A non-probabilist principle of higher-order reasoning

Abstract

The author uses a series of examples to illustrate two versions of a new, nonprobabilist principle of epistemic rationality, the special and general versions of the metacognitive, expected relative frequency (MERF) principle. These are used to explain the rationality of revisions to an agent’s degrees of confidence in propositions based on evidence of the reliability or unreliability of the cognitive processes responsible for them—especially reductions in confidence assignments to propositions antecedently regarded as certain—including certainty-reductions to instances of the law of excluded middle or the law of noncontradiction in logic or certainty-reductions to the certainties of probabilist epistemology. The author proposes special and general versions of the MERF principle and uses them to explain the examples, including the reasoning that would lead to thoroughgoing fallibilism—that is, to a state of being certain of nothing (not even the MERF principle itself). The author responds to the main defenses of probabilism: Dutch Book arguments, Joyce’s potential accuracy defense, and the potential calibration defenses of Shimony and van Fraassen by showing that, even though they do not satisfy the probability axioms, degrees of belief that satisfy the MERF principle minimize expected inaccuracy in Joyce’s sense; they can be externally calibrated in Shimony and van Fraassen’s sense; and they can serve as a basis for rational betting, unlike probabilist degrees of belief, which, in many cases, human beings have no rational way of ascertaining. The author also uses the MERF principle to subsume the various epistemic akrasia principles in the literature. Finally, the author responds to Titelbaum’s argument that epistemic akrasia principles require that we be certain of some epistemological beliefs, if we are rational.

Appendices

Appendix

1.1 The general MERF principle

To state the general version of the MERF principle I need to introduce two complications to the special MERF principle in the text. The special version of the principle presupposed that, at the meta-level, the agent’s MCS has beliefs about the relevant factors. This simplified the exposition. The general version of the MERF principle replaces meta-level beliefs with meta-level confidence assignments.

Also, I may have given the impression that there is a division in the cognitive self—that there are two different selves involved in two different levels of cognitive processing. This is only a useful fiction. There is only one self. That single, unified self can evaluate not only the reliability of its ground-level processes; it can evaluate the reliability of its meta-level processes, though of course, it can’t evaluate them all at once. The difference in levels is just a heuristic to remind us that whenever the self evaluates its cognitive processes, it steps back and brackets their outputs, rather than just reasserting them.

Here then is the general principle:

• (General metacognitive expected relative frequency (MERF) Principle) An agent’s confidence assignment of c to a proposition P is in disequilibrium if: The agent assigns higher-order confidence to propositions of the following form: There exists a reliability-relevant category of cognitive processes ${ CP}_{1}$ that includes the cognitive processes responsible for the agent’s confidence assignment of c to P such that: ${ ERF}({ T_r/conf=c}, { CP}_{1}) = x$;

• and the weighted average of these higher-order confidence assignments (the sum of her confidence in each of them weighted by x) is equal to d;

• and it is not the case that $d \approx c$;

• UNLESS [Narrower Reference Class Exception] the agent assigns higher-order confidence to propositions asserting that there is a reliability-relevant categorization ${ CP}_{2}$ of the causal processes responsible for her confidence assignment to P, ${ CP}_{2} \le { CP}_{1}$, such that:

• ${ ERF}({ T_r/conf=c}, { CP}_{2}) = y$;

• and the weighted average of these higher-order confidence assignments (the sum of her confidence in each of them weighted by y) is approximately equal to c. ³⁰

This general version of the MERF principle makes it possible to explain how it could be rational for an agent, Van, to be certain about nothing, for it makes it possible for Van’s MCS to reduce all of his confidence assignments of 1.0, both ground-level and meta-level, to a value between 0 and 1.0 (and, similarly, to increase all of his confidence assignments of 0 to a value between 0 and 1.0). Here is a simple example meant only to illustrate the main idea: Let $\Phi$ be the set of all propositions to which Van assigns confidence of 1.0. I explain how the general MERF principle could lead him to adopt a confidence assignment in which every member of $\Phi$ is assigned confidence of .998, with the result that no proposition is assigned confidence of 1.0 (or 0). For simplicity, I assume that there is no relevant narrower subclass $\Phi$ for which Van’s MCS projects a different expected relative frequency of truth than for $\Phi$. In the cases of interest, Van will not assign confidence of 1.0 to the proposition that the expected relative frequency of truth of the members of $\Phi$ is .998. Van will divide his confidence among various alternative values for that expected relative frequency of truth. Here is a simple example: Van assigns confidence of .5 to each of two possibilities: that the relevant relative frequency of truth is .999 and that the relevant relative frequency of truth is .997. His confidence assignment of 1.0 to a proposition P (in $\Phi$) will be in disequilibrium when, as in this case, the weighted average of his various estimates of the expected relative frequency of truth in the members of $\Phi$ (in this case, .998) is not equal to his confidence in the individual members of $\Phi$ (in this case, 1.0). In the simplest case, the general MERF principle will require him to reduce his confidence assignment to each of the members of $\Phi$ to .998.

1.2 A proof that probabilist epistemologies are immodest

To show that all probabilist epistemologies imply that we are rationally required to assign confidence of 1.0 to at least some substantive epistemological claims, I carry out the argument for strict classical probabilism. It is easily modified to apply to non-strict classical probabilism, and even to non-classical probabilism, simply by making suitable substitutions for the variable ‘L,’ because all of these views require certainty in some propositions, and those requirements support implications to the conclusion that some substantive epistemological claims must be certain. Here is the argument for strict, classical probabilism:

• Let RC(P) = x be the relation: Rationality requires assigning confidence of x to proposition P. The central principle of any form of probabilism is:

  • (1) [prob (P) = x] iff [RC(P) = x].

