Justification logics with counterfactual and relevant conditionals
Meghdad Ghari
TL;DR
The paper develops a family of conditional justification logics by combining Artemov’s justification logic with Chellas-style conditional logic, yielding $ extsf{LPC}^+$ as a central system. It constructs relational semantics that unify justification and conditional notions via formula-indexed and term-indexed state relations, and proves completeness via canonical relational models; several extensions and fragments, including internalization ($ extsf{LPC}^{int}$), a 4′-based variant ($ extsf{LPC}^{ extprime}$), a conditioned knowledge logic with RCEA ($ extsf{LPCK}^+$), and a belief fragment ($ extsf{J4C}^+$), are developed with corresponding model theories and completeness results. The framework is then applied to formalize Nozick’s counterfactual conditions, explore their connections to Aumann knowledge, and analyze Gettier and McGinn counterexamples; a Gettier-resolving model and the famous sheep-in-the-field example are presented to illustrate limitations of Nozick and JTB within this setting. A tableau system for a relevant-counterfactual variant (JRC) with the variable-sharing property is introduced and shown complete relative to Routley relational models, further enriching the toolkit for epistemic analysis under counterfactual and relevant conditionals. Overall, the work provides a robust, modular formal apparatus for investigating knowledge, justified belief, and counterfactual reasoning across multiple epistemic viewpoints.
Abstract
The purpose of this paper is to introduce justification logics based on conditional logics. We introduce a new family of logics, called conditional justification logics, which incorporates a counterfactual conditional in its language. For the semantics, we offer relational models that merge the selection-function semantics of conditional logics with the relational semantics of justification logics. As an application, we formalize Nozick's counterfactual conditions in his analysis of knowledge and investigate their connection to Aumann's concepts of knowledge. Additionally, we explore Gettier's counterexamples to the justified true belief analysis, as well as McGinn's counterexamples to Nozick's analysis of knowledge. Furthermore, we introduce a justification logic that includes a relevant counterfactual conditional and we demonstrate the variable-sharing property for this conditional. We also develop a tableau system for this logic and establish its completeness with respect to Routley relational models. Finally, we formalize Nozick's counterfactual conditions using this relevant counterfactual conditional and represent the sheep in the field example of Chisholm within this logic.