Cory Juhl

Learning Theory and the Philosophy of Science

This paper places formal learning theory in a broader philosophical context and provides a glimpse of what the philosophy of induction looks like from a learning-theoretic point of view. Formal learning theory is compared with other standard approaches to the philosophy of induction. Thereafter, we present some results and examples indicating its unique character and philosophical interest, with special attention to its unified perspective on inductive uncertainty and uncomputability.

The speed-optimality of Reichenbach's straight rule of induction

In his Theory of Probability, Hans Reichenbach made a bold and original attempt to ‘vindicate’ induction. He proposed a rule, the ‘straight rule’ of induction, which would guarantee inductive success if any rule of induction would. A central problem facing his attempt to vindicate the straight rule is that too many other rules are just as good as the straight rule if our only constraint on what counts as ‘success’ for an inductive rule is that it is ‘asymptotic’, i.e. that it converges in the limit to the true limiting frequency (of some type of outcome O in a sequence of events) whenever such a limiting frequency exists. In this paper I consider the consequences of requiring speed-optimality of asymptotic methods, that is, requiring that inductive methods must get to the truth as quickly as possible. Two main results are proved: (1) the straight rule is speed-optimal; (2) there are (uncountably) many non-speed-optimal asymptotic methods. A further result gives a sufficient but not necessary condition for speed-optimality among asymptotic methods. Some consequences and open questions are then discussed.

Objectively reliable subjective probabilities

Subjective Bayesians typically find the following objection difficult to answer: some joint probability measures lead to intuitively irrational inductive behavior, even in the long run. Yet well-motivated ways to restrict the set of “reasonable” prior joint measures have not been forthcoming. In this paper I propose a way to restrict the set of prior joint probability measures in particular inductive settings. My proposal is the following: where there exists some successful inductive method for getting to the truth in some situation, we ought to employ a (joint) probability measure that is inductively successful in that situation, if such a measure exists. In order to do show that the restriction is possible to meet in a broad class of cases, I prove a “Bayesian Completeness Theorem”, which says that for any solvable inductive problem of a certain broad type, there exist probability measures that a Bayesian could use to solve the problem. I then briefly compare the merits of my proposal with two other well-known proposals for constraining the class of “admissible” subjective probability measures, the “leave the door ajar” condition and the “maximize entropy” condition.

Is Gold-Putnam diagonalization complete?

Diagonalization is a proof technique that formal learning theorists use to show that inductive problems are unsolvable. The technique intuitively requires the construction of the mathematical equivalent of a “Cartesian demon” that fools the scientist no matter how he proceeds. A natural question that arises is whether diagonalization iscomplete. That is, given an arbitrary unsolvable inductive problem, does an invincible demon exist?The answer to that question tunas out to depend upon what axioms of set theory we adopt. The two main results of the paper show that if we assume Zermelo-Fraenkel set theory plus AC and CH, there exist undetermined inductive games. The existence of such games entails that diagonalization is incomplete. On the other hand, if we assume the Axiom of Determinacy, or even a weaker axiom known as Wadge Determinacy, then diagonalization is complete.In order to prove the results, inductive inquiry is viewed as an infinitary game played between the scientist and nature. Such games have been studied extensively by descriptive set theorists. Analogues to the results above are mentioned, in which both the scientist and the demon are restricted to computable strategies.The results exhibit a surprising connection between inductive methodology and the foundations of set theory.

Realism, Convergence, and Additivity

In this paper, we argue for the centrality of countable additivity to realist claims about the convergence of science to the truth. In particular, we show how classical sceptical arguments can be revived when countable additivity is dropped.

On the empirical inaccessibility of higher-level modality and its significance for cosmological fine-tuning

In this paper we propose that cosmological fine-tuning arguments, when levied in support of the existence of Intelligent Designers or Multiverses, are much less interesting than they are thought to be. Our skepticism results from tracking the distinction between merely epistemic or logical possibilities on one hand and nonepistemic possibilities, such as either nomological or metaphysical possibilities, on the other. We find that fine-tuning arguments readily conflate epistemic or logical possibilities with nonepistemic possibilities and we think that this leads to treating the search for an explanation of fine-tuning as analogous to standard empirical theorizing about first-order nomological matters, when in fact the two investigational enterprises are profoundly different. Similar conflation occurs when fine-tuning arguments do not carefully distinguish between different interpretations of probabilities within the arguments. Finally, these arguments often rely on spatial analogies, which are often misleading precisely in that they encourage the conflation of epistemic and nonepistemic possibility. When we pay attention to the distinctions between merely epistemic versus nonepistemic modalities and probabilities, the extant arguments in favor of intelligent designers or multiverses, or even for the nonepistemic improbability of fine-tuning, consist of empirically unconstrained (beyond what is entailed by facts about the actual universe) speculation concerning relevant nonepistemic modal facts.