1. The Claims

This article investigates the measure theoretic consistency of what we call the “Abstract Principal Principle.” The consistency expresses that the Abstract Principal Principle is in harmony with the basic structure of measure theoretic probability theory. This type of consistency is tacitly assumed in the literature on the Principal Principle, although we will see that the consistency in question is not trivial. The main philosophical significance of proving such a consistency is that without making sure that such a consistency obtains, the Abstract Principal Principle would be inconsistent as a general norm that guides forming subjective degrees of belief (credences): without such consistency a Bayesian agent would not always be able to adjust his degrees of belief to objective probabilities (e.g., chances) in a Bayesian manner, via Bayesian conditionalization.

After stating the Abstract Principal Principle informally in section 2, we define formally the weak and strong consistency of theAbstract Principal Principle (definitions 1 and 2) in section 3 and state weak and strong consistency of the Abstract Principal Principle (propositions 1 and 2). We then argue that it is very natural to strengthen the Abstract Principal Principle by requiring it to satisfy a stability property, which expresses that conditional degrees of belief in events already equal (in the spirit of the Abstract Principal Principle) to the objective probabilities of the events do not change as a result of conditionalizing them further on knowing the objective probabilities of other events (in particular, of events that are independent with respect to their objective probabilities). We call this amended principle the Stable Abstract Principal Principle (if stability is required only with respect to further conditionalizing on values of probabilities of independent events: Independence-Stable Principal Principle). This stability requirement leads to suitably modified versions of both the weak and strong consistency of the (Independence-)Stable Abstract Principal Principle (definitions 3 and 4). We will prove that the Stable Abstract Principal Principle is weakly consistent (proposition 3). This entails weak consistency of the Independence-Stable Abstract Principal Principle (proposition 4). The strong consistency of both the stable and the Independence-Stable Abstract Principal Principle remain open problems, however; we conjecture that both consistencies hold.Footnote ¹

Until section 6, few references are given. Section 6 puts the results into context: here we discuss the relevance of strong consistency of the Stable Abstract Principal Principle from the perspective of Lewis’s Principal Principle and its “debugged” versions. The details of all the proofs are in the appendix.

2. The Abstract Principal Principle Informally

The Abstract Principal Principle regulates probabilities representing the subjective degrees of belief $p_{\mathrm{subj}}\left(A\right)$ of an abstract Bayesian agent by stipulating that $p_{\mathrm{subj}}\left(A\right)$ are related to the objective probabilities $p_{\mathrm{obj}}\left(A\right)$ as

(1)$\begin{array}{}{p}_{\mathrm{subj}}\left(A\right|⌜p_{\mathrm{obj}}\left(A\right)=r⌝)=p_{\mathrm{obj}}(A),\end{array}$

where $⌜p_{\mathrm{obj}}\left(A\right)=r⌝$ denotes the proposition “the objective probability, $p_{\mathrm{obj}}\left(A\right)$, of A is equal to r.”

The formulation (1) of the Abstract Principal Principle presupposes that both $p_{\mathrm{subj}}$ and $p_{\mathrm{obj}}$ are probability measures: additive maps defined on a σ-algebra taking values in $[0,1]$. Probability $p_{\mathrm{obj}}$ is supposed to be defined on a σ-algebra $S_{\mathrm{obj}}$ of random events, and $p_{\mathrm{subj}}$ is supposed to be a map with a domain of definition being a σ-algebra $S_{\mathrm{subj}}$.

It is crucial to realize that the σ-algebras $S_{\mathrm{obj}}$ and $S_{\mathrm{subj}}$ cannot be unrelated: for the conditional probability $p_{\mathrm{subj}}\left(A\right|⌜p_{\mathrm{obj}}\left(A\right)=r⌝)$ in equation (1) to be well defined via Bayes’s rule, the σ-algebra $S_{\mathrm{subj}}$ must contain both the σ-algebra $S_{\mathrm{obj}}$ of random events and with every random event A also the proposition $⌜p_{\mathrm{obj}}\left(A\right)=r⌝$—otherwise the formula $p_{\mathrm{subj}}\left(A\right|⌜p_{\mathrm{obj}}\left(A\right)=r⌝)$ cannot be interpreted as an expression of conditional probability specified by Bayes’s rule.

