argument for GAR fails. It is just hard to see how something like the routine presence of a nominally probabilistic alternative at a context could make the alternative less relevant at this same context. Moss endorses ATT. If a possibility is routinely present for us it should be safe to say that we are routinely aware of it. With ATT it follows that the same possibility is routinely relevant for us. In this light, Moss's observation of nominally probabilistic alternatives being routinely present relative to our thoroughly probabilistic beliefs should leave us even more concerned about the possibility of acquiring and maintaining probabilistic knowledge rather than placate the worries raised by probabilistic AIs.

Moss offers a second argument for GAR. Following Weatherson (Reference Weatherson2011) she argues that there is a difference between AIs that raise merely abstract possibilities and those that make us aware of more ordinary possibilities that we have not ruled out yet, such that the former arguments appear to be less effective than the latter. Moss holds that raising nominally probabilistic alternatives against thoroughly probabilistic beliefs is a merely abstract move. Nominally probabilistic contents are therefore less challenging relative to thoroughly probabilistic beliefs. But this reasoning presupposes the required asymmetry without explaining it. Why are nominally probabilistic alternatives merely abstract possibilities relative to thoroughly probabilistic contents? As we have seen, the possibility of Thilo's ticket (certainly) winning does not appear to be skeptical, far-off, or abstract in any way similar to a BIV-hypothesis. Furthermore, the possibility of winning surely does not constitute an abstract alternative relative to the belief that Thilo (certainly) lost. So why should it be abstract in the context of believing that Thilo's ticket probably lost? We are returned to the initial asymmetry challenge.

I conclude that GAR is implausible. There are no obvious relevant contextual differences invoked by transitioning from believing a thoroughly probabilistic to believing a corresponding nominally probabilistic content. If the possibility of winning is relevant at the context of Thilo's belief that his ticket has (certainly) lost, it should also be relevant at the context of his belief that his ticket probably lost. ATT, as well as ACT and RES do not suggest that we should find any asymmetry here. Nonetheless, I am convinced that we do not have to bite the bullet and conclude that we (almost) never are able to obtain probabilistic knowledge on merely statistical evidence. A plausible story can be told about how ordinary nominally probabilistic alternatives are always relevant and yet can be ruled out on purely statistical evidence relative to thoroughly probabilistic beliefs. Such a story demands an inflated account of probabilistic truth. The general idea is the following: by saying more about probabilistic truth we are revealing what our probabilistic beliefs aim at. An inflation of probabilistic truth will thus uncover important asymmetries and provide us with a better grip on the concept of probabilistic knowledge. In the following section I introduce such inflation and argue that it is compatible with Moss's overall semantic theory of probabilistic contents, beliefs, and knowledge claims. I then show how the account allows us to meet the challenge posed by probabilistic AIs without running into the asymmetry challenge.

3. Inflating Probabilistic Truth

An important step towards exacting the concept of probabilistic truth is provided by Rich's (Reference Rich2020) introduction of Kripke models for probabilistic knowledge and, consequently, probabilistic factivity. Put briefly, Rich's model M =  〈Ω, R, V ^P〉 is a tuple consisting of a set of possible worlds, Ω, a valuation function, V ^P, mapping atomic propositions onto sets of possible worlds, and an accessibility relation, R, specifying which probability spaces the agent does not rule out at each world ω ∈ Ω.Footnote ¹⁹

Let us define S (P) as the set of all probability spaces, s, for which the domain of possibilities is a subset of, or identical to, Ω as defined in M. Let ψ ⊆ S(P). Probabilistic truth is then modeled as follows:

$$( {M, \;{\rm \;}\psi } ) {\rm \models }T\varphi {\rm \;iff\;}( {M, \;{\rm \;}\psi } ) {\rm \;}{\rm \models }\varphi {\rm \;where\;}( {M, \;{\rm \;}\psi } ) {\rm \models }\varphi {\rm \;iff\;}\psi {\rm \;}\subseteq \varphi $$

Footnote ²⁰

According to probabilistic Kripke models, the factivity of a probabilistic content φ is dependent on a probabilistic reference point ψ. φ is true relative to reference point ψ iff ψ implies φ. ψ implies φ iff it is a subset of, or equivalent to, φ. Saying more about probabilistic truth, therefore, amounts to determining conditions for choosing reference points.

