Formula-Preferential Systems for Paraconsistent Non-Monotonic Reasoning (an extended abstract)

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1 Formula-Preferential Systems for Paraconsistent Non-Monotonic Reasoning (an extended abstract) Abstract We rovide a general framework for constructing natural consequence relations for araconsistent and lausible nonmonotonic reasoning. The framework is based on referential systems whose references are based on the satisfaction of formulas in models. The framework encomasses different tyes of referential systems that were develoed from different motivations of araconsistent reasoning and non-monotonic reasoning, and reveals an imortant link between them. 1 Introduction For a long time the research efforts on araconsistency and on nonmonotonic reasoning were searated. The former research dealt with the question of how to revent the inference of every fact from an inconsistent source of knowledge, and how to isolate inconsistent arts of the knowledge and yet work in the usual way with the consistent arts. The latter dealt with the question of how to jum to conclusions based on artial knowledge of the domain (this is needed since having comlete knowledge is often unrealistic), and how to revise revious hasty conclusions in the face of new and fuller information. However, in recent years the formal connections between these two areas have begun to be revealed. It is only natural that such a connection would exist, because conclusions that are drawn based on artial knowledge may contradict new and more reliable information, and each new iece of information may contradict revious information and hence force us to revise some of our knowledge. As the famous examle goes, if we conclude that Tweety can fly based on the sole fact that it is a bird, the new iece of information that Tweety is a enguin and enguins cannot fly forces us not only to revise revious conclusions but also to deal with the fact that we now have a contradiction in our knowledge. Both goals of handling contradictions and reasoning nonmonotonically require some selection between alternatives: which arts of the knowledge to retain and which to discard or change. A central tool in both fields has been referential systems, meaning that only a subset of the models should be relevant for making inferences from a theory. These models are the most referred ones according to some criterion. Arnon Avron and Iddo Lev School of Comuter Science Tel-Aviv University Ramat Aviv 69978, Israel faa,iddolevg@ost.tau.ac.il In the research on araconsistency, referential systems were used for constructing logics which are araconsistent but stronger than substructural araconsistent logics. The references in these systems were defined in different ways. Some were based on checking which abnormal formulas (such as ^ : ) are satisfied in the models of a theory (see e.g. [Priest, 1991; Batens, 1998]). Others were based on references between the truth values that are assigned to formulas (see e.g. [Kifer and Lozinskii, 1992; Arieli and Avron, 2000a]). Preferential systems were also used for roviding semantics for nonmonotonic consequence relations (see e.g. [Shoham, 1987; Kraus et al., 1990; Makinson, 1994]). It was discovered, however, that in order for them to satisfy all the desired theoretical roerties that lausible nonmonotonic relations should have (see e.g. [Lehmann, 1992]), referential systems need to satisfy a further condition called stoeredness or smoothness. The roblem is that this condition is usually not easy to verify. In this aer we rovide a general framework for constructing natural consequence relations for araconsistent and lausible nonmonotonic reasoning. The main technique is using referential systems in which the reference between models is made according to a certain set of formulas which are satisfied in them. The framework encomasses different tyes of referential systems that were used for constructing useful araconsistent consequence relations. Moreover, these natural referential systems that were originally designed for araconsistent reasoning satisfy the stoeredness condition as well, and hence have also the desired theoretical roerties of nonmonotonic consequence relations. As we said, the theoretical research on nonmonotonic reasoning and the research on araconsistent reasoning have been conducted searately at first. Nevertheless, formulareferential systems, which are a generalization of methods used in the latter, solve a key issue in the former, and hel to bridge the ga between the two directions of research and to combine them under a unified framework. This rovides strong evidence for their imortant rule in non-classical reasoning. 2 Preliminaries In what follows L is a language, W is its set of wffs, ; ; denote formulas of L, and?; denote sets of formulas.