It is far from obvious, however, that, given any σ-algebra $S_{\mathrm{obj}}$ of random events with any probability measure $p_{\mathrm{obj}}$ on $S_{\mathrm{obj}}$, there exists a σ-algebra $S_{\mathrm{subj}}$ meeting these algebraic requirements in such a way that a probability measure $p_{\mathrm{subj}}$ satisfying the condition (1) also exists on $S_{\mathrm{subj}}$. If there exists a σ-algebra $S_{\mathrm{obj}}^{*}$ of random events with a probability measure $p_{\mathrm{obj}}^{*}$ giving the objective probabilities of events for which there exists no σ-algebra $S_{\mathrm{subj}}$ on which a probability function $p_{\mathrm{subj}}$ satisfying (1) can be defined, then the Abstract Principal Principle would be inconsistent as a general norm: in this case the agent, being in the epistemic situation of facing the objective facts represented by $(S_{\mathrm{obj}}^{*},p_{\mathrm{obj}}^{*})$, cannot have degrees of belief satisfying the Abstract Principal Principle for fundamental structural reasons inherent in the basic structure of classical probability theory. We say that the Abstract Principal Principle is weakly consistent if it is not inconsistent in the sense described. (The adjective “weakly” will be explained shortly.)

Remark. One can construe the Principal Principle differently: taking it as a norm that regulates internal consistency of the agent.Footnote ² Under this construal, the subjective degrees of belief should satisfy

(2)$\begin{array}{}{p}_{\mathrm{subj}}\left(A\right|⌜p_{\mathrm{obj}}\left(A\right)=r⌝)=r\mathrm{for}\mathrm{all}r\in [0,1]\mathrm{.}\end{array}$

Here $⌜p_{\mathrm{obj}}\left(A\right)=r⌝$ is the proposition that the agent believes that the objective probability of A is equal to r, and (2) requires that the agent’s subjective degrees of belief conditional on this belief should be equal to r—otherwise the agent is inconsistent in his thinking. The difference between (1) and (2) is that r on the right-hand side of (2) need not be equal to the real objective probability $p_{\mathrm{obj}}\left(A\right)$. The difference between these two interpretations plays no role, however, from the perspective of the consistency problem we investigate here: because of the universal quantification over $p_{\mathrm{obj}}$ in the consistency definitions and because of the universal quantification over r in (2), the two construals lead to the same consistency problem.

3. Weak and Strong Consistency of the Abstract Principal Principle

A classical probability measure space is denoted $(X,S,p)$, where $S$ is a σ-algebra of (some) subsets of X, and p is a probability measure on $S$. Given two σ-algebras $S$ and $S\prime$, the injective map $h:S\to S\prime$ is a σ-algebra embedding if it preserves all Boolean σ-operations. The probability space $(X^{\prime },S\prime ,p^{\prime })$ is called an extension of $(X,S,p)$ with respect to h if h is a σ-algebra embedding of $S$ into $S\prime$ that preserves the probability measure p:

(3)$\begin{array}{}{p}^{\prime }\left(h\right(A\left)\right)=p\left(A\right)A\in S\mathrm{.}\end{array}$