Additionally, the model allows us to identify a first asymmetry between factivity for propositional and probabilistic contents. According to standard Kripke models a propositional content p is true at a world ω iff ω ∈ V(p).Footnote ²¹ Since for Moss propositions are directly translatable into nominally probabilistic contents, it is natural to assume that nominally probabilistic contents are true relative to the probabilistic analog of a possible world. This analog is what Moss calls a divine probability space.Footnote ²² Divine probability spaces are probability spaces with only one world in their domain. The probabilistic equivalent of the actual world would thus be a probability space containing only the actual world. In this way there is still a pretty clear sense in which nominally probabilistic contents are true at worlds. Remember that the simple content that p can be displayed as a set of probability spaces, φ, according to which p is certain (i.e. true at every world of the domain of φ). Whether p is true at ω then depends on whether the probability space that has only ω in its domain is a subset of φ. This, in turn, boils down to the question whether ω is a member of the domain of φ which amounts to the standard Kripke interpretation of propositional factivity.

Thoroughly probabilistic contents do not correspond to simple propositions. If we were to treat them like nominally probabilistic contents and standardly assess them relative to divine probability spaces, they could only ever be trivially true. For instance, Thilo's belief that his ticket has probably lost could only be true if the world in which Thilo holds this belief would be such that Thilo's ticket indeed lost. As a further consequence, moderate credences could never be true and knowledge. Moss argues that we can know probabilistic contents as specific as ‘Jones is .6 likely to smoke.’ I hold that, quite intuitively, Thilo does not only know that his ticket has probably lost, but is also in a position to know that it is .999 likely that his ticket has lost.Footnote ²³

The question is whether we can find conditions to pick out reference points which allow for thoroughly probabilistic contents to be non-trivially true. Rich suggests three possible kinds of reference points that could play the desired role: objective chances, evidential probabilities, or unique rational credences. Furthermore she observes that the correct choice of reference point is likely to depend (in some way or other) on contextual features.Footnote ²⁴ With this in mind we can further specify the questions an inflated account of probabilistic truth for thoroughly probabilistic contents is supposed to answer:

1. 1. Which features at which context are relevant for picking out reference points?

2. 2. Of which kind are the reference points suggested by these features?

4. Evidential Probability Relativism and Probabilistic Gettier Cases

We might not want to stop here. Though EPR as an inflated account of probabilistic truth does well in defying probabilistic AIs and the asymmetry challenge, it blocks us from a natural assessment of probabilistic Gettier situations. To see this consider the following case:

Here is a dilemma for the relativist who picks evidential probabilities as reference points. It seems that Alice does not know that she probably lost, because it appears to be a matter of pure luck that Alice's lottery draw was indeed standard and not easy to win. If the latter would have been the case, intuitively, Alice's probabilistic belief would have been misled. The only way a proponent of EPR can account for this verdict, as far as I see, is this: she can argue that while relative to Alice's perspective she ends up with a true belief in the standard as well as the easy lottery scenario simply by following her total evidence concerning the lottery, relative to our perspective and our superior knowledge of the situation Alice ends up with a false belief if she follows the evidence from the letter in an easy lottery situation. That is, relative to our perspective Alice receives misleading evidence in the easy lottery scenario, where misleading evidence is defined as evidence that leads her to believe something false. Since, relative to our perspective, it appears to be a matter of pure luck that Alice was not misled; her true, justified probabilistic belief is Gettiered.

While this at first might seem like a good explanation of why we hold that Alice's belief does not constitute knowledge, it ultimately leads us to problematic conclusions concerning what counts as misleading evidence in connection with probabilistic beliefs. Revisit, once again, Lottery. Imagine that we already know that Thilo is the lucky winner, while Thilo is oblivious to this fact. For all he knows his ticket probably lost. Given the above explanation of why it is true that Alice could easily have been misled, we would have to conclude that Thilo in believing that his ticket probably lost is misled by his evidence. After all, he is led to believe something that is false relative to our perspective. But this seems wrong. Thilo has only correct, unambiguous information about the constitution and procedure of the lottery. I hold that what we would like to conclude in his case is that, though his belief is false, the evidence upon which he formed his high credence did not lead him astray either. Again, truth-relativism allows us to consider multiple perspectives. For some questions – like truth simpliciter – the assessor's perspective will be decisive, while for other questions – like whether a subject was misled by her own evidence – the perspective from the context of use will be key.

If we accept that Thilo is not misled by his statistical evidence, because relative to his perspective he, epistemically speaking, ends up in the correct place, the same has to hold for Alice. In the easy lottery scenario Alice is still receiving a letter which indicates that she participated in a standard lottery. Though this information is false in this case, relative to Alice's perspective her probabilistic beliefs are not led astray when they conform with her evidential probabilities. And her evidential probabilities indeed indicate that she has probably lost. The dilemma for the proponent of EPR now takes the following form: either she concludes that just as Alice is misled by the information in the letter in the easy lottery scenario, Thilo is misled by the statistical evidence when he has in fact won the lottery, or she concedes that neither Alice nor Thilo are misled in these cases. Both answers are not satisfactory.

