1. Objective probability theory, direct inference, and the Oscar Seminar's argument for thirdism

One version of the theory of objective probability, advocated by Pollock, invokes a distinction between ‘definite’ probabilities [indicated with small caps ‘PROB’ as in ‘PROB(P/Q)’ or ‘PROB(P)’] which attach to propositions, and ‘indefinite’ probabilities [indicated with lower case ‘prob’ as in ‘prob(Fx/Gx)’] which attach to properties (including n-adic relations) or propositional functions. What is called ‘a theory of direct inference’ within objective probability theory becomes, within this version of the theory, an account of the conditions under which one is justified in inferring definite probabilities from indefinite probabilities. In ‘prob(Fx/Gx)’ G is called ‘the reference property’ and F ‘the consequent property’. Following the pioneering work of Reichenbach (Reference Reichenbach1949: 374), a core principle is that one should identify the probability that a given particular, a, is F (PROB(Fa)), with prob(Fx/Gx), where G is the logically strongest evidentially relevant property such that one knows prob(Fx/Gx) and one knows Ga.

The Oscar Seminar's argument invokes direct inference in defense of thirdism, as follows. Where ‘a Sleeping Beauty scenario’ is an instance of the Sleeping Beauty problem, the Oscar Seminar's members use ‘B(t,s)’ to mean ‘s is a Sleeping Beauty scenario and t is a time during s’ and ‘Toss(x,s)’ to mean ‘x is a (the) coin toss involved in s’. Where x is a coin toss, they use ‘Hx’ to mean ‘x lands heads’. And they use ‘W(t,s)’ to mean ‘Sleeping Beauty is awakened in the scenario s during an interval Δ since t, without remembering being awakened at any prior time during s’. According to the Oscar Seminar, the following indefinite probability claims are both true:

1. (1) prob(Hx/B(t,s) & Toss(x,s)) = 1/2

2. (2) prob(Hx/W(t,s) & B(t,s) & Toss(x,s)) = 1/3

where ‘x’, ‘t’, and ‘s’ are free variables within the formulas to which the operator ‘prob(/)’ applies, and are bound within (1) and (2) by ‘prob’ itself.

Let σ be a particular Sleeping Beauty scenario and let τ be the coin toss in σ. On Sunday in σ, according to the Oscar Seminar, Beauty knows B(now, σ) and Toss(τ,σ) and so can conclude by direct inference from (1) that PROB(Hτ) = 1/2. But upon being awakened during the experiment, Beauty comes to know W(now,σ) & B(now,σ) and Toss(τ,σ). Since (2) involves ‘a more specific reference property than (1)’, the Oscar Seminar claims that Beauty should, by Reichenbach's principle, conclude by direct inference deploying (2) that PROB(Hτ) = 1/3 (Reference Seminar2008: 152).

2. The Pust/Thorn/Draper dialectic regarding the Oscar Seminar's argument

Joel Pust (Reference Pust2011) and Kaila Draper (Reference Draper2017) both maintain that the Oscar Seminar's argument for the thirdist answer to the Sleeping Beauty problem is unsound. Neither author contests statement (2), and neither author launches a general objection to direct inference as a technique for drawing conclusions about single-case probabilities. Rather, they both argue that the Oscar Seminar's argument violates the constraints on direct inference that are imposed by the conceptual machinery of objective probability theory itself. Pust's reasoning appeals to the following indefinite probability claim:

1. (3) prob(Hx/Toss(x,s)) = 1/2

Pust argues as follows:

In response to Pust, Paul Thorn (Reference Thorn2011) contends that ‘Sleeping Beauty should disregard Pust's direct inference [viz., the prima facie justified direct inference based on (3)], and accept the direct inference to the 1/3-conclusion’ (Thorn Reference Thorn2011: 662). The heart of Thorn's reasoning is a proposed account of comparative logical strength for two reference properties that is applicable even when one of the reference properties has different ‘arity’ than the other. Thorn writes:

In reply to Thorn, Kaila Draper (Reference Draper2017) argues that Thorn's defense of the Oscar Seminar's argument for thirdism fails because Thorn's proposed definition of ‘logically stronger reference property’ is unacceptable. Draper writes:

Draper goes on to offer a new argument of her own, different from Pust's, aimed at establishing that the direct inference in the Oscar Seminar's argument for thirdism is unsound. I will address this further argument in §5 below.

