# Circumscribing Inconsistency

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2 BESNARD & SCHAUB 151

3 tion v, given in the first two rows, assigns o to (the conjunction however just an indication and should not be confused with the actual ordering relation on models which is based on set inclusion! A preferred model is indicated by boldface typesetting. For a complement, take a look at clause set Moreover, we can show that truthful parts are never polluted by contradictions: As illustrated below, the last theorem extends in some cases This theory induces the truth-values given in Table 2. Among is neither expected to carry over to the case where 1 is inconsistent. A salient property of our approach is that it is monotonic on inconsistent premises: For those familiar with [Priest, 1989], we note that this approach has {A : o, B : /} as a second preferred model, which denies conclusion B. See Section 4 for details. The example illustrates further the aforementioned extendibility of Theorem 2.4: Despite the inconsistency of A, we derive B from the consistent premises A and ->A V B. Actually, things do not necessarily change by orienting the above disjunctions as implications: 152 AUTOMATED REASONING

4 Semantically, the move from l- to ll- amounts to minimizing the set of premises with truth-value o. That is, we prefer models that assign truth-value o to a minimal set of premises. We can turn this idea into the syntax by using a connective indicating that a formula has a truth value which is less than the one of another formula. As anticipated in Section 2, such a connective can be defined as follows: 3 Proof theory This section presents a formal proof system for our approach to circumscribing inconsistency. In analogy to the semantics, we first axiomatize li- and then we account for minimization by providing a syntactic axiom schema, so that the resulting system axiomatizes ll- The axiomatization of lh consists of modus ponens as inference rule and the following axiom schemas: This induces the following truth table corresponding to the poset of truth-values on the right hand side. With this connective, we are now ready to express the following circumscription schema providing a syntactic account for BESNARD & SCHAUB 153

5 For illustration, let us return to our initial example 4 Related work There are a number of proposals addressing inconsistent information. At first, there is the wide range of paraconsistent logics [Priest et a/., 1989]. As opposed to our approach, such logics usually fail to identify with classical logic when the set of premises is consistent. There are also many approaches dealing with classical reasoning from consistent subsets. In a broader sense, this includes also belief revision and truth maintenance systems. A comparative study of the aforementioned approaches in general is given in [Besnard, 1991]. hood is then limited to models containing a minimal number of prepositional variables being assigned o. As our approach, this allows for drawing "all classical inferences except where inconsistency makes them doubtful anyway" [Priest, 1989]. There are two major differences though: First, the aforementioned restriction of modelhood focuses on models as close as possible to 2-valued interpretations, while the one in our approach aims at models next to 2-valued models of the considered formula. The effects of making the formula select its The difference between our approach and "reasoning from maximal consistent subsets of the premises" is that we still pay attention to one objection motivating relevant logics [Anderson and Belnap, 1975] and that is applying disjunctive syllogism to contradictory premises. However, we do not go as far as sanctioning any classical inference not using inconsistent subformulas. That is, we still follow the principle of relevant logics that an inference rule is a priori applicable to any premise. This is in contrast with the idea of restricted access logic [Gabbay and Hunter, 1993], where all classical inference rules are admitted with some special application conditions. Among others, logic programming with inconsistencies was addressed in [Blair and Subrahmanian, 1988; 1989]. [Wagner, 1991] describes a procedural framework for handling contradictions that relies on the notions of "support" and "acceptance". The former avenue of research is further developed in [Grant and Subrahmanian, 1995], where it is shown how the approach of [Blair and Subrahmanian, 1988] can be extended by classical inferences, like reasoning by cases. Intuitively, the corresponding entailment relations amount to logic programming in a 3-valued (and 4-valued, respectively) logic. The major difference to our approach is that compared to classical entailment, these approaches are sound but not complete (even when the set of premises is consistent). As with other approaches, this is because they aim at paraconsistent reasoning in a logic programming setting that does not necessarily coincide with classical logic. Our approach is clearly semantical in contrast to many other proposals to paraconsistency: (i) the idea of "forgetting" literals [Kifer and Lozinskii, 1989; Besnard and Schaub, 1996]; (ii) the idea of stratified theories [Benferhat et al., 1993]; (iii) the idea of reliability relation [Roos, 1992], (iv) and more generally the idea of reasoning from consistent sub- 154 AUTOMATED REASONING

