Belnap, N. D. 1977. A useful four-valued logic. In Modern uses of multiple-valued logic, G. Epstein and J. M. Dunn, Eds. Reidel, Dordrecht, NL, 5--37.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....a literal, then it is forced to not satisfy the complement of the literal. Since we have decoupled the link between a formula and its negation at the level of the model, we have the basis for a semantics for paraconsistent reasoning. This intuition coincides with that of four valued logics [Bel77] However, we will not follow the four valued lattice theoretic interpretation of the connectives given in [Bel77] but instead provide a significantly different semantics. First, we define strong satisfaction. Definition 3.4 Let j= s be a satisfiability relation, called strong satisfaction, ....

....between a formula and its negation at the level of the model, we have the basis for a semantics for paraconsistent reasoning. This intuition coincides with that of four valued logics [Bel77] However, we will not follow the four valued lattice theoretic interpretation of the connectives given in [Bel77] but instead provide a significantly different semantics. First, we define strong satisfaction. Definition 3.4 Let j= s be a satisfiability relation, called strong satisfaction, such that j= s (O) Theta L clauses . For X 2 (O) we define j= s as follows, and ff 1 ; ff n are literals in ....

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N Belnap. A useful four-valued logic. In G Epstein, editor, Modern Uses of Multiple-valued Logic, pages 8--37. Reidel, 1977.

....and truth value assignment functions [6, 9] Possible worlds and truth value assignment functions reflect the knowledge contained in objects. The accessibility relations captures structural relationships between objects. Four truth values are used to capture incompleteness and inconsistency [4]. We follow two routes to formalise the knowledge augmentation process. The first approach defines a different set of modal operators, referred to as augmented knowledge modal operators, one for each of the objects forming a structured document. An augmented knowledge operator is used to model ....

BELNAP, N. A useful four-valued logic. In Modern Uses of Multiple-valued Logic, J. Dunn and G. Epstein, Eds. Reidel, Dordrecht, 1977.

....A number of specific propositional multi valued algebras have been proposed and studied. For example, L ukasiewicz [L uk70] first introduced a 3 valued logic to allow for propositions whose truth values are unknown . and Kleene [Kle52] introduced several alternative 3 valued algebras. Belnap [Bel77] proposed a 4 valued logic that also introduces the value both (i.e. TRUE and FALSE) to handle inconsistent assertions in database systems. Each of these logics can be generalized to allow for additional levels of uncertainty or disagreement. The class of quasi boolean algebras considered in ....

....a b means b is more true than a . Further, of the lattice are interpreted as values TRUE and FALSE of the logic, respectively. A quasi boolean algebra defined over the lattice in Figure 2. 2(d) is the Belnap s 4 valued logic which has been used for reasoning about inconsistent databases [Bel77, AB75] Here, N and = B. The underlying lattice is isomorphic to the one in Figure 2.2(c) but the resulting quasi boolean algebras are not. The lattice in Figure 2.2(e) represents the set of propositional formulas over p , referred to as PF ( p ) The logic defined over this lattice is ....

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N.D. Belnap. "A Useful Four-Valued Logic". In Dunn and Epstein, editors, Modern Uses of Multiple-Valued Logic, pages 30--56. Reidel, 1977. 83

.... for a 2 A [ s) s) s) s) u [ s) s) s) t [ s) EX ] s) t2S (R(s; t) u [ t) The semantics of the EX operator comes from extending the notion of existential quantification for multi valued reasoning through the use of disjunction [1, 16]. The other operators are defined as their CTL counterparts (see Section 2.1) where and are interpreted as lattice t and u, respectively. 6 We also introduce bounded versions of EU and EG operators. E[ U i ] s) s) if i = 0 [ EXE[ U i 1 ] s) if i 0 [ EG i ....

N.D. Belnap. "A Useful Four-Valued Logic". In Dunn and Epstein, editors, Modern Uses of Multiple-Valued Logic, pages 30--56. Reidel, 1977.

.... degree of truth ( true is more true than false ) and in their degree of knowledge (to know that a fact is false is more information than to know that it is default true ) The smallest non trivial bilattice is the well known lattice of four valued logic FOUR, first introduced by Belnap [Belnap, 77] has been called consensus operator, gullible operator [Fitting, 91] These operators match two values wrt their degree of information. The values of example 2.1 can be structured in D 1 , the reader who is familiar with bilattices will realize the bilattice of default reasoning) figure 2. ....

Belnap N.D., jr.: "A useful four-valued logic". J.M. Dunn and G. epstein (eds.) Modern uses of Multiple-valued logics. Reidel, 8-37, 1977.