3. Draper vs. Thorn on comparative logical strength between reference properties of different arity

I have several comments about Thorn's proposed definition of ‘logically stronger reference property’ and about Draper's objections to it.

First, the natural way to try characterizing comparative logical strength for properties of different arity would be in two steps: first, offer a definition of logical entailment between properties, in a way that allows this relation to hold between properties of differing arity. Second, define greater logical strength as follows: property R′ is logically stronger than property R just in case R′ logically entails R but R does not logically entail R′.

Second, it is also very natural to construe Thorn's own proposed definition of greater logical strength as actually constituting a proposed definition of logical entailment between an n-ary property R′ and an m-ary property R (where n ≥ m) – and to construe Thorn as having mistakenly conflated the task of defining logical entailment for properties with the distinct task of defining the ‘logically stronger than’ relation between properties.

Third, this means that if Thorn's proposed definition does indeed adequately characterize logical entailment between properties, then Draper's objection is easily overcome – by saying that property R′ is logically stronger than property R just in case R′ entails R but not conversely.

However, fourth, Thorn's proposal (as thus reconstrued) does not constitute an adequate definition of logical entailment between properties. The problem is that the proffered definition has the following (probably unintended) consequence:

This putatively necessary condition on the logical entailment relation (and on the ‘logically stronger than’ relation) is far too strong, in two respects. First, the specific items from {i ₁,…, i _n} that are logically guaranteed to satisfy R, given that < i ₁,…, i _n> satisfies R′, need not be the first m items in the n-tuple < i ₁,…, i _n>; rather, they could be any m items from the set {i ₁,…, i _n}. Second, the order in which the R-satisfying items from {i ₁,…, i _n}are logically guaranteed to satisfy R, given that < i ₁,…, i _n> satisfies R′, need not be the same as the order in which those items occur, left to right, in the n-tuple < i ₁,…, i _n>; rather, it could be any successive ordering whatever of those items.Footnote ²

Fifth, the problem just noted appears, prima facie, to be a mere technical problem. There ought to be a natural and appropriate way to define logical entailment between properties, as a relation that can hold even when the related properties differ in arity from one another. (Technical problems have technical solutions!) If such a definition can be given, then it will turn out that greater logical strength, construed as one-way logical entailment, can indeed obtain between properties of differing arity. This would vindicate the underlying idea behind Thorn's reply to Pust.

4. Defining logical entailment, and greater logical strength, between properties

I will now offer a definition of logical entailment, as a relation that can hold between two properties of differing arity. Greater logical strength will then be easily definable, as one-way logical entailment. The two definitions together will elucidate the content of the intuitive thought that the greater logical strength of property R′, vis-à-vis property R, is a matter of R′ being ‘informationally richer’ than R – the idea behind Reichenbach's principle. If satisfactory, the proposed definition of logical entailment for properties will render the reference property in claim (2) logically stronger than the reference property in claim (3), thereby defusing Pust's objection to the Oscar Seminar's argument for thirdism. And the definition will be a contribution to the theory of direct inference, as regards the proper construal of Reichenbach's principle.

Two criteria of adequacy should be met by the sought-for definition of logical entailment for properties. First, the definition should avoid the problems, described in §3, encountered by Thorn's own proposal. His is inadequate, either as a definition of greater logical strength between properties or as a definition of logical entailment between properties, because it is not sufficiently permissive about the various ways that the elements of an n-tuple < i ₁,…, i _n> might be related to the elements of an m-tuple < j ₁,…, j _m> in instances where it is logically true that if < i ₁,…, i _n> satisfies property R′ then < j ₁,…, j _m> satisfies property R.

Second, the definition should specify some uniform connection that must obtain between (i) an n-tuple of items < i ₁,…, i _n> that satisfies the n-ary property R, and (ii) the specific corresponding m-tuple of items, all from the set {i ₁,…, i _n}, that must satisfy the m-ary property R′. The idea here is that if R′ really logically entails R, then the pertinent, definitive, connection between an n-tuple that satisfies R′ and the corresponding m-tuple that must satisfy R should be the same regardless of the specific items that satisfy R′ and R, rather than depending upon which particular items are involved. The connection should be a matter of logic alone.

I begin with two preliminary definitions. First is the notion of a position-pairing function, which pairs element-positions in n-tuples with element-positions in m-tuples. (There are n respective positions in an n-tuple, and m respective positions in an m-tuple.)