    Consider L, a truth of classical logic. Strict classical probabilism requires:

  • (2) prob(L) = 1.

    From (1):

  • (3) [prob(L) = 1] iff [RC(L) = 1].

    From (2) and (3):

  • (4) RC(L) = 1 [Rationality requires assigning confidence of 1.0 to L.]

    What is the probability of (4)? To answer this question, we need to determine the probability of (2)—that is, the probability of a probability, which is a higher-order probability. In a formalism rich enough to coherently model higher-order probabilities (e.g., Skyrms 1980), from (2) it follows that:

  • (5) prob[prob(L) = 1] =1.

    We also need one more probability theorem:

  • (6) [P iff Q] $\rightarrow$ [prob(P) = prob (Q)].

    Then from (3), (5), and (6):

  • (7) prob[RC(L) = 1] = 1.

    And from (1) and (7):

  • (8) RC([RC(L) = 1] = 1.

So strict classical probabilism requires an assignment of confidence of 1.0 to it’s own non-trivial epistemological claim [RC(L) = 1]—that is, it requires a confidence assignment of 1.0 to the claim that we are rationally required to assign confidence of 1.0 to L. We can continue the construction to generate a potentially infinite list of non-trivial claims of strict classical probabilist epistemology to which strict classical epistemology requires us to assign confidence of 1.0. This shows that strict classical probabilism is an immodest epistemology.³¹ Parallel arguments show that any non-strict classical probabilist epistemology, such as Garber (1983), and even any non-classical probabilist epistemology must be immodest. So all probabilist theories, which include all Bayesian theories, are immodest. They all require rational degrees of confidence of 1.0 in at least some of their own substantive (i.e., non-trivial) epistemological claims.

A proof that departures from MERF-Defined equilibria increase expected inaccuracy

In the text, I assert that when an agent S’s confidence assignment satisfies the special or general MERF Principle and S’s MCS has opinions about the relevant relative frequencies of truth, changes to S’s confidence assignment increase its expected inaccuracy (and thus decrease its expected accuracy). Here I prove this result for the most commonly used measure of inaccuracy, the Brier score, according to which the inaccuracy of a confidence assignment of x to P equals the square of its distance from the truth value of P (i.e., 1.0 if p is true; 0 if p is not true).

The key step in the proof is to define the probabilities to be used in the definition of expected inaccuracy. Joyce correctly points out that expected inaccuracy cannot be consistently defined using non-probabilist confidence assignments in the role of probabilities (1998, pp. 589–590). I do not do so. The probabilities I use are the probabilities defined by the MCS’s expected relative frequency of truth for the narrowest relevant reference class that includes the proposition of interest. The MCS must have opinions about those relative frequencies for these probabilities to exist. So there can be no determination of expected inaccuracy without them.

I use prob to refer to the MCS’s relevant estimates of the expected relative frequencies of truth. Then the expected inaccuracy (EI) of an agent S’s confidence assignment of z to proposition P can be defined as the weighted sum of its inaccuracy if P is true $([\hbox {1}-\mathrm{z}]^{2})$, weighted by the probability that P is true [prob(P)] and its inaccuracy if P is not true $(\hbox {z}^{2})$, weighted by the probability that P is not true (prob($-$P) = [1$-$prob(p)]).³²

In the case in which S’s confidence assignment of x to P is in equilibrium, prob(P) = conf(P) = x. Therefore, the expected inaccuracy of the confidence assignment of x to P is:

• (1) EI(conf(p) = x) = $\hbox {x}[1-\mathrm{x}]^{2}$ + $\hbox {(1}-\mathrm{x)x}^{2}$ = x$\,-\,\hbox {x}^{2}$

I compare (1) with what the expected inaccuracy of S’s confidence assignment to P would be if S’s confidence in P were increased from x to [x + y] (where $y > 0$ and $0 \le [\hbox {x}\,+\,\hbox {y}] \le 1$). (The proof of the case in which S’s confidence assignment to p is decreased is exactly parallel.) Intuitively, increasing S’s confidence assignment to P will decrease the inaccuracy of the assignment if P is true and increase the inaccuracy of the assignment if P is not true. The expected inaccuracy of S’s confidence assignment of [x + y] to P is again a weighted sum of two components: the inaccuracy of the assignment of [x + y] to P if P is true $([1-[\hbox {x}\,+\,\hbox {y}]]^{2})$, weighted by the probability that P is true (in this case, x), and the inaccuracy of the assignment of [x + y] to P if P is not true $([\hbox {x}\,+\,\hbox {y}]^{2})$, weighted by the probability that P is not true (1$-$x). Thus:

• (2) EI(conf(p) = [x + y]) = $\hbox {x}(1 - [\hbox {x}+\mathrm{y}])^{2}$ + $(\hbox {1}-\mathrm{x})[\hbox {x}\,+\,\mathrm{y}]^{2} = x {-} \hbox {x}^{2}$ + $\hbox {y}^{2}$

The expected inaccuracy of the confidence assignment of $[\hbox {x}\,+\,\hbox {y}]$ to P is greater than the expected inaccuracy of the confidence assignment of x to P by the amount $\hbox {y}^{2}$. So the confidence assignment of x to P minimizes expected inaccuracy, and the farther S’s confidence to P departs from x (i.e., ${\vert }\hbox {y}{\vert }$), the greater its expected inaccuracy.