home >> research >> induction and confirmation: formal approaches
The probability calculus is not the universal logic of induction; there is no such thing. An axiom system disassembles the probability calculus into distinct notions about induction, which it is urged, may be invoked independently to tailor a logic of induction to the problem at hand. The probability calculus fails as the inductive logic of certain indeterministic systems. "Probability Disassembled," British Journal for the Philosophy of Science, 58 (2007), pp. 141-171. Download.
Dual Venn diagram Forming the dual is a familiar operation in logic and mathematics. Truth is the dual of falsity; and (A or B) is the dual of (A and B). Here I develop the corresponding notion for additive measures, such as probability measures. The resulting dual additive measures are degrees of disbelief and turn out to obey their own peculiar calculus. An ignorance state is conveniently characterized as one that is self-dual. "Disbelief as the Dual of Belief," International Studies in the Philosophy of Science, 21(2007), pp. 231-252. Download.
graph What if, like me, you don't think that the probability calculus is the One, True Logic of Induction? Then you want to know what other logics are possible. Here I map out a large class of inductive logics that originate in the idea that the inductive support B affords A, that is "[A|B]," is defined in terms of the deductive relations among propositions. I demonstrate some very general properties for these logics. In large algebras of propositions, for example, inductive independence is generic in all of them. A no-go result forces all the logics to supplement the deductive relations among propositions with intrinsically inductive structures. "Deductively Definable Logics of Induction," Journal of Philosophical Logic. 39 (2010), pp. 617-654. Download.

For a less formal development, see "What Logics of Induction are There?" in Goodies.
mona lisa Non-trivial calculi of inductive inference are shown to be incomplete. That is, it is impossible for a calculus of inductive inference to capture all inductive truths in some domain, no matter how large, without resorting to inductive content drawn from outside that domain. Hence inductive inference cannot be characterized merely as inference that conforms with some specified calculus. "A Demonstration of the Incompleteness of Calculi of Inductive Inference" British Journal for the Philosophy of Science70 (2019), pp. 1119–1144. Download.

"The Ideal of the Completeness of Calculi of Inductive Inference: An Introductory Guide to its Failure" Draft
jumping flea An infinite lottery machines chooses without favor among a countable infinity of outcomes. This sort of selection creates well-known problems for probability theory. But is it really physically possible to construct such a machine?. "How to Build an Infinite Lottery Machine" 8 (2018), pp. 71-95.
(with Alexander R. Pruss) Correction to John D. Norton “How to Build an Infinite Lottery Machine, ” European Journal for Philosophy of Science. 8 (2018), pp. 143-44.
Dial fragment All efforts to design an infinite lottery machine using ordinary probabilistic randomizers fail. This failure is not a result of a lack of imagination in design. It is assured by a familiar problem in set theory: we know no way to construct probabilistically nonmeasurable sets. "How NOT to Build an Infinite Lottery Machine." Studies in History and Philosophy of Science. 82(2020), pp. 1-8. Download.
Lottery ticket fragment
Matt Parker and I disagree on whether a multiple of four is less likely than an even number in a drawing from an infinite lottery.
"An Infinite Lottery Paradox" Axiomathes 32, supplement issue 1 (2022), (Special Issue Epistemologia 2022), pp. 1-6. Download.

Here is an illustration of how it is possible in a principled way to devise a strict weakening of the probability calculus that is non-additive and not prone to the Bayesian's notorious problem of the priors. It represents an interesting technical exercise, but I regard it as superceded by the analyses of the later papers above. "The Theory of Random Propositions," Erkenntnis, 41, 1994, pp. 325-352.