For natural numbers n and m, an n-to-m position-pairing function is a 1–1 function f that meets the following conditions:

1. (1) if n > m, then (i) the domain of f is a set containing m members of the set of all n-tuple positions, (ii) the range of f is the set of all m-tuple positions, and (iii) for each distinct m-tuple position q there is a distinct n-tuple position p such that f(p) = q, and

2. (2) if n ≤ m, then (i) the domain of f is the set of all n-tuple positions, (ii) the range of f is a set containing n members of the set of all m-tuple positions, and (iii) for each distinct n-tuple position p there a distinct m-tuple position q such that f(p) = q.

Second is the notion of conformity between an m-tuple and an n-tuple, relative to an n-to-m position-pairing function.

For natural numbers n and m, n-tuple < i ₁,…, i _n>, m-tuple < j ₁,…, j _m>, and n-to-m position-pairing function f, the m-tuple < j₁,…, j_m> f-conforms to the n-tuple < i₁,…, i_n> just in case the following condition obtains:

Using these preliminary definitions, I now propose the following definition of logical entailment, as a relation between an n-ary property R′ and an m-ary property R.

This definition meets the first criterion of adequacy, the permissiveness criterion, because it allows any m of the elements in an R′-satisfying n-tuple < i ₁,…, i _n>, in any order, to constitute the m successive elements in the corresponding R-satisfying m-tuple < j ₁, …, j _m>. And the definition meets the second criterion of adequacy, the uniqueness criterion, by requiring the pairing function f to select certain specifically-positioned elements of < i ₁,…, i _n> as the successive elements of < j ₁, …, j _m> solely on the basis of the n-tuple positions of the selected items from < i ₁,…, i _n>, independently of what the items themselves are.

Of course, comparative logical strength for properties is now easily defined this way: n-ary property R′ is logically stronger than property R just in case (i) R′ logically entails R and (ii) R does not logically entail R′.

Return now to the guiding intuition behind the idea that an n-ary property R′ can be logically stronger than an m-ary property R even when n > m. That intuition can be formulated this way: R′ is logically stronger than R only if the claim R′(i ₁,…, i _n), about the items i ₁,…, i _n, says (or entails) everything about some of those items (about m specific ones of them) that is said by predicating R of those items; and in addition, the claim R′(i ₁,…, i _n) also says more besides. My proposed definition of ‘logically stronger than’, I submit, does full justice to this intuition.Footnote ³

The upshot, as regards the Oscar Seminar's argument for the thirdist answer to the Sleeping Beauty problem, is that the reference property in claim (2) above is indeed logically stronger than the reference property in claim (3). Thorn's response to Pust's objection to the argument is thereby vindicated in spirit – even though Draper is correct in claiming that Thorn's own proposed definition of ‘logically stronger than’ is not acceptable, and even though it also is not an acceptable definition of logical entailment between properties.

The wider upshot is the licensing of preference for direct inferences based on logically stronger reference properties, regardless of arity – in alignment with existing treatments of direct inference that also purport to have this feature (Pollock Reference Pollock1990; Kyburg and Teng Reference Kyburg and Teng2001: 216; Thorn Reference Thorn2011, Reference Thornin press).Footnote ⁴ This greatly enhances the scope and power of direct inference as a method for drawing conclusions about single-case epistemic probabilities.

5. Draper's new objection to the Oscar Seminar's argument

Draper raises an objection of her own to the Oscar Seminar's argument for thirdism about the Sleeping Beauty problem – an objection that does not involve properties of different arity. If we let ‘Mt’ abbreviate ‘t is a time on Monday’, she says, then it is clear that

1. (4) prob(Hx/W(t,s) & B(t,s) & Toss(x,s) & Mt) = 1/2.

It also is clear that the reference property in (4) is logically stronger than the reference property in (2) – even in the ordinary sense, since both reference properties have the same, tertiary, arity. She now argues as follows:

At first blush, this argument might seem very difficult to fault. After all, the reference property in (4) is indeed logically stronger than the reference property in (3), even in the noncontroversial sense of ‘logically stronger’ involving two properties with the same arity. Also, since Beauty knows both (4) itself and that W(ttm,σ) & B(ttm,σ) & Toss(τ,σ) & M(ttm), it would seem that she can indeed arrive by direct inference at the prima facie conclusion that PROB(Hτ) = 1/2. How, then, could one possibly fault Draper's contention that this prima facie direct inference rebuts and thereby defeats the prima facie direct inference that relies on (2)?