Definition 1 says: given the “objective” probability space $(X_{\mathrm{obj}},S_{\mathrm{obj}},p_{\mathrm{obj}})$, the σ-algebra $S_{\mathrm{subj}}$ in $(X_{\mathrm{subj}},S_{\mathrm{subj}},p_{\mathrm{subj}})$ contains the “copies” $h\left(A\right)$ of all the random events $A\in S_{\mathrm{obj}}$ and also an element $A^{\prime }$ to be interpreted as representing the proposition “the objective probability, $p_{\mathrm{obj}}\left(A\right)$, of A is equal to r” (this proposition we denoted by $⌜p_{\mathrm{obj}}\left(A\right)=r⌝$). If $A\overline{)=}B$, then $A^{\prime }\overline{)=}{B}^{\prime }$ must hold because $⌜p_{\mathrm{obj}}\left(A\right)=r⌝$ and $⌜p_{\mathrm{obj}}\left(B\right)=s⌝$ are different propositions—this is expressed by ii in the definition. The main content of the Abstract Principal Principle is then expressed by condition (4), which states that the conditional degrees of beliefs $p_{\mathrm{subj}}\left(h\right(A\left)\right|A^{\prime })$ of an agent about random events $h\left(A\right)\leftrightarrow A\in S_{\mathrm{obj}}$ are equal to the objective probabilities $p_{\mathrm{obj}}\left(A\right)$, where the condition $A^{\prime }$ is that the agent knows the values of the objective probabilities.

The above proposition follows from proposition 3 stating the weak consistency of the Stable Abstract Principal Principle, which we state later.

The content of this additional requirement is that the agent’s prior probability function $p_{\mathrm{subj}}$ restricted to the random events can be equal to probability measure $p_{\mathrm{subj}}^0$ on $S_{\mathrm{obj}}$ that can differ from the objective probabilities of the random events given by $p_{\mathrm{obj}}$.

4. The Stable Abstract Principal Principle

Once the agent has adjusted his subjective degree of belief by conditionalizing, $p_{\mathrm{subj}}\left(h\right(A\left)\right|⌜p_{\mathrm{obj}}\left(A\right)=r⌝)=r$, he may then learn the value of another objective probability, $⌜p_{\mathrm{obj}}\left(B\right)=s⌝$, in which case he must conditionalize again. What should be the result of this second conditionalization? Since the agent’s conditional degrees of belief $p_{\mathrm{subj}}\left(h\right(A\left)\right|⌜p_{\mathrm{obj}}\left(A\right)=r⌝)$ in A are already correct (equal to the objective probabilities), it would be irrational to change his already correct degree of belief about A upon learning an additional truth, namely, the value of the objective probability $p_{\mathrm{obj}}\left(B\right)$. So a rational agent’s conditional subjective degrees of belief should be stable in the sense of satisfying the following condition:

(6)$\begin{array}{}{p}_{\mathrm{subj}}\left(h\right(A\left)\right|⌜p_{\mathrm{obj}}\left(A\right)=r⌝)=p_{\mathrm{subj}}(h\left(A\right)|⌜p_{\mathrm{obj}}(A)=r⌝\cap ⌜p_{\mathrm{obj}}(B)=s⌝)\forall B\in S_{\mathrm{obj}}\mathrm{.}\end{array}$

If A and B are independent with respect to their objective probabilities $p_{\mathrm{obj}}(A\cap B)=p_{\mathrm{obj}}\left(A\right)p_{\mathrm{obj}}\left(B\right)$, then, if the conditional subjective degrees of belief are stable in the sense of (6), then (assuming the Abstract Principal Principle) one has

(7)$\begin{array}{}\begin{array}{c}{p}_{\mathrm{subj}}\left(h\right(A)\cap h(B\left)\right|⌜p_{\mathrm{obj}}\left(A\right)=r⌝\cap ⌜p_{\mathrm{obj}}\left(B\right)=s⌝\cap ⌜p_{\mathrm{obj}}(A\cap B)=t⌝)\\ =p_{\mathrm{subj}}\left(h\right(A\cap B\left)\right|⌜p_{\mathrm{obj}}\left(A\right)=r⌝\cap ⌜p_{\mathrm{obj}}\left(B\right)=s⌝\cap ⌜p_{\mathrm{obj}}(A\cap B)=t⌝)\\ =p_{\mathrm{subj}}\left(h\right(A\cap B\left)\right|⌜p_{\mathrm{obj}}(A\cap B)=t⌝)\\ =p_{\mathrm{obj}}(A\cap B)\\ =p_{\mathrm{obj}}\left(A\right)p_{\mathrm{obj}}\left(B\right)\\ =p_{\mathrm{subj}}\left(h\right(A\left)\right|⌜p_{\mathrm{obj}}\left(A\right)=r⌝\left)p_{\mathrm{subj}}\right(h\left(B\right)|⌜p_{\mathrm{obj}}(B)=s⌝)\\ =p_{\mathrm{subj}}\left(h\right(A\left)\right|⌜p_{\mathrm{obj}}\left(A\right)=r⌝\cap ⌜p_{\mathrm{obj}}\left(B\right)=s⌝\cap ⌜p_{\mathrm{obj}}(A\cap B)=t⌝)\\ \times p_{\mathrm{subj}}\left(h\right(B\left)\right|⌜p_{\mathrm{obj}}\left(A\right)=r⌝\cap ⌜p_{\mathrm{obj}}\left(B\right)=s⌝\cap ⌜p_{\mathrm{obj}}(A\cap B)=t⌝)\mathrm{.}\end{array}\end{array}$

