Justification logics, logics of knowledge, and conservativity
type: Item
instance of: scholarly article
title: Justification logics, logics of knowledge, and conservativity
zbMATH Open document ID: 1173.03013
author: Melvin Fitting
published in: Annals of Mathematics and Artificial Intelligence
publication date: 2009-07-06T00:00:00Z
review text: The author first reminds the reader of the axiomatizations of the main logics of knowledge, namely, \textbf{T} (in which agents have the ability to apply logical reasoning to their beliefs, which are necessarily true), \textbf{S4} (in which agents also know they know what they know), and \textbf{S5} (in which agents also know they don't know what they don't know). He then points to the problematic fact that they exhibit the property of logical omniscience (agents know all tautologies and know all logical consequences of what they know), and introduces justification logics, originating from Artemov's work on the arithmetic semantics of propositional intuitionistic logic, that make use of justification terms, to build formulas of the form \(t:X\) to intuitively express that \(t\) is a justification for proposition \(X\). Four operators can be applied to justification terms, with the following intended meanings: if \(t\) and \(u\) are justifications for \(X\) and \(X\supset Y\), respectively, then \(t\cdot u\) is a justification for \(Y\); if \(t\) and \(u\) are justifications for \(X\) and \(Y\), respectively, then \(t+u\) is a justification for both \(X\) and \(Y\); if \(t\) is a justification for \(X\) then \(!t\) is a justification for \(t:X\); and if \(t\) is not a justification for \(X\) then \(? t\) is a justification for \(\neg(t:X)\). Justification terms include constants, intended to justify what one wants to justify without further analysis, and their applicability determined by one of 3 axioms (depending on whether they can justify axioms only or also theorems, and on whether they can justify justified propositions). Capturing the intended meanings of these operators gives rise to axiomatic systems that parallel \textbf{T}, \textbf{S4} and \textbf{S5}. The author outlines the possible worlds semantics for justification logics he proposed in 2005, and discusses the difference between \textit{explicit} and \textit{implicit} knowledge: replacing in a formula o…
Mathematics Subject Classification ID: 03B42
Mathematics Subject Classification ID: 03B45
Mathematics Subject Classification ID: 03F45
zbMATH DE Number: 5576023
zbMATH Keywords: justification logic
zbMATH Keywords: logic of knowledge
zbMATH Keywords: modal logic
zbMATH Keywords: conservativity
reviewed by: Éric Martin
MaRDI profile type: Publication
full work available at URL: s10472-009-9112-2
OpenAlex ID: W2072402123
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DOI: 10.1007/S10472-009-9112-2
description: scientific article; zbMATH DE number 5576023