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Determining Maximal Entropy Functions for Objective Bayesian Inductive Logic

Landes, Juergen, Rafiee Rad, Soroush, Williamson, Jon (2022) Determining Maximal Entropy Functions for Objective Bayesian Inductive Logic. Journal of Philosophical Logic, . ISSN 0022-3611. E-ISSN 1573-0433. (doi:10.1007/s10992-022-09680-6) (KAR id:97427)

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Official URL:
https://doi.org/10.1007/s10992-022-09680-6

Abstract

According to the objective Bayesian approach to inductive logic, premisses inductively entail a conclusion just when every probability function with maximal entropy, from all those that satisfy the premisses, satisfies the conclusion. When premisses and conclusion are constraints on probabilities of sentences of a first-order predicate language, however, it is by no means obvious how to determine these maximal entropy functions. This paper makes progress on the problem in the following ways. Firstly, we introduce the concept of a limit in entropy and show that, if the set of probability functions satisfying the premisses contains a limit in entropy, then this limit point is unique and is the maximal entropy probability function. Next, we turn to the special case in which the premisses are categorical sentences of the logical language. We show that if the uniform probability function gives the premisses positive probability, then the maximal entropy function can be found by simply conditionalising this uniform prior on the premisses. We generalise our results to demonstrate agreement between the maximal entropy approach and Jeffrey conditionalisation in the case in which there is a single premiss that specifies the probability of a sentence of the language. We show that, after learning such a premiss, certain inferences are preserved, namely inferences to inductive tautologies. Finally, we consider potential pathologies of the approach: we explore the extent to which the maximal entropy approach is invariant under permutations of the constants of the language, and we discuss some cases in which there is no maximal entropy probability function.

Item Type: Article
DOI/Identification number: 10.1007/s10992-022-09680-6
Subjects: B Philosophy. Psychology. Religion > BC Logic
Q Science > QA Mathematics (inc Computing science) > QA273 Probabilities
Divisions: Divisions > Division of Arts and Humanities > Department of Philosophy
Funders: [247] The Leverhulme Trust
Deutsche Forschungsgemeinschaft (https://ror.org/018mejw64)
Depositing User: Jon Williamson
Date Deposited: 13 Oct 2022 21:16 UTC
Last Modified: 14 Oct 2022 08:52 UTC
Resource URI: https://kar.kent.ac.uk/id/eprint/97427 (The current URI for this page, for reference purposes)
Williamson, Jon: https://orcid.org/0000-0003-0514-4209
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