Probabilistic Justification Logic

Joseph Lurie

doi:10.3390/philosophies3010002

Abstract

Justification logics are constructive analogues of modal logics. They are often used as epistemic logics, particularly as models of evidentialist justification. However, in this role, justification (and modal) logics are defective insofar as they represent justification with a necessity-like operator, whereas actual evidentialist justification is usually probabilistic. This paper first examines and rejects extant candidates for solving this problem: Milnikel’s Logic of Uncertain Justifications, Ghari’s Hajek–Pavelka-Style Justification Logics and a version of probabilistic justification logic developed by Kokkinis et al. It then proposes a new solution to the problem in the form of a justification logic that incorporates the essential features of both a fuzzy logic and a probabilistic logic.

Highlights

There are justification analogues of normal justification logics that contain axioms not found in LP; these are created by extending the justification logic syntax with additional operators on proof polynomials, which are used to formulate justification analogues of the required modal axioms
One is generally considered justified in believing a proposition if the probability given by the available evidence exceeds a certain epistemic threshold. This understanding of evidence is widely accepted, but the question of how this threshold is determined has spawned a major debate in epistemology, with at least three major competing views: contextualism, which claims that the threshold is set by the conversational context of an utterance of “I know that p”; subject-sensitive invariantism, which claims that conversational contexts do not affect the epistemic threshold, but that the threshold does vary depending on the real-world situation of an epistemic agent; and pure invariantism, which fixes a single epistemic threshold for all agents in all epistemic circumstances
Kokkinis et al [21,22] develop a class of genuinely probabilistic justification logics using methods analogous to Milnikel’s

Summary

Justification Logic

Justification logics are constructive analogues of modal logics. Syntactically, justification logics are built from the following constituents:. Simple axiom justification If φ is an instance of one of the axiom schemas included in a particular system, there is a proof constant c such thatc : φ. An evidence relation for the system LP is a relation E between the set of proof polynomials and the set of sentences, which satisfies the following: If φ is an instance of any axiom schema in the axiomatic presentation of LP given above, there is a proof constant c such that E (c, φ). Justification analogues of other normal modal logics are well known; Artemov [6] provides a good overview of how to construct these systems. There are justification analogues of normal justification logics that contain axioms not found in LP; these are created by extending the justification logic syntax with additional operators on proof polynomials, which are used to formulate justification analogues of the required modal axioms. The same changes that are used to transform LP into an analogue of any other desired modal logic can be applied directly to the system presented in Section 3 below

Epistemology

Probability Theory and Fuzzy Logic

Milnikel’s Logic of Uncertain Justifications

Kokkinis’ Probabilistic Justification Logic6

Ghari’s Hájek-Pavelka-Style Justification Logics

Probabilistic Justification Logic