2 When the language is roositional, A denotes its set of roositional variables, and ; q; r denote such variables. Definition A semantic structure for a language L is a air S = hm S ; j= S i, where j= S M S W. M S is a set of models and j= S is called a satisfaction relation. A model m 2 M S satisfies a formula if m j= S. m is a model of? (m j= S?) if it satisfies every formula in?. The set of the models of? is denoted by mod(?; S). is a consequence of? in S (? `S ) if for every m 2 mod(?; S), m j= S for some 2. It is easy to verify that every semantic structure induces a monotonic consequence relations (mcr in short, i.e. it satisfies reflexivity: if?\ 6= ; then? `, monotonicity: if? ` and?? 0, 0 then? 0 ` 0, and cut: if? ` ; and? 0 ; ` 0 then?;? 0 ` ; 0 ). A common tye of semantic structures for roositional logics is the class of multi-valued matrices. In these structures the value that a valuation assigns to a comlex formula is uniquely determined by the values that it assigns to its subformulas. However, an agent acting in the real world often has only incomlete or inconsistent knowledge to guide its decisions. One ossible aroach for dealing with this roblem is to borrow the idea of non-deterministic comutations from automata and comutability theory, and aly it for assigning truth-values to comlex formulas. Here we use a natural generalization of the logical concet of a matrix the value that a valuation assigns to a comlex formula can be chosen non-deterministically from a certain nonemty set of otions: Definition 2.2 [Avron and Lev, 2000] A non-deterministic matrix (Nmatrix for short) for a roositional language L is a tule S = ht ; D; Oi, where T is a non-emty set of truth values, D is a non-emty roer subset of T (its designated values), and for every n-ary connective, O includes a corresonding n-ary function e from T n to 2 T? f;g. A valuation in S is a function v : W! T that satisfies the condition: if is an n-ary connective, and 1 ; : : : ; n 2 W, then v(( 1 ; : : : ; n )) 2 e(v( 1 ); : : : ; v( n )). V S denotes the set of valuations of S. The satisfaction relation j= S V S W is defined: v j= S iff v( ) 2 D. We identify the Nmatrix S with the semantic structure hv S ; j= S i. Every (deterministic) matrix can be identified with an Nmatrix whose functions in O always return singletons. In addition to their obvious otential for reasoning under uncertainty and for secification and verification of nondeterministic rograms, N-matrices have considerable ractical technical alications. It is well known that every roositional logic can be characterized semantically using a multivalued matrix ([Łos and Suszko, 1958]). However, there are imortant logics whose characteristic matrices necessarily consist of an infinite number of truth values, and are thus of little hel in roviding decision rocedures for their logics, or in getting real insight into them. Our generalization of the concet of a matrix allows us to relace in many cases an infinite characteristic matrix for a given roositional logic by a characteristic finite structure that automatically rovides a decision rocedure. 1 See e.g. [Makinson, 1994; Lehmann, 1992]. A rime examle for such a case is the Nmatrix S >. It is defined in the classical roositional language with the connectives f^; _; ;:; fg. The interretation of all the connectives excet for : is the classical one (e.g. xe^y = ftg if x = y = t and ffg otherwise), whereas negation is a non-deterministic oeration: e:f = ftg but e:t = ft; f g (a formula and its negation may both be assigned t in a valuation). The mcr induced by S > can be characterized using the Gentzen-tye calculus that is obtained from Gentzen s original calculus (in [Gentzen, 1969]) for classical logic (including cut) by omitting the rule [: )] for introducing negation on the left. The consequence relation induced by S > cannot be induced by any finite matrix. Moreover, any two-valued Nmatrix which has at least one roer nondeterministic oeration does not have an equivalent finite matrix. We shall later mention nonmonotonic logics which are based on the underlying araconsistent monotonic logics induced by S > as well as the well-known four-valued matrix S 4 of [Belna, 1977] with the truth-values ft; f; >;?g (in the classical roositional language) and its submatrix S > with the values ft; f; >g (t; > are designated). S > is the maximal araconsistent logic that contains `os (ositive classical logic), whereas S > is the minimal logic that contains `os 2 and in which `> ; : for all. The following result will be imortant for our framework in section 4: Theorem 2. Every finite Nmatrix is finitary. Nonmonotonic Consequence Relations In recent years there has been a wide study of theoretical roerties that nonmonotonic consequence relations should satisfy (see e.g. [Arieli and Avron, 2000b] for a list of such works). Here we shall use the following notion: Definition.1 4 Let ` be an mcr. A binary relation j between sets of formulas and sets of formulas is called `- lausible if it satisfies the following conditions: Ext `-extension: for every?; 6= ;, if? ` then? j. RM right monotonicity: if? j and 0 then? j 0. LCM left cautious monotonicity: if? j for every 2? 0, and? j then?;? 0 j. LCC left cautious cut: if? j ; for every 2 and?; j then? j. RCC right cautious cut: if?; j for every 2 and? j ; then? j. A central method for roviding semantics to lausible nonmonotonic consequence relations has been the use of referential systems. The idea of referential systems (which began in [Shoham, 1987]) is that instead of using all the models of a given theory for checking which conclusions follow from 2 The mcr induced by S > is the same as CLuN from [Batens et al., 1999] and the logic K=2 of [Béziau, 1999]. See [Avron and Lev, 2000]. 4 After [Lehmann, 1992; Arieli and Avron, 2000b], with slightly different names and conditions.

