M. Fitting "Kleene's three-valued logics and their children". Fundamenta Informaticae, 20, 113-131, 1994  Home/Search   Document Not in Database   Summary   Related Articles   Check

.... val (l) val (l) Also, for every propositional constant x 2 ft; f; c; ug that is associated with an element x2ft; f; g in FOUR, we de ne i (x) x (i = 0; 1; The basic operator used here for handling contradictory information is the k join, As it is noted in [21,24], this operator may be associated with a gullibility ( accept all ) function that computes the combined knowledge of its arguments. The choice of this operator may be intuitively justi ed, then, by the need to do the following: a) record cases in which there is an evidence for a speci c ....

....a uniform approach for a diversity of applications in arti cial intelligence. In particular, he treated rst order theories and their consequences, truth maintenance systems, and formalisms for default reasoning. The algebraic structure of bilattices has been further investigated by Fitting [21,24] and Avron [9] In a series of paper Fitting has also shown that bilattices are very useful tools for providing semantics for logic programs. He proposed an extension of Smullyan s tableauxstyle proof method to bilattice valued programs, and showed that this method is sound and complete with ....

M.Fitting, Kleene's three-valued logics and their children, Fundamenta Informaticae 20 (1994) 113-131.

....that do not contradict any previously drawn conclusions are allowed. For achieving the goals above we consider an algorithmic approach that is based on a four valued semantics [3, 4] Using a multiple valued semantics is a common way to overcome the shortcomings of classical calculus (see, e.g. [3, 6, 7, 12 14]) and as we shall see in what follows, four valued semantics is particularly suitable for our purpose. A similar algorithmic approach for recovering stratified knowledge base, which is also based on a four valued semantics, was introduced in [1, 2] Here we generalize and improve that approach ....

M.Fitting. Kleene's three-valued logics and their children. Fundamenta Informaticae 20, pages 113--131, 1994.

.... the truth value of formulas about which there is inconsistent data (such a formula is both true and false ) while is the truth value of formulas on which no data at 7 Belnap s structure is nowadays known also as the basic bilattice, and its logic as the basic bilattice logic (see [31, 32, 26, 25, 27, 28, 3, 4, 5]) all is available ( neither true nor false ) With these intuitions, the interpretations of the connectives are exactly the same as in the 3 valued case (so : is defined exactly as above, and , are interpreted as the max and min operations according to the following partial ....

M. Fitting. Kleene's three-valued logics and their children. Fundamenta Informaticae, 20:113--131, 1994.

....The notion of a bilattice was introduced by Ginsberg in [Gi88] as a general framework for a diversity of applications, such as truth maintenance systems, default inferences and others. The notion was further investigated and applied for logic programming by Fitting ( Fi89] Fi90] Fi91] [Fi94]) The main idea behind bilattices is to use structures in which there are two partial order relations, having different interpretations. There should, of course, be a connection between the two relations. Ginsberg uses for this an operation of negation which is order preserving w.r.t. one, an ....

....(2) 3.7 Proposition Suppose B is a (complete, distributive) IBL with negation. Then there exist a (complete, distributive) bounded lattice L, so that hB; i is equivalent to LfiL, equipped with Ginsberg s negation (i.e. x; y) y; x) 9 Proof The proof is practically identical to that in [Fi94] for the distributive case: The function g defined in the proof of 3.3 (g(x) x ; x ) is an isomorphism of B on L B fi R B . On the other and the function x:x is easily seen to be an isomorphism between the lattices L B and R B . Hence the function h defined by h(x; y) x; y) is an ....

Fitting M., Kleene's three-valued logics and their children. Fundamenta Informaticae Vol.20 (pp. 113-131); 1994.

....that some well known plausible nonmonotonic logics can be constructed using this method. Most of these logics are paraconsistent as well (these include some logics that we have considered in previous works [2, 3, 5] 2 This is a common method for dealing with inconsistent theories see, e.g. [13, 14, 15, 20, 21, 23, 34, 35, 39, 42, 43]. General Patterns for Nonmonotonic Reasoning 121 2 Preferential systems from an abstract point of view In this section we investigate preferential reasoning from an abstract point of view. First we briefly review the original treatments of Makinson [28] and Kraus, Lehmann, and Magidor [24] ....

M.Fitting. Kleene's three-valued logics and their children. Fundamenta Informaticae, Vol.20, pages 113--131, 1994.

....then OE j 0 . cautious cut: if j 0 OE and OE j 0 , then j 0 . left logical equivalence: if cl OE and j 0 , then OE j 0 . right weakening: if cl , OE and j 0 , then j 0 OE. 2 This is a common method for dealing with inconsistent theories see, e.g. [13, 14, 15, 20, 21, 23, 34, 35, 39, 42, 43]. 3 A conditional assertion in terms of [24] 3 Definition 2 [24] A cumulative relation j 0 is called preferential if it is closed under the following rule: introduction (Or) if j 0 and OE j 0 , then OE j 0 . Note In order to distinguish between the rules of Definitions 1, ....