The worry for EPR, then, is the following. Regarding the standard lottery case, it seems plausible that although we would consider Thilo's probabilistic belief to be false if he actually won, we would not go so far as to judge that Thilo was misled by the statistical evidence, since the lottery was indeed designed to make it very unlikely that he would win. This suggests that in assessing whether our probabilistic beliefs are led astray by our total evidence, our perspective prevails. EPR, however, does not leave any room for such a view of being misled by one's evidence. Under EPR, (rationally) following our total evidence can never lead us astray, no matter whether our evidence consists in obviously misleading information (as for Alice in the easy lottery scenario) or not (as for Thilo in the standard lottery case). EPR thereby blocks us from the intuitive explanation that Alice's probabilistic belief is Gettiered because her belief could easily have been misled.

The good news is that the problem does not lie within relativism itself, but only with EPRs choice of the type of reference points. We do not think that Thilo gets something right merely because he correctly follows his evidence. The same would be true for Alice, but in her case we believe that she easily could have been misled. The important difference between Thilo and Alice is that Thilo's statistical evidence properly functions as an indicator of the actual objective chances of his ticket losing before the draw, while Alice's evidence in the easy lottery scenario leads her away from what is in fact objectively likely. To accommodate for this difference, we need to provide a theory of probabilistic truth that centers on objective chances rather than evidential probabilities as reference points.

5. The Objective Chance Analog

A simple suggestion on how to conceive of probabilistic truth in terms of objective chance is given by Hájek (Reference Hájek2011) who presents the following analogy (from now on referred to as the objective chance analog or OCA): “[t]ruth is to belief, as agreement with objective chance is to degree of belief.” The claim is intuitively appealing. As Hájek points out, full beliefs are quite obviously governed by a norm of veracity. Imagine two agents, each entertaining a consistent set of beliefs. While one agent entertains mainly false beliefs, the other has mainly true ones. A natural verdict is that the latter agent is doing better, epistemically speaking. Something similar applies to our credences. For instance, we think that some weather forecasters are better than others without putting into question that the probabilistic beliefs that they communicate are coherent. Furthermore, adopting Hájek's simple approach puts us in a position to make out an important difference between Lottery and Fake Lottery Letters. As stated earlier, Thilo's evidence is not misleading concerning the objective chances of his ticket being a loser, while Alice's evidence could have easily indicated something false concerning the objective chances of her being a loser in the communal lottery. There are, however, obstacles to this approach.

The first arises from the fact that OCA in its rather loose formulation does not tell us enough about which objective chances are relevant for determining probabilistic truth. A natural clarification to the demand is the following: Credences (and other probabilistic beliefs) possess the probabilistic analog of truth iff they match (or are consistent with) the objective chances at the world and time which the belief in question is about. This response yields two major problems.

First, if probabilistic beliefs are true at a world in virtue of matching certain objective chances at this world, it is not clear what makes them different from simple propositional beliefs about objective chances. However, as Moss (Reference Moss2018b: 2) points out, probabilistic beliefs “are not full beliefs about objective chance facts.” Moss shows that in using probabilistic or epistemic vocabulary we frequently do express something different than mere beliefs about chances. The naive version of the objective chance analog is, in this sense, a variety of contextualism. It, therefore, falls prey to the same general objection against contextualist theories of probabilistic truth discussed in §3.1; it inadvisably conflates the two notions of probabilistic and full beliefs, leaving no room for any substantial differences.

Secondly, objective chances of past events are often assumed to be extreme. From this it follows that our probabilistic beliefs about the past could only ever be trivially true, while our moderate credences about the past could never be true at all. This is false. If credences are ever fit for constituting knowledge, Thilo's rational .999 credence that his ticket has lost should be a clear instance of a knowledge-apt probabilistic belief. But in believing that it is .999 likely that his ticket has lost, Thilo entertains a probabilistic belief about a past event. The draw already took place and his ticket was either chosen to be the winner or not.Footnote ³⁵ Nonetheless, Thilo's credence appears to be correct. A proper theory of probabilistic truth should allow for this correctness to reflect in the veracity of Thilo's belief.

In the next section I present a version of the objective chance analog for which both these issues do not arise. My proposal is simple. We should adopt a variant of truth-relativism which centers on objective chance rather than evidential probabilities. I call this kind of relativism objective chance relativism (OCR).

5.1. Objective Chance and Relative Truth

Remember that one of our initial observations was the following: Probabilistic claims are claims that include epistemic vocabulary, and epistemic vocabulary is sensitive to a subject's epistemic state or evidence. We further saw that it was advisable to determine the truth of probabilistic contents from the context of assessment rather than from the context of use. So far we have only considered the possibility of an assessor's, A's, total evidence determining a reference point that mirrors A's evidential probabilities. This is not the only viable option we have, though it might be the most obvious. I argue that objective chance reference points, similar to evidential probability reference points, are sensitive to A's total evidence in such a way that a change in A's total evidence potentially invokes shifts from one objective chance reference point to another.