3 it, the models are ordered by a reference relation, and only the most referred models are used as relevant for making inferences from the theory. Notation.2 If A is a set with a re-order, x y denotes x y and y 6 x. Min (A) = fx 2 A j 8y 2 A: y 6 xg. Definition. 5 Let S be a semantic structure. 1. A referential system in S is a air P = hs; i, where is a re-order on M S. 2. A model m 2 mod(?; S) is a P-referential model of? if m 2 mod(?; P) = Min (mod(?; S)).. A set of formulas? P-referentially entails a set of formulas (notation:? `P ) if for every m 2 mod(?; P) there is a 2 s.t. m j= S. 6 `P is called the consequence relation 7 induced by P. Definition.4 Let A be a set with a re-order. A is stoered under if every x 2 A has x 0 2 Min (A) s.t. x 0 x. Definition.5 8 A referential system P = hs; i is stoered if for all?, mod(?; S) is stoered under. Theorem.6 9 If P is a stoered referential system in S then `P is `S-lausible. Note: The stoeredness condition is introduced because some referential systems which are not stoered do not satisfy the condition LCM of Definition.1 (although the other conditions are always satisfied by all referential systems). As noted in [Kraus et al., 1990; Makinson, 1994], it is usually not easy to check whether a referential system is stoered. Preferential systems were originally develoed as a framework for roviding semantics for nonmonotonic inference relations. They were also used, aarently indeendently at first, for constructing systems for reasoning with inconsistencies (and other abnormalities) in a way which is on the one hand non-trivial and on the other hand not as weak as monotonic substructural logics (see e.g. [Priest, 1991; Kifer and Lozinskii, 1992; Batens, 1998]). Interestingly, these ideas, which were develoed from motivations different from stoeredness will rovide us with methods for constructing stoered referential systems. 4 Formula-Preferential Systems Formula-referential systems are a generalization of the minimal-abnormality strategy from [Batens, 1998]. That aer uses a secific selection of models from S >. Denoting K(v) = f 2 W cl j v( ^ : ) = tg, a model v of? is selected iff there is no other model v 0 of? s.t. K(v 0 ) K(v). In this way the minimal-abnormality strategy minimizes the 5 Following [Makinson, 1994; Lehmann, 1992]. 6 Note that we do not require that m 2 mod(fg; P), or that m 2 mod(? [ fg; P). 7 The term consequence relation here is more general than an mcr. In articular, we do not assume monotonicity. 8 Following [Makinson, 1994]. In [Kraus et al., 1990; Lehmann, 1992] this is called smoothness. 9 A Generalization of a result in [Arieli and Avron, 2000b]. abnormalities (here inconsistencies) in the models of a theory (by abnormality we mean a formula that leads to triviality w.r.t. a desired logic, here classical logic). Our generalization is to choose some set G of formulas in the language, and to have the referential system select those models of a theory that minimize the satisfaction of formulas from G. Formula-referential systems can be defined w.r.t. any set G of formulas, and also in any semantic structure, since what is imortant for the reference relation between the models is which formulas from G they satisfy, and not their inner structure. Formally: Notation 4.1 Let S be a semantic structure and let G W. For m 2 M S denote: SatS;G(m) = f 2 G j m j= S g. Definition 4.2 Let G W. A formula-referential system based on G is a referential system P = hs; i that satisfies: for all m 1 ; m 2 2 M S, m 1 m 2 iff SatS;G(m 1 ) SatS;G(m 2 ). P is called in short a G-referential system. In this way formula-referential systems rovide a natural source of stoered referential systems. The formulas in G exress the undesired roerties which we would like to minimize in the referred models, and the nonmonotonic consequence relations that these systems induce satisfy the conditions of Definition.1 whenever they are based on a finitary semantic structure: Theorem 4. If P is a formula-referential system in a finitary semantic structure then P is stoered. Corollary 4.4 If P is a formula-referential system in a finitary semantic structure S then `P is `S-lausible. Corollary 4.5 If P is a formula-referential system in a finite Nmatrix S then `P is `S-lausible. The last Corollary follows from Theorem 2. and Corollary 4.4. Since in ractice one usually works with finite structures, this means that this result has great ractical significance. We mention now some known systems from the literature which can be constructed using formula-referential systems. All of them are based on finite Nmatrices, so by Corollary 4.5 their induced consequence relations are lausible. Whenever the underlying monotonic logics are araconsistent, so are the induced nonmonotonic relations. Closed-World Assumtion In the Closed-World