Equation (7) means that if the conditional subjective degrees of belief are stable, then, if A and B are objectively independent, they (their isomorphic images $h\left(A\right),h\left(B\right)$) are also subjectively independent—independent also with respect to the probability measure that represents conditional subjective degrees of belief, where the condition is that the agent knows the objective probabilities of all of A, B, and $(A\cap B)$. In this case, the conditional subjective degrees of belief properly reflect the objective independence relations of random events—they are independence faithful. Note that for the subjective degrees of belief to satisfy the independence-faithfulness condition expressed by equation (7), it is sufficient that stability (6) only holds for the restricted set of elements B in the σ-subalgebra $S_{\mathrm{obj}}^{A,\mathrm{ind}}$ of $S_{\mathrm{obj}}$ generated by the elements in $S_{\mathrm{obj}}$ that are independent of A with respect to $p_{\mathrm{obj}}$.

This motivates us to amend the Abstract Principal Principle by requiring stability of the subjective probabilities, resulting in the “Stable Abstract Principal Principle”:

Stable Abstract Principal Principle. The subjective probabilities $p_{\mathrm{subj}}\left(A\right)$ are related to the objective probabilities $p_{\mathrm{obj}}\left(A\right)$ as required by equation (1); furthermore, the subjective probability function is stable in the sense that the following condition holds:

(8)$\begin{array}{}{p}_{\mathrm{subj}}\left(h\right(A\left)\right|⌜p_{\mathrm{obj}}\left(A\right)=r⌝)=p_{\mathrm{subj}}(h\left(A\right)|⌜p_{\mathrm{obj}}(A)=r⌝\cap ⌜p_{\mathrm{obj}}(B)=s⌝)\forall B\in S_{\mathrm{obj}}\mathrm{.}\end{array}$

If the subjective probability function is only independence stable in the sense that (8) above holds for all $B\in S_{\mathrm{obj}}^{A,\mathrm{ind}}$, then the corresponding Stable Abstract Principal Principle is called the Independence-Stable Abstract Principal Principle.

5. Is the Stable Abstract Principal Principle Strongly Consistent?

The above proposition entails

The problem of strong consistency of both the stable and the Independence-Stable Abstract Principal Principle remains open (see Bana Reference Bana2016).

6. Relation to Other Works

Lewis (Reference Lewis1986) introduced the term “Principal Principle” to refer to the principle linking subjective beliefs to chances. In the context of the Principal Principle, $p_{\mathrm{subj}}\left(A\right)$ is called the “credence,” $C{r}_t\left(A\right)$, of the agent in event A at time t; $p_{\mathrm{obj}}\left(A\right)$ is the chance, ${\mathrm{Ch}}_t\left(A\right)$, of the event A at time t; and the Principal Principle is the stipulation that credences and chances are related as

(9)$\begin{array}{}{\mathrm{Cr}}_t\left(A\right|⌜{\mathrm{Ch}}_t\left(A\right)=r⌝\cap E)={\mathrm{Ch}}_t(A)=r,\end{array}$

where E is any admissible evidence the agent has at time t in addition to knowing the value of the chance of A.

