Pure Inductive Logic

Abstract

Pure inductive logic is the study of rational probability treated as a branch of mathematical logic. This monograph, the first devoted to this approach, brings together the key results from the past seventy years plus the main contributions of the authors and their collaborators over the last decade to present a comprehensive account of the discipline within a single unified context. The exposition is structured around the traditional bases of rationality, such as avoiding Dutch Books, respecting symmetry and ignoring irrelevant information. The authors uncover further rationality concepts, both in the unary and in the newly emerging polyadic languages, such as conformity, spectrum exchangeability, similarity and language invariance. For logicians with a mathematical grounding, this book provides a complete self-contained course on the subject, taking the reader from the basics up to the most recent developments. It is also a useful reference for a wider audience from philosophy and computer science.

The Theory of Spectrum Exchangeability

“…The context described in the introduction, and based on that considered in [15], consists of a first order language L containing finitely many (possibly polyadic) relation symbols {R 1 , R 2 , . .…”

“…All the standard properties of probability functions readily follow from (P1-3), see for example [11, Proposition 2.1], [15].…”

“…Given spectram,ñ and a state description Θ with spectrumm it can be shown, see [6], [9], [10], [15], that the number of state descriptions with spectrumñ extending Θ depends only onm,ñ and not on the particular choice of Θ. We denote this number by N (m,ñ).…”

“…The framework of this paper is that of Pure Inductive Logic, PIL, as presented in [12], [15], according to which PIL is the mathematical formalization of "assigning logical, as opposed to statistical, probabilities by attempting to formulate the underlying notions, such as symmetry, irrelevance, relevance on which they appear to depend ".…”

“…Several other principles are known to have non-trivial theories, in particular the Invariance Principle INV (see [15]) and, for purely unary languages Johnson's Sufficient Postulate JSP (see [5] for an explicit description of T h(JSP)). A feature of such principles seems to be that they are rather powerful, suggesting, rather reasonably, that this power comes at the price of the agent accepting a non-tautological 'creed'.…”