Assumtion method [Reiter, 1978], a roositional variable that cannot be roved is assumed to be false. A corresonding formula-referential system is defined in the classical two-valued matrix and is based on A cl (the roositional variables of the language). The obtained consequence relation is nonmonotonic but not araconsistent. Preferential systems for handling contradictions `> is araconsistent, but it is too weak for adequate reasoning, e.g. the Disjunctive Syllogism (from, : _ infer ) is not valid in it, even on classically consistent sets. A consequence relation that is located between `> and classical logic can be obtained by using the formula-referential system P = hs > ; i that is based on G = f ^ : j 2 A clg. `P is the same as LPm of [Priest, 1991] (when S > is without ) and ACLuNs2 of [Batens, 1998]. It is nonmonotonic,

4 araconsistent, and in contrast to `>, it is the same as classical logic on classically consistent sets. Adative Logics Adative logics [Batens, 1998; 2000] were originally introduces by dynamic roof systems that are designed to mimic some asects of human reasoning with inconsistencies, esecially the fact that conclusions that are drawn at a certain stage may be rejected at a later stage because of other conclusions, and then even acceted again. The name adative is due to the fact that these logics adat their rules to the given set of remises. E.g. the Disjunctive Syllogism is not valid in `>. Its use is not allowed by ACLuNs2 on? = fr; :r; :r _ s; ; : _ qg for inferring s (since r behaves inconsistently) but it is allowed for inferring q (since there is no reason to suose that behaves inconsistently). Adative logics that are based on the minimal-abnormality strategy are a secial case of the formula-referential systems where the set G is taken as a set of abnormal formulas. For examle, ACLuN2 (note: not ACLuNs2) is induced by the formula-referential system in S > that is based on G = f ^ : j 2 W cl g. ACL;2 from [Batens, 1999] is based on the two-valued Nmatrix S 0 in which all the connectives of the classical language are weakened: for an n-ary connective 2 cl = f^; _; ;:; fg and any x 2 ft; fg n, e(x) = ft; fg. S 0 has in addition the connectives and & which function in S 0 as classical negation and conjunction. P 0 is the formulareferential system in S 0 that is based on the set G 0 : the set that includes all formulas which exress the fact that a certain formula ( 1 ; : : : ; n ) and one or more of 1 ; : : : ; n are assigned values that are illegal in a classical valuation, e.g. & :, ( & ) & ( ^ ), ( & ) & ( ), etc. In comarison to ACLuN2, ACL;2 is adative on all the connectives in cl, not only :. Other adative logics (see [Batens, 2000]) use a formulareferential system P in a more comlicated way: the definition of the adative logic j is:? j iff T r(?) `P T r(), where T r is some re-rocessing of the formulas. 5 Pointwise-referential systems [Arieli and Avron, 2000b] suggests another method for constructing referential systems that are stoered. The method is based on a tye of referential systems called ointwise referential systems. The underlying idea is to have a reference between the truth values of a multile-valued structure and to base the reference between the valuations on this reference. For examle, in the (N)matrix S 4, we might refer the classical values t and f over > and?, since a valuation satisfies exactly one of and : iff it assigns a classical value to. If there are two models for a given set of remises and they assign the same values to all atomic formulas excet that one assigns t to and the other >, we might refer the first. This is the underlying idea of the following definition: Definition Let S be an Nmatrix with a set of truth values T, and let be a re-order on T. A ointwise referential 10 A generalization of ointwise referential systems from [Arieli and Avron, 2000b], which are in our notations strongly ointwise referential systems in matrices. system (in S) based on is a referential system P = hs; i that satisfies the condition: for all v 1 ; v 2 2 V S, v 1 v 2 iff for every roositional variable, v 1 () v 2 (). If is a artial-order, P is called strongly ointwise. P will be called in short a -referential system. [Arieli and Avron, 2000b] shows that ointwise referential systems that are based on well-founded artial orders are stoered and hence induce lausible relations. Pointwise referential systems are in general a different tye of systems than formula-referential systems. Nevertheless, by adding certain connectives to the language, we can construct for each ointwise referential system a formulareferential system that induces the same consequence relation and, in a certain sense, has the same reference relation. A consequence of this embedding is that the finitariness of the underlying semantic structure