I maintain that one can, and one should, fault this contention. To see the problem, consider some of the pertinent statements that Beauty knows to obtain. She knows both (2) and (4). She also knows the following two statements concerning the scenario σ (the one she is now in) and the coin toss τ in this scenario.

1. (5) W(now,σ) & B(now,σ) and Toss(τ,σ)

2. (6) W(ttm,σ) & B(ttm,σ) & Toss(τ,σ) & M(ttm)

Now, the reference property in (4) is clearly logically stronger than the reference property in (3). But how do (5) and (6) stack up against one another in terms of comparative logical strength? Well, the statement ‘M(ttm)’ is trivially true, since it says ‘This time on Monday is a time on Monday’. (It is logically guaranteed to be true, given that the indexical expression ‘this time’ is logically guaranteed to refer to the current time and the statement ‘B(ttm,σ)’ entails that there is unique referent of ‘Monday’ within the scenario.) Thus, (5) is logically equivalent to

1. (5*) W(now,σ) & B(now,σ) and Toss(τ,σ) & M(ttm)

But (5*) is logically stronger than (6), because (5*) entails (6) but not conversely. (This is because (6) is consistent with the statement ‘¬W(now,σ) & B(now,σ) and Toss(τ,σ)’, which would have been true had today been Tuesday and had the coin toss come up Heads – in which case Sleeping Beauty would have slept dreamlessly all day today rather than being awakened today by the experimenters.) Thus, since (5) and (5*) are logically equivalent, (5) is logically stronger than (6).Footnote ⁵

So something odd and curious is at work in Draper's argument: although the reference property in (4) is indeed logically stronger than the reference property in (2), nevertheless the pertinent instantiation of the reference property in (4) – the instantiation that is deployed in Draper's prima facie direct inference relying on (4) – is logically weaker than the pertinent instantiation of the reference property in (2) that is deployed in the Oscar Seminar's direct inference relying on (2).Footnote ^6, Footnote ^7, Footnote ⁸

What moral should be drawn from this curious inversion in comparative logical strength? The answer, I suggest, is this: the Oscar Seminar's direct inference from (2) and (5) to the conclusion PROB(Hτ) = 1/3 defeats, but is not itself defeated by, Draper's prima facie direct inference from (4) and (6) to the conclusion PROB(Hτ) = 1/2. This is so because the reference-property instantiation invoked in the Oscar Seminar's direct inference – viz., statement (5) – is logically stronger than the reference-property instantiation invoked in Draper's prima facie direct inference – viz., statement (6).

The immediate upshot is that Draper's objection to the Oscar Seminar's argument for thirdism concerning the Sleeping Beauty problem is unsuccessful, because she is mistaken to claim that ‘the direct inference that relies on (4) rebuts and thereby defeats the direct inference that relies on (2)’. On the contrary, the rebut-and-defeat relation obtains only in the other direction: the inference that relies on (2) rebuts and thereby defeats the direct inference that relies on (4), but not conversely.

The more general upshot concerns the proper formulation and proper application of Reichenbach's principle. The intuitive idea behind the principle is that a proper direct inference will be one that deploys one's strongest pertinent evidence. It turns out, however, that this intuitive idea is not perfectly captured by the idea that a proper direct inference to a conclusion PROB(Fa₁,…, a_n) = r will deploy the logically strongest evidentially relevant reference property G such that a known, or justifiably believed, instantiation of G, plus a known, or justifiably believed, indefinite-probability claim whose reference property is G, together provide a prima facie direct inference to the conclusion PROB(Fa₁,…, a_n) = r. Trouble can arise in cases where the instantiation of G is a conjunctive statement in which at least one conjunct is degenerate (as I will put it) – i.e., this conjunct is an atomic statement whose singular-term constituent(s) is/are so constructed, vis-à-vis its predicate-constituent, as to render the statement trivially true.

This is exactly what happens in Draper's prima facie direct inference deploying statements (4) and (6). Statement (6), the pertinent instantiation of the reference-property in (4), contains a degenerate conjunct – viz., the conjunct ‘M(ttm)’, which symbolizes ‘This time on Monday is a time on Monday’. This statement is trivially true, because the indexical expression ‘ttm’ is constructed so as to trivially satisfy the predicate ‘M’. And that is why statement (6) is logically weaker than statement (5).