To arrive at this conclusion it is important to notice that objective chances are indexed to worlds and times. An event can have multiple objective chances at different worlds and, most importantly, different times. Intuitively, in Lottery Thilo can know that his ticket has probably lost even after the draw has already taken place. Though the objective chances of Thilo having lost are either 1 or 0 after the draw, Thilo's belief appears to be non-trivially true. One immediate conjecture as to how this result could be realized on a version of OCA is to postulate that objective chance reference points are in some way connected to our (the assessor's) total evidence about Thilo's case, such that, given our total evidence, the appropriate reference point is the reference point that mirrors the non-extreme objective chances of Thilo's ticket losing shortly before the draw.

The upshot is that postulating a connection between a subject's total evidence concerning a certain event and the objective chance of the event's occurrence at a certain time is in no way ad hoc. To the contrary, it can be directly read off of Lewis's (Reference Lewis1987: 87) independently plausible Principal Principle:

Principal Principle (PP)

Let P ₀ be the credence function of an agent at the beginning of her epistemic life. Let t be any time. Let ω be any world. Let x be a real number in the unit interval. Let C _t,  ω(A) = x be the proposition that the chance at time t and world ω of A's holding equals x. Let E be the agent's total evidence that is admissible at time t. Then

$$P_0( A{\rm \;\vert }C_{t, {\rm \;}\omega }( A ) = x{\rm \;}\& {\rm \;}E) = x$$

PP relates time-dependent objective chance to time-dependent admissible evidence via the concept of rational credence.Footnote ³⁶ Admissible evidence, in turn, is defined as follows:

Evidence E is admissible at t concerning outcome A iff it contains the sort of information whose impact on credence about A, if any, comes entirely by way of credence about the chances of A at t.Footnote ³⁷

Concerning Lottery, let L be the proposition that Thilo's ticket loses. Let t ₋₁ be the time shortly before the draw and $C_{t_{{-}1}, \;\omega }( L ) = .999$ the proposition that at world ω and time t ₋₁ the objective chances of Thilo's ticket losing are .999. Let E be our total evidence. We can now investigate the following question: Which credence P ₀ in L should we adopt at the present instance t ₀? Following PP we should reason as follows:

(P1) If our present total evidence E is admissible at t ₋₁ concerning L, then the present rational credence for us is $P_0( L\;{\rm \vert }\;C_{t_{{-}1}, \;\omega }( L ) = .999\;\& \;E) = .999$. (according to PP)

(P2) E is admissible at t ₋₁ concerning L.

(C) Therefore, the present rational credence for us is $P_0( L\;{\rm \vert }\;C_{t_{{-}1}, \;\omega }( L ) = .999\; \amp\; E) =$ .999.Footnote ³⁸

E is admissible at t ₋₁ because it contains only information about the objective chances of L at t ₋₁. E does not include any information concerning the specific outcome of the draw. If we had such information, E would no longer be admissible at t ₋₁, but only at a later point in time after the draw.

Together with PP it follows that it is rational for us to have a .999 credence that Thilo lost, though the draw already took place. The relativized version of the objective chance analog (OCR) simply adds that we should adjust our credences in accordance with PP, because given that we are not misled about the objective chances of Thilo losing being .999 at t ₋₁, adjusting our credence in this way will satisfy the probabilistic analog of truth (at least from our perspective). Consequently, we are able to judge that Thilo's probabilistic belief is non-trivially true. It is consistent with the objective chance reference point determined by our total evidence concerning whether L.

My suggestions, therefore, amount to the subsequent relativistic principle for determining objective chance reference points: objective chance reference points shift relative to the admissibility of a subject's total evidence concerning a certain question at the context of assessment of a probabilistic belief, assertion, or content. One might worry that a subject's total evidence is admissible at multiple points in time concerning any outcome A. For instance, our total evidence E is admissible at t ₋₁ concerning L. But according to the above definition of admissibility it is also admissible at later times, specifically at the present time t₀ after the draw. What is it then that connects E to the objective chances at t ₋₁ rather than t ₀? As a rule, a subject's total evidence with regard to a certain outcome is admissible during some interval [t _e, ∞) that begins at some time t _e and extends continuously into the future. This implies that for each outcome A there is a certain point in time where a subject's total evidence starts being admissible. Adopting Lewis's terminology, call the earliest point in the interval the endpoint time of admissibility of S's total evidence in relation to A. It is this endpoint time of admissibility that determines the correct objective chance reference point. We end up with a probability space mirroring the chances of A at t _e.