Proposition $⌜{\mathrm{Ch}}_t\left(A\right)=r⌝$ is clearly admissible evidence for (9), and, substituting $E=⌜{\mathrm{Ch}}_t\left(A\right)=r⌝$ into equation (9), we obtain

(10)$\begin{array}{}{\mathrm{Cr}}_t\left(A\right|⌜{\mathrm{Ch}}_t\left(A\right)=r⌝)={\mathrm{Ch}}_t(A)=r,\end{array}$

which, at any given time t, is an instance of the Abstract Principal Principle if we make the identifications $p_{\mathrm{obj}}\left(A\right)={\mathrm{Ch}}_t\left(A\right)$, $p_{\mathrm{subj}}\left(A\right)={\mathrm{Cr}}_t\left(A\right)$. By proposition 2 we know that, for any time parameter t, relation (10) is consistent with probability as measure.

If, however, admissibility of evidence E is defined in such a way that propositions stating the values of chances of other events B at time t (i.e., propositions of the form $⌜{\mathrm{Ch}}_t\left(B\right)=s⌝$) are admitted as E, then (9) together with (10) entails that we also should have

(11)$\begin{array}{}{\mathrm{Cr}}_t\left(A\right|⌜{\mathrm{Ch}}_t\left(A\right)=r⌝\cap ⌜{\mathrm{Ch}}_t\left(B\right)=s⌝)={\mathrm{Ch}}_t(A)=r\mathrm{.}\end{array}$

The relation (11) together with equation (10) is, at any given time t, an instance of the Stable Abstract Principal Principle if we make the identifications $p_{\mathrm{obj}}\left(A\right)={\mathrm{Ch}}_t\left(A\right)$, $p_{\mathrm{subj}}\left(A\right)={\mathrm{Cr}}_t\left(A\right)$, and $p_{\mathrm{obj}}\left(B\right)={\mathrm{Ch}}_t\left(B\right)$. Thus, whether relations (11) and (10) can hold at all is exactly the question whether the Stable Abstract Principal Principle is strongly consistent. If one allows as evidence E in (11) only propositions stating the value of objective chances of events B that are objectively independent of A, then the question whether relations (11) and (10) can hold in general is exactly the question whether the Independence-Stable Abstract Principal Principle is strongly consistent. Since Lewis regarded admissible all propositions containing information that is “irrelevant” for the chance of A (Reference Lewis1986, 91), for him, admissible evidence should include propositions about values of chances of events that are independent of A with respect to the probability measure describing their chances. Under this interpretation of “irrelevant” information, the consistency of Lewis’s Principal Principle as a general norm needs proven consistency of the Independence-Stable Abstract Principal Principle. It should be emphasized that this kind of consistency has nothing to do with any metaphysics about chances or with the concept of natural law that one may have in the background of the Principal Principle; in particular, this inconsistency is different from the one related to “undermining” (see below). This consistency expresses a simple but fundamental compatibility of the Principal Principle with the basic structure of probability theory.

Lewis himself saw a consistency problem in his Principal Principle (he called it the “big bad bug”): if A is an event in the future of t that has a nonzero chance $r>0$ of happening at that later time but we have knowledge E about the future that entails that A will in fact not happen, $E\subset A^{\perp }$, then substituting this E into (9) leads to contradiction if $r>0$. Such an A is called an “unactualized future that undermines present chances”—hence, the phrase “undermining” to refer to this situation. Since certain metaphysical arguments led Lewis to think that one is forced to admit such an evidence E, he tried to “debug” the Principal Principle (Lewis Reference Lewis1994); the same sort of debugging was proposed simultaneously by Hall (Reference Hall1994) and Thau (Reference Thau1994). Other debugging attempts have followed (Black Reference Black1998; Roberts Reference Roberts2001; Hall Reference Hall2004; Loewer Reference Loewer2004; Hoefer Reference Hoefer2007; Ismael Reference Ismael2008; Glynn Reference Glynn2010; Meacham Reference Meacham2010; Nissan-Rozen Reference Nissan-Rozen2013; Pettigrew Reference Pettigrew2013; Frigg and Hoefer Reference Frigg and Hoefer2015), and to date no consensus has emerged as to which of the debugged versions of the Principal Principle is tenable: Vranas (Reference Vranas2004) claims that there was no need for a debugging in the first place; Briggs (Reference Briggs2009) argues that none of the modified principles work; Pettigrew (Reference Pettigrew2012) provides a framework that allows one to choose the correct Principal Principle depending on one’s metaphysical concept of chance.

