Measuring Inconsistency in Some Logics with Modal Operators
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Measuring Inconsistency in Some Logics with Modal Operators In Memoriam Boleslaw Soboci´ nski (1906–1980)
Abstract. The first mention of the concept of an inconsistency measure for sets of formulas in first-order logic was given in 1978, but that paper presented only classifications for them. The first actual inconsistency measure with a numerical value was given in 2002 for sets of formulas in propositional logic. Since that time, researchers in logic and AI have developed a substantial theory of inconsistency measures. While this is an interesting topic from the point of view of logic, an important motivation for this work is also that some intelligent systems may encounter inconsistencies in their operation. This research deals primarily with propositional knowledge bases, that is, finite sets of propositional logic formulas. The goal of this paper is to extend the concept of inconsistency measure in a formal way to sets of formulas with the modal operators “necessarily” and “possibly” applied to propositional logic formulas. We use frames for the semantics, but in a way that is different from the way that frames are commonly used in modal logics, in order to facilitate measuring inconsistency. As a set of formulas may have different inconsistency measures for different frames, we define the concept of a standard frame that can be used for all finite sets of formulas in the language. We do this for two languages. The first language, AM P L, contains formulas where a prefix of operators is applied to a propositional logic formula. The second language, CM P L, adds connectives that can be applied to AM P L formulas in a limited way. We show how to extend propositional logic inconsistency measures to such sets of formulas. Finally, we define a new concept, weak inconsistency measure, and show how to compute it. Keywords: Inconsistency measure, Modal operator, Standard frame.
1.
Introduction
The first mention of the concept of measuring the inconsistency of a set of formulas in logic was given in 1978 [4]. However, that paper defined not a specific inconsistency measure, but classifications for theories (sets of statements) in first-order logic. The first specific inconsistency measure was given I am grateful to Professor Soboci´ nski, who as the Editor-In-Chief of the Notre Dame Journal of Formal Logic in the 1970s helped me publish several papers including reference [4].
Presented by Heinrich Wansing; Received December 12, 2019
Studia Logica https://doi.org/10.1007/s11225-020-09917-y
c Springer Nature B.V. 2020
J. Grant
in 2002 [5] for finite sets of propositional logic formulas. Since that first attempt, logic and AI researchers developed a theory of propositional logic inconsistency measures, including new measures, properties of such measures, as well as expressibility and complexity issues. We refer to the survey [7] for additional information, as it contains much of this research and has 46 references. Some of the researchers working in this field are motivated by the fact that intelligent systems, as created in AI, ma
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