M.Fitting. Kleene's three-valued logics and their children. Fundamenta Informaticae, Vol.20, pages 113--131, 1994.

....the generality. Lemma 2.2 [3] For every formula there is a finite set S of clauses such that for every valuation , 2f ; tg iff (OE) 2f ; tg for every OE 2S. 1 (ft; f; g; k ) is also a lattice, and so a k meet and a k join operations might be defined on FOUR as well (see, e.g. [2, 4, 6, 7, 13, 14, 16]) 2 2.3 Measurement of consistency Notation 2.3 Inc( fp j (p) g. Definition 2.4 Let 1 ; 2 2V. a) 1 is more consistent than 2 iff Inc( 1 ) ae Inc( 2 ) b) M 2mod(KB) is a most consistent model of KB (mcm, for short) if there is no other model of KB which is more consistent than M . ....

....is to use more than just four values. This will allow us, e.g. to view truth values as representing probabilities, confidence factors, etc. One possible way of doing so is to use bilattices [16] which are algebraic structures, with arbitrary number of truth values, that naturally generalize FOUR [2, 13, 14]. Such extensions will be considered in a future work. Acknowledgment I would like to thank Arnon Avron for helpful discussions on the topics of this paper. 10 Recall that here j= W is either j= cc W or j= ci W . 11 ....

M.Fitting. Kleene's three-valued logics and their children. Fundamenta Informaticae 20, pages 113--131, 1994.

....[Gi88] to the general concept of a bilattice. He proposed Bilattices as a basis for a general framework for many applications. Bilattices were further investigated by Fitting, who used them for extending some well known logics (like Kleene 3 valued logics) and for logic programming (see, e.g. [Fi90, Fi91, Fi94]) In [AA96] the set D is also generalized to what is called there a bifilter, and bilattices based logics are introduced. It turned out, however, that from a logical point of view, FOUR has among bilattices the same role that the two valued Boolean algebra has among Boolean algebras. It is ....

M. Fitting, Kleene's three-valued logics and their children. Fundamenta Informaticae, Vol.20, pp. 113--131; 1994.

....does not reduce the generality. Lemma 2.2 [3] For every formula there is a finite set S of clauses such that for every valuation , 2 f ; tg iff (OE) 2f ; tg for every OE 2S. 1 (ft; f; g;k ) is also a lattice, and so a k meet and a k join operators might be defined as well (see, e.g. [2, 5, 6, 7, 13, 14, 16, 17]) 2.3 Measurement of consistency Definition 2.3 Inc( fp j (p) g. Definition 2.4 Let 1 ; 2 2V. a) 1 is more consistent than 2 iff Inc( 1 ) ae Inc( 2 ) b) M 2 mod(KB) is a most consistent model of KB (mcm, for short) if there is no other model of KB which is more consistent than M . ....

....next natural step is to allow more than just four values. This will allow us, 10 Recall that j=W stands here for either j= cc W or j= ci W . e.g. to use truth values that represent probabilities, confidence factors, etc. One possible way of doing so is to use bilattices [16, 17] see also [2, 13, 14]) which are algebraic structures that naturally generalize FOUR. The idea is to consider arbitrary number of truth values, and to arrange them (as in FOUR) in two closely related partial orders, each forming a lattice. Such extensions will be considered in a future work. Acknowledgment I would ....

M.Fitting. Kleene's three-valued logics and their children. Fundamenta Informaticae, Vol.20, pp. 113--131, 1994.

....by Ginsberg in [Gi88] as a general framework for applications in AI. Specifically, Ginsberg considered their use in first order theories, truth maintenance systems, and formalisms for default reasoning. The algebraic structure of bilattices has been further investigated by Fitting and Avron [Fi90b, Fi94, Av96]. Fitting has shown that bilattices are very useful tools for providing semantic to logic programs [Fi90a, Fi91, Fi93] He proposed an extension of Smullyan s tableauxstyle proof method to bilattice valued programs, and showed that this method is sound and complete with respect to a natural ....

M.Fitting. Kleene's three-valued logics and their children. Fundamenta Informaticae, Vol.20, pages 113-131, 1994.

....then OE j 0 . Note: In order to distinguish between the rules of Definitions 2.1, 2. 2, and their generalized versions that will be considered in the sequel, the condition above will usually be preceded by the 1 This is a common method for dealing with inconsistent theories see, e.g. [Gi87, Gi88, Pr89, Su90, Fi90, Fi91, Pr91, KL92, Fi94, Su94, Sc96]. 2 A conditional assertion in terms of [KLM90] string KLM . Also, a relation that satisfies the rules of Definition 2.1 [Definition 2.2] will sometimes be called KLM cumulative [KLM preferential] The conditions above might look a little bit ad hoc. For example, one might ask himself why ....

