Classifications for inconsistent theories.
Abstract: In [2] N. C. A. da Costa surveys some interesting results about inconsistent formal systems. A formal system is said to be inconsistent if there is a formula φ such that both φ and~<ρ are theorems. The approach in [2] towards the study of inconsistent systems is basically syntactical. In this paper we investigate inconsistent theories from a model-theoretical point of view. However we do not analyze semanticaily the calculi presented in [2] as suggested on Page 508. Instead we define a notion of structure which allows for the possibility of built-in inconsistencies. These structures may then be models of inconsistent theories. We classify theories in 3 different ways. Intuitively, the higher a theory is in a classification, the more inconsistent it is. This way we obtain measures of inconsistency for theories.
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Citations
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TL;DR: This work provides a measure to quantify the inconsistency of a knowledgebase, thereby allowing for the comparison of the consistency of various knowledgebases, represented as first-order logic formulas, using quasi-classical (QC) logic.
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TL;DR: The main aim of this paper is to review the measures of information and contradiction, and to study some potential practical applications, which have significant potential in developing intelligent systems that can be tolerant to inconsistencies when reasoning with real-world knowledge.
Combining Knowledge Bases Consisting of First Order Theories
Chitta Baral,Sarit Kraus,Jack Minker,V. S. Subrahmanian +3 more
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References
On the theory of inconsistent formal systems.
Abstract: Introduction This is an expository work,* in which we shall treat some questions related to the theory of inconsistent formal systems. The exposition will be neither rigorous nor complete. For details, the reader may consult the works cited in the References. (With reference to the historical aspects of the theory, see specially [l].) In general, the terminology, the notations, etc., are those of Kleene's book [17], with evident adaptations. A formal system (deductive system, deductive theory, . . .) S is said to be inconsistent if there is a formula A of S such that A and its negation, lA, are both theorems of this system. In the opposite case, S is called consistent. A deductive system S is said to be trivial if all its formulas are theorems. If there is at least one unprovable formula in S, it is called non-trivial. If the underlying logic of a system S is the classical logic (the intuitionistic logic, . . .), then S is trivial if, and only if, it is inconsistent. Hence, employing such a category of logics, the inconsistent systems do not present any proper logico-mathematical interest. Usually, we try to change the inconsistent theories to transform them into consistent ones. It is clear that under this transformation, some characteristic properties of a given inconsistent theory must be preserved; for instance, the common formal systems of set theory preserve certain traits of inconsistent naive set theory. Nonetheless, there are certain cases in which we might think of studying directly an inconsistent theory. For example, a set theory containing RusselΓs class (the class of all classes which are not members of themselves) as an existing set, or a theory whose aim be the systematization of Meinong's theory of objects. Apparently, it would be as
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