When Do Introspection Axioms Matter for Multi-Agent Epistemic Reasoning?


Authors
Yifeng Ding
University of California, Berkeley
Wesley H. Holliday
University of California, Berkeley
Abstract
The early literature on epistemic logic in philosophy focused on reasoning about the knowledge or belief of a single agent, especially on controversies about "introspection axioms" such as the 4 and 5 axioms. By contrast, the later literature on epistemic logic in computer science and game theory has focused on multi-agent epistemic reasoning, with the single-agent 4 and 5 axioms largely taken for granted. In the relevant multi-agent scenarios, it is often important to reason about what agent A believes about what agent B believes about what agent A believes; but it is rarely important to reason just about what agent A believes about what agent A believes. This raises the question of the extent to which single-agent introspection axioms actually matter for multi-agent epistemic reasoning. In this paper, we formalize and answer this question. To formalize the question, we first define a set of multi-agent formulas that we call agent-alternating formulas, including formulas like Box_a Box_b Box_a p but not formulas like Box_a Box_a p. We then prove, for the case of belief, that if one starts with multi-agent K or KD, then adding both the 4 and 5 axioms (or adding the B axiom) does not allow the derivation of any new agent-alternating formulas—in this sense, introspection axioms do not matter. By contrast, we show that such conservativity results fail for knowledge and multi-agent KT, though they hold with respect to a smaller class of agent-nonrepeating formulas.
Keywords epistemic logic  modal logic  multi-agent  higher-order knowledge  introspection axioms  KK principle  positive introspection  negative introspection  uncertainty
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References found in this work BETA

Modal Logic.James W. Garson - 2009 - Stanford Encyclopedia of Philosophy.
People with Common Priors Can Agree to Disagree.Harvey Lederman - 2015 - Review of Symbolic Logic 8 (1):11-45.

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