M.Fitting. Kleene's three-valued logics and their children. Fundamenta Informaticae, Vol.20, pages 113--131, 1994.

....orders, each forming a lattice. The original motivation of Ginsberg for introducing bilattices was to provide a uniform approach for a diversity of applications in AI. Bilattices were further investigated by Fitting, who showed that they are useful also for providing semantic for logic programs [Fi90a, Fi91, Fi93, Fi94]. In [AA94, AA96] we presented bilattice based logics and corresponding proof systems. These logics turned out to have many desirable properties (like paraconsistency) In the present paper we proceed with this logical approach. In particular, we consider bilattice based logics that are ....

....logics and Fitting s guard connective) The meet and the join in FOUR with respect to t correspond to the conjunction and disjunction of strong Kleene s logic. In order to represent the connectives of the other Kleene s three valued logics (weak Kleene 3 and sequential Kleene 4 ) Fitting [Fi94] introduces a new connective, called the guard connective. This connective is denoted p : q, and is evaluated as follows: if p is assigned a designated value (t or ) the value of p : q has the value of q, otherwise p : q has the value . The guard connective has the following simple and natural ....

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M.Fitting. Kleene's three-valued logics and their children. Fundamenta Informaticae, Vol.20, pages 113-131, 1994.

....cut elimination procedure (like natural deduction or free deduction) could give a framework for studying the algorithmic meaning of Belnap s logic. ffl Comparison with other sequent calculi for three valued logic [10, 14] ffl Extension to the case of truth spaces that are (general) bilattices [6, 7]. 2 PROPOSITIONAL BELNAP S LOGIC 2.1 THE SET OF TRUTH VALUES The set of truth values is the complete bilattice Four = ff; t; g together with the two orderings t and k as presented in the Introduction. Complete Sets of Connectives. In classical logic, the connectives : and form a ....

FITTING m., Kleene's Three-Valued Logics and their Children. Fundamenta Informaticae (1993) forthcoming.

....that contain two such partial orders simultaneously (see definition 2.1) His motivation was to present a general framework for many applications, like truth maintenance systems and default inferences. This notion was further investigated and applied for various properties by Fitting (see [Fit1] [Fit6] The present paper has two main goals: The first is to develop proof systems, which correspond to bilattices in an essential way. For this purpose we have found it useful to introduce and investigate the notion of a logical bilattice. All the bilattices which were actualy proposed for ....

....k corrsponds to differences in our knowledge about formulae and not to their truth values. see [Gins] for further discusion) Definition 2.2 A bilattice is called distributive [Gins] if all the twelve possible distributive laws concerning , Omega , and Phi hold. It is called interlaced [Fit1] if each one of , Omega , and Phi, is monotonic with respect to both t and k . Lemma 2.3 [Fit1] Every distributive bilattice is interlaced. Example 2.4 The bilattices FOUR and NINE (figure 1) are both distributive bilattices 1 , while Ginsberg s DEFAULT [Gins] figure 2) is not even ....

[Article contains additional citation context not shown here]

M.Fitting. Kleene's Three-Valued Logics and Their Children. Proc. of the Bulgarian Kleene 90 Conference; 1990.

....a basis for a general framework for many applications such as first order theorem provers, truth maintenance systems, and implementations of default inferences. This notion was further developed by Fitting, who used bilattices for extending some well known logics (like Kleene 3 valued logics; see [Fi90a, Fi94]) and introduced bilattice based logic programming methods ( Fi89, Fi90b, Fi91, Fi93] In bilattices the elements (which are also referred to as truth values ) are arranged in two partial orders simultaneously: one, t , may intuitively be understood as a measure of the degree of truth that ....

....FOUR (if this indeed is the case) does not exclude the potential usefulness of other logical bilattices (This point, of course, is not unrelated to the first one) 3. The framework of bilattices opens the door for various nonclassical connectives (like Fitting s conflation and guard connectives [Fi94], or the nonmonotonic implications of [AA96] It is doubtful that with these extra connectives FOUR will still be sufficient for defining j= con . The discussions in this section and in the previous ones leads to several directions of research: ffl Determine the exact role of FOUR with respect ....

M.Fitting. Kleene's three-valued logics and their children. Fundamenta Informaticae, Vol.20, pages 113--131, 1994.

....as we have implied, is to switch into a special multi valued framework. For this, we use a special algebraic structures called bilattices. Bilattices were first proposed by Ginsberg (see [Gi88] as a basis for a general framework for many applications. This notion was further developed by Fitting ([Fi90a, Fi94]) who showed that bilattices are most suitable for logic programming ( Fi89, Fi90b, Fi91, Fi93] In bilattices the elements (which are also referred to as truth values ) are simultaneously arranged in two partial orders: one, t , may intuitively be understood as a measure of the degree of ....