The following claim is very plausible: if property G is logically stronger than property G, but a given instantiation I_G of property G is logically weaker than a given instantiation I*_H of property H, then this is because there is at least one degenerate conjunct in I_G. And the following further claim also is very plausible: if a statement PROB(Fa₁,…, a_n) = r can be legitimately derived from a given body of evidence by direct inference deploying the reference property G, then it can be derived by a G-deploying direct inference in which the pertinent instantiation of G is a statement containing no degenerate conjuncts.

In light of these two highly plausible claims, a simple and natural way to amend Reichenbach's principle is to rule out as improper any putative direct inference employing a reference-property instantiation containing a degenerate conjunct. The revised principle is this: a proper direct inference to a conclusion PROB(Fa₁,…, a_n) = r should deploy the logically strongest evidentially relevant reference property G such that a known, or justifiably believed, G-instantiation containing no degenerate conjunct, plus a known, or justifiably believed, indefinite-probability claim whose reference property is G, together provide a prima facie direct inference to the conclusion PROB(Fa₁,…, a_n) = r. (Reichenbach's principle also can be amended in a way that does not appeal to the notion of a degenerate conjunct, although the alternative revised principle will be more cumbersome.Footnote ⁹)

6. The Oscar Seminar's argument and the nature of probability

Could opponents of thirdism grant my replies to Pust and Draper, but then resist the Oscar Seminar's argument by repudiating certain assumptions about the nature of probability to which the argument is committed? The Oscar Seminar's paper begins with the following remarks:

Shortly thereafter the paper says, “Most objective approaches to probability tie probabilities to relative frequencies in some way” (p. 150). These passages might encourage the thought that one way to resist the Oscar Seminar's argument for thirdism would be to repudiate all philosophical views that construe probabilities as relative frequencies of some kind.

It is true enough that the sort of position typically espoused by advocates of objective probability theory themselves asserts that (i) objective probability (so-called “chance”) is relative frequency of some kind, (ii) indefinite probability is chance as thus construed, and (iii) definite probability is a derivative attribute that is determined by known, or justifiably believed, indefinite probability. But one need not embrace any of this just by virtue of embracing the Oscar Seminar's argument for thirdism. Instead one could construe the pertinent conceptual machinery of objective probability theory in some other way.

Here is one alternative construal, for example. (I myself find it very attractive.) So-called indefinite probability, as expressed by ‘prob’, is not really a distinct kind of probability from the kind expressed by ‘PROB’. Rather, a statement ‘prob(Fx│Gx) = r’ is just a notational variant of the universally quantified statement ‘∀x(PROB(Fx│Gx) = r)’. Moreover, epistemic probability, as expressed by ‘PROB’, is quantitative degree of evidential support, relative to one's total available evidence (cf. Horgan Reference Horgan2017a, Reference Horgan2017b: section 6). Thus, ‘prob(Fx│Gx) = r’ is to be understood as asserting that for any arbitrary item x, considered as such (and apart from any additional knowledge one might happen to have about this x specifically), one's degree of evidential support for x's being F, given that x is G, is equal to r. Direct inference is a transition, from an evidentially sanctioned premise ‘∀x(PROB(Fx│Gx) = r)’ plus an evidentially sanctioned premise ‘Gτ’, to a conclusion ‘PROB(Fτ) = r’. (Multiple variables can be involved, instantiated by multiple singular terms.) Quantifiers in statements of the form ‘∀x(PROB(Fx│Gx) = r)’ range over possible instances of the reference property G; hence such a statement, when evidentially sanctioned itself, thereby determines a rationally expectable hypothetical long run of G-instances – a long run in which the relative frequency of F-instances is r. But relative frequency across such a hypothetical long run is a derivative matter, rather than being either fundamental or a kind probability itself.

Other construals of the pertinent conceptual machinery of objective probability theory seem available too. In particular, there is no obvious reason why one could not adopt a “Bayesian” construal. This would be similar to the construal described in the preceding paragraph, but would identify epistemic probability with quantitative degree of partial belief rather than with quantitative degree of evidential support.

The upshot is that those who wish to repudiate the Oscar Seminar's argument for thirdism bear a substantial burden of proof. Since the conceptual machinery used in the argument looks to be construable in conformity with most any familiar way of interpreting the kind of probability that is at issue in the Sleeping Beauty problem, opponents of the Oscar Seminar's argument cannot repudiate it just by rejecting frequentism. Instead they must confront the argument on its own terms.