ensures the stoeredness roerty: Definition 5.2 Let S = ht ; D; Oi be a Nmatrix for a roositional language L, and let L 0 be a roositional language with the same variables as L but with additional logical connectives. An extension of S to L 0 is a Nmatrix S 0 = ht ; D; O 0 i for L 0 s.t. O 0 O. A valuation v 0 in S 0 is an extension of a valuation v in S to L 0 if v and v 0 agree on W. Definition 5. Let S = ht ; D; Oi be an Nmatrix for L and let P = hs; i be a -referential system. A formulareferential system associated with P is P 0 = hs 0 ; 0 i for the language L 0, where L 0 is like L but with the added or defined connectives fi x j x 2 T g, S 0 is an extension of S to L 0 with the same truth values s.t. for every x; y 2 T, I e x y D if y x and I e x y T? D otherwise, and P 0 is based on G = fi x j x 2 T ; 2 Ag. Note: For all valuations v in S 0, v j= S 0 I x iff v( ) x. Theorem 5.4 Let P = hs; i be a -referential system and let P 0 = hs 0 ; 0 i be an associated formula-referential system. 1. For all?; W,? `P iff? `P For all v 1 ; v 2 2 V S, v 1 v 2 iff for each of their (resective) extensions v 0 1 ; v0 2 2 V S 0 to L 0, v v 0 2. Note: for each x 2 T that is a least element (x y for all y 2 T ), defining G without any formula I x will give the same result, since such x guarantees that v j= S 0 I x for all v, and so the resence of these formulas in G does not influence the reference relation. Corollary 5.5 If P is a ointwise referential system in a finitary Nmatrix S then P is stoered and `P is `Slausible. The ointwise referential systems from [Arieli and Avron,, so they 2000a] are based on the finite matrices S 4 and S > can be embedded in formula-referential systems, and their induced consequence relations are therefore lausible. The first tye of systems is based on the idea of minimal knowledge. In S 4, it is based on the artial order k :? < k (t; f) < k >. The corresonding formula-referential system (in S 4 ) is based on the set that includes I > = ^ :, I t =

5 , I f = :, I? =, for all 2 A cl. According to the remark above, I? is redundant. In this articular case I > is also redundant, i.e. it is enough to take G = A cl [ f: j 2 A cl g. Note that contrary to the original motivation behind the minimal-abnormality strategy of [Batens, 1998], in this system (as well as in CWA of section 4) we do not regard the formulas in G as abnormal (in articular, all the variables are in G), but rather as the formulas whose satisfaction we want to minimize in the models. The second tye of systems from [Arieli and Avron, 2000a] is based on the idea of minimal inconsistency: it is based on choosing a subset of inconsistent truth values I, and defining the re-order I : x 1 I x 2 iff x 1 2 T? I or x 2 2 I. The referential system from section 4 that corresonds to ACLuNs2 can be defined as the I -referential system in S > where I = f>g. If T = ft; f; >g, this is the only inconsistency set. In the general case, there may be other in- = f>g and consistency sets. For examle, in S 4, both I 1 I 2 = f>;?g are inconsistency sets, and they induce different consequence relations. Let P i (i = 1; 2) be the ointwise Ii -referential system in S 4. P 1 can be embedded in the formula-referential system (in S 4 ) that is based on the set G that includes I > = ^ : for all 2 A cl and I t = I f = I? =. According to the remark above, only the formulas I > are necessary. For P 2, the set includes I > = I? = ( :) ^ (: ) for all 2 A cl (and I t = I f = are redundant). 6 Conclusion Our main goal in this aer was to demonstrate the central role of formula-referential systems in non-classical reasoning. We have shown how different systems from the literature for reasoning in the face of inconsistencies and other abnormalities, can be constructed in this framework. Moreover, although most of these systems were not originally art of the theoretical research of nonmonotonic consequence relations, the generalization of their reference relations to the idea of formula-referential systems rovides us with a method for ensuring the condition of stoeredness: formula-referential systems that are based on finitary semantic structures are stoered, and hence satisfy theoretical desiderata for a lausible nonmonotonic logic. All the examles from the literature that we have given are of this kind since they are based on finite non-deterministic matrices. References [Arieli and Avron, 2000a] O. Arieli and A. Avron. Bilattices and araconsistency. In [Batens et al., 2000], ages [Arieli and Avron, 2000b] O. Arieli and A. Avron. General atterns for nonmonotonic reasoning: from basic entailments to lausible relations. Logic Journal of the IGPL, 8(2): , [Avron and Lev, 2000] A. Avron and I. Lev. Nondeterministic matrices Submitted. [Batens et al., 1999] D. Batens, K. 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