M.Fitting. Kleene's three-valued logics and their children. Fundamenta Informaticae Vol.20 (pp. 113-131); 1994.

.... = inf(L) inf(L) LfiL = sup(L) sup(L) t LfiL = sup(L) inf(L) and f LfiL = inf(L) sup(L) The original motivation of Ginsberg for using bilattices was to provide a uniform approach for a diversity of applications in AI (see [Gi88] Fitting has further investigated these structures [Fi90a, Fi94] and showed that they are useful for providing semantic to logic programs [Fi90a, Fi91, Fi93] In [AA94, AA96] we presented a preliminary development of bilattice based logics and corresponding proof systems. These logics turned out to have a proof theory with many desirable properties. In ....

M.Fitting. Kleene's three-valued logics and their children. Fundamenta Informaticae, Vol.20, pages 113--131, 1994.

.... Aviv 69978, Israel The notion of a bilattice was first introduced by Ginsburg (see [Gin] as a general framework for a diversity of applications (such as truth maintenance systems, default inferences and others) The notion was further investigated and applied for various purposes by Fitting (see [Fi1] [Fi6] The main idea behind bilattices is to use structures in which there are two (partial) order relations, having different interpretations. The two relations should, of course, be connected somehow in order for the mathematical structure to be useful. It is not clear, however, what this ....

....all the agreed upon particular cases. The second to show that every such bilattice can be represented by a diagram as described above. This will show, I believe, that the suggested definition is adequate. At the same time it will justify this general method of representation. 2 Definition 1 [Fi1]. A prebilattice is a structure B = hB; t ; k i such that B is a nonempty set containing at least two elements, and both hB; t i and hB; k i are (complete) lattices. Notation. Following Fitting, we shall use and for the lattice operations which correspond to t , and Omega and Phi for ....

[Article contains additional citation context not shown here]

M. Fitting. Kleene's Three-Valued Logics and Their Children. Proc. of the Bulgarian Kleene 90 Conference, 1990.

....our degree of belief by the degree to which we did not doubt. There are many other examples of distributive bilattices of considerable interest. Indeed, there is a general method of construction, from [12] and also discussed and extended to take negation and conflation into account, in [6] 8] [9] and [10] We do not have space here to present it again. If B is a distributive bilattice with negation, a B valuation is a mapping from ground atoms to members of the bilattice, mapping the atom false to the truth value false. Valuations are given pointwise orderings. That is, we set v 1 ....

Melvin C. Fitting. Kleene's three-valued logics and their children. In Proceedings of the Bulgarian Kleene 90 Conference, 1990. Forthcoming.

....natural subsystems can be extracted that are analogous to the classical sublogic or the Kleene three valued sublogic of Belnap s fourvalued logic. One uses an operation called conflation for this it plays the role for # k that negation plays for # t . We do not give details here, see [13]. These generalizations of the classical or Kleene logics continue to have many of the key properties of the logics they generalize, and are interesting objects of study in themselves. 10 Metric Spaces Consider again the program of Example 16, which we repeat here for convenience. even(0) # ....

Fitting, M. C. Kleene's three-valued logics and their children. Fundamenta Informaticae 20 (1994), 113--131.

....entirely general, in the sense that every complete, infinitely distributive bilattice with a negation and a conflation that commute is isomorphic to T#Tfor some complete lattice T with a de Morgan complement. A proof of this, generalizing a representation theorem of Ginsberg, can be found in [1, 4]. 5 Annotated Revision Programs and Bilattices In section 3, T valuations were defined. It will be more convenient to work with valuations in a bilattice, and connections are easy to make. First, by a (T#T) valuation we mean a mapping from atoms (not revision atoms) to members of the ....

Fitting, M. C. Kleene's three-valued logics and their children. Fundamenta Informaticae 20 (1994), 113--131.

....B is a distributive bilattice, it is isomorphic to L 1 # L 2 , where L 1 and L 2 are distributive lattices. If B has a negation, L 1 and L 2 can be taken to be the same lattice. If B has a negation and a conflation that commute, L 1 is a de Morgan lattice. Proofs of this can be found in [19, 11, 16]. In fact, the result can be looked at as a variation on the Polarities Theorem of Dunn, which goes back to his dissertation of 1966 (see [5] A bilattice with conflation and negation that commute with each other contains some natural sublogics within it. Suppose B is such a bilattice. Call a ....

Fitting, M. C. Kleene's three-valued logics and their children. In Proceedings of the Bulgarian Kleene `90 Conference (Forthcoming).

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M. Fitting "Kleene's three-valued logics and their children". Fundamenta Informaticae, 20, 113-131, 1994

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