Papers aiming at “debugging” Lewis’s Principal Principle typically combine the following three moves a, b, or c:

• a) Restricting the admissible evidence in (9) to a particular class $A_A$ of propositions in order to avoid “undermining” (Hoefer Reference Hoefer2007).

• b) Modifying the Principal Pinciple by replacing ${\mathrm{Ch}}_t\left(A\right)$ on the right-hand side of (9) with a value $F\left(A\right)$ given by a function F different from the objective chance function (new principle by Hall [Reference Hall1994]; general Principal Principle by Lewis [Reference Lewis1980] and by Roberts [Reference Roberts2001]).

• c) Modifying the Principal Principle by replacing the conditioning proposition $⌜{\mathrm{Ch}}_t\left(A\right)=r⌝\cap E$ on the left-hand side of (9) by a different conditioning proposition $C_A$, which is a conjunction of some propositions from $S_{\mathrm{obj}}$, $A_A$, and propositions of the form $⌜p_{\mathrm{obj}}\left(B\right)=r⌝$ (conditional principle and general principle by Vranas [Reference Vranas2004]; general recipe by Ismael [Reference Ismael2008]).

To establish a theory of chance along a debugging strategy characterized by a combination of a, b, and c, it is not enough to show that undermining is avoided: one has to prove that the debugged Principal Principle is consistent in the sense of definition 5 below, which is in the spirit of the notion of consistency investigated in this article:

We say that the “$(A_A,C_A,F)$-debugged” Principal Principle is weakly consistent if i, ii, and iv hold.

Taking specific $C_A$ and F, one obtains particular definitions expressing the consistency of specific debugged Principal Principles. For instance, stipulations

(15)$\begin{array}{}{C}_A=B\cap ⌜p_{\mathrm{obj}}\left(A\right|B)=r⌝\end{array}$

(16)$\begin{array}{}F\left(A\right)=p_{\mathrm{obj}}\left(A\right)\end{array}$

yield Vranas’s conditional principle (Reference Vranas2004, 370), whereas Hall’s new principle (Reference Hall1994, 511) can be obtained by

(17)$\begin{array}{}{C}_A=H_{t,w}\cap T_w\end{array}$

(18)$\begin{array}{}F\left(A\right)=p_{\mathrm{obj}}\left(A\right|T_w),\end{array}$

where $H_{t,w}$ is “the proposition that completely characterizes w’s history up to time t” and $T_w$ is the “proposition that completely characterizes the laws at w” (w being a possible world; 506).

Proving consistency of the $(A_A,C_A,F)$-debugged Principal Principles is necessary for the respective debugged Principal Principles to be compatible with measure theoretic probability theory. To our best knowledge such consistency proofs have not been given: it seems that this type of consistency is tacitly assumed in the works analyzing the modified Principal Principles, although, as the propositions and their proofs presented in this article show, the truth of these types of consistency claims is far from obvious.

The problem of strong consistency of the Stable Abstract Principal Principle is also relevant from the perspective of existence of particular models of the axioms of higher-order probability theory (HOP) suggested by Gaifman (Reference Gaifman1988). If one regards HOP as an axiomatic theory, then the question arises whether models of the theory exist. Gaifman provides a few specific examples that are models of the axioms (208–10), but he does not raise the general issue of what kinds of models exist. What one would like to know is whether any objective probability theory can be made part of a HOP in such a way that the objective probabilities are related to the subjective ones in the manner required by the HOP axioms. Proving the existence of such HOPs entails that the Stable Abstract Principal Principle is strongly consistent.