6 sets of the premises. In contrast to [Tlirner, 1990], where the baseline is to analyze propositions (so as to resolve paradoxes about truth, for instance), we simply apply a system of truthvalues so that we can have non-trivial inconsistent premises. Moreover, our approach is purely deductive, as opposed to argumentation-based frameworks, like [Wagner, 1991; Elvang and Hunter, 1995]. An unusual approach to reasoning from inconsistency is due to [Lin, 1996], who introduces the notion of consistent belief by means of modal operators. This approach fails to satisfy reflexivity (not every premise is concluded). 5 Conclusion We presented a semantical approach to dealing with inconsistent knowledge bases that is founded on the minimization of three-valued models. This was complemented by a formal proof system accomplishing model minimization by appeal to a circumscription axiom. The distinguishing features of our approach are (i) its desire to provide models making true (instead of true and false) as many as possible items of the knowledge base, (ii) its centering on inferences drawn by modus ponens by means of a primitive implication connective, and (iii) its property of restricted monotonicity. A major further development will be lifting the approach to the first-order case. In this context, we draw the reader's attention to the fact that our approach (unlike [Priest, 1989]) does not rely on the notion of an atomic proposition, which is always problematic when passing from the propositional case to the first-order case. References [Anderson and Belnap, 1975] A. Anderson and N. Belnap. Entailment: The Logic of Relevance and Necessity. Princeton University Press, [Arieli and Avron, 1994] O. Arieli and A. Avron. Logical bilattices and inconsistent data. In Logic in Computer Science Conf, pp ,1994. [Arieli and Avron, 1996] O. Arieli and A. Avron. Automatic diagnoses for properly stratified knowledge-bases. In Int. Conf. on Tools with Artificial Intelligence, pp IEEE Press, [Belnap, 1977] N. Belnap. A useful four-valued logic. In J. Dunn and G. Epstein, eds, Modern Uses of Multiple- Valued Logic. Reidel, [Benferhat et al., 1993] S. Benferhat, D. Dubois, & H. Prade. Argumentative inference in uncertain and inconsistent knowledge bases. In Int. Conf on Uncertainty in Artificial Intelligence, pp ,1993. [Besnard and Schaub, 1996] P. Besnard and T. Schaub. A simple signed system for paraconsistent reasoning. In European Workshop on Logics in Artificial Intelligence, pp Springer Verlag, [Besnard, 1991] P. Besnard. Paraconsistent logic approach to knowledge representation. In World Conf on Fundamentals of Artificial Intelligence, [Blair and Subrahmanian, 1988] H. Blair and V.S. Subrahmanian. Paraconsistent foundations of logic programming." Journal of Non-Classical Logics, 5(2):45-73,1988. [Blair and Subrahmanian, 1989] H. Blair and V.S. Subrahmanian. Paraconsistent logic programming. Theoretical Computer Science, 68(2): ,1989. [Carnielli etal, 1991] W. Carnielli, L. Fariftas del Cerro, and M. Lima Marques. Contextual negations and reasoning with contradictions. In Int. Joint Conf on Artificial Intelligence, pp Morgan Kaufmann, [Elvang and Hunter, 1995] M. Elvang and A. Hunter. Argumentative logics: reasoning with classically inconsistent information. Journal of Knowledge and Data Engineering, 16: ,1995. [Gabbay and Hunter, 1993] D. Gabbay and A. Hunter. Restricted access logics for inconsistent information. In European Conf on Symbolic and Quantitative Approaches to Reasoning and Uncertainty. Springer Verlag, [Grant and Subrahmanian, 1995] J. Grant and V.S. Subrahmanian. Reasoning in inconsistent knowledge bases. IEEE Transactions on Knowledge and Data Engineering, 7(1): ,1995. [Kifer and Lozinskii, 1989] M, Kifer and E. Lozinskii. RI: A logic for reasoning with inconsistency. In Logic in Computer Science, pp ,1989. [Lin, 1987] F. Lin. Reasoning in the presence of inconsistency. In AAAI Nat. Conf on Artificial Intelligence, pp AAAI/MIT Press, [Lin, 1996] J. Lin. A semantics for reasoning consistently in the presence of inconsistency. Artificial Intelligence, 86(1-2):75-95, [McCarthy, 1980] J. McCarthy. Circumscription a form of nonmonotonic reasoning. Artificial Intelligence, 13(1-2):27-39,1980. [Priest etal, 1989] G. Priest, R. Routley, and J. Norman, editors. Paraconsistent Logics. Philosophica Verlag, [Priest, 1979] G. Priest. Logic of paradox. Journal of Philosophical Logic, 8: ,1979. [Priest, 1989] G. Priest. Reasoning about truth. Artificial Intelligence, 39: ,1989. [Roos, 1992] N.Roos. A logic for reasoning with inconsistent knowledge. Artificial Intelligence, 57:69-103,1992. [Sandewall, 1985] E. Sandewall. A functional approach to non-monotonic logic. Computational Intelligence, 1:80-87,1985. [Turner, 1990] R. Tbrner. Truth and Modality for Knowledge Representation. Pitman, [Wagner, 1991] G. Wagner. Ex contradictione nihil sequitur. In Int. Joint Conf on Artificial Intelligence, pp Morgan Kaufmann, BESNARD & SCHAUB 155

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