....the least upper bound. As usual in the lattice theory, lub imposes a partial order on BSL: a b i# b = lub(a, b)anda bi# a b and a is di#erent from b.Twotypicalexamples of BSL (which happen to be complete lattices) are shown in Figure 1. In both Lattice with Defaults [8] 4 valued Lattice [2,3] dt df d# t f # # # t f Fig. 1. Typical Belief Semilattices of them, the lattice elements are ordered upwards. The specific BSL used in this paper is introduced later, in Figure 2. Thus, the only syntactic di#erence between APC and classical predicate logic is that the atomic formulas of ....

N. Belnap. A Useful Four-Valued Logic. In M. Dunn and G. Epstein, editors, Modern Uses of Multi-Valued Logic, pages 8--37. Reidel Publ. Co., 1977.

....logics for giving semantics to logic programs is discussed and justi ed in Sections 2.2 and 4.2. The rest of this paper is organized as follows: in the next section we represent our framework. In particular, we consider some semantical aspects such as showing that Belnap four valued structure [11,12] is particularly suitable for representing the kind of information we intend to decode in logic programs. In Section 3 we introduce our xpoint theory, rst for logic programs without negation as failure, and then for the general case. In Section 4 we further generalize our formalism to cases in ....

.... literals (i.e. literals that may be preceded by not) The complement of a literal l is denoted by l (that is, if l =p for some atom p then l = p, and if l = p then l =p) As usual in the context of logic programming, we shall deal with formulae in a clausal form, as de ned below: See, e.g. [5,6,11,12,32,34,39] for some non classical methods for reasoning with partial or contradictory information. De nition 2.1. Let n m 0. A positive clause is a formula of the form p p 1 ; p n A standard clause is a formula of the form p p 1 ; p m ; not p m 1 ; not p n A normal ....

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N.D.Belnap, A useful four-valued logic, in: Modern Uses of Multiple-Valued Logic, eds. G.Epstein and J.M.Dunn, Reidel Publishing Company, 1977, pp. 7-37.

....b( F . So, if the above axioms are taken together with (C1) C2) then we obtain a natural dyadic semantics for LFI1. Also in [9] two slightly di erent (nongentzenian) bivaluation semantics for LFI1 were explored. 10 Example 4. 4 Belnap s paraconsistent and paracomplete 4 valued logic (cf. [2]) B 4 = hf0; 3 , 3 ; 1g; f: g; f 3 ; 1gi, can be presented by way of the following matrices: 3 2 : 0 3 1 0 3 2 3 1 0 0 0 0 0 3 2 3 2 3 2 0 3 2 3 1 3 2 3 1 3 1 3 2 1 1 1 1 1 Clearly, p separates 1 and 3 and also separates 3 and 1, so: ....

N. D. Belnap. A useful four-valued logic. In J. M. Dunn, editor, Modern uses of multiple-valued logic, pages 8-37. D. Reidel Publishing, Boston, 1977.

.... [155] they are based on model enumeration and do not yield a notion of proof (besides making difficult to re use past computations) Notice that we should distinguish between logics or deduction for approx imate reasoning, such as fuzzy logic by Ying [183] or the multi valued logic of Belnap [14] from approximating deduction. Multi valued logics aim at mod eling uncertainty and imprecise concepts, and the corresponding satisfiability problem has been proven NP complete by Mundici [131] Beside the limitation to clausal normal form, which is common to almost all tractability results in ....

.... Socratic completeness . Here the problem is to identify the formulae that must be used with the cut rule. The dual approach is to weaken the logical by accepting the omniscience in a weaker logic than classical logic. The (now standard) approach is to use a variant of Belnaps four valued logic [14]. This approach has been initially proposed by Levesque [104] has been refined by a number of researchers such as Delgrande [48] Lakemeyer [99, 100] Fagin et al. 56] This technique yields (sometimes) tractable, sound incomplete (or unsound complete) inference procedures for classical logics. ....

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N. Belnap. A useful four-valued logic. In G. Epstein and J. Dunn, editors, Modern Uses of Multiple-Valued Logic, pages 8-37. Reidel, 1977.

....and special non logical imperative predicates and are considered undesirable. There is certainly no conceptual framework within logic programming for allowing only those actions which have logical meaning. Some researchers have come close to touching upon the imperative reading of logic. Belnap [Bel77] and the later so called data semantics school regard a formula j as generating an action on a model M, and changing it. In logic programming and deductive databases the handling of integrity constraints borders on the use of logic imperatively. Integrity constraints have to be maintained. Thus ....

N. D. Belnap. A Useful Four-Valued Logic. In J. M. Dunn and G. Epstein, editors, Modern Uses of Multiple-Values Logic, pages 5--37. D. Reidel Publishing Company, 1977.

....approaches to logic programming semantics, of which we will discuss 22 two in more detail: the algebraic approach via bilattices due to Fitting, and the work of Dix. Bilattice based semantics has a long tradition in logic programming theory, starting out from the four valued logic of Belnap [Bel77]. The underlying set of truth values, a fourelement lattice, was recognized to admit two ordering relations which can be interpreted as truth and knowledge order. As such it has the structure of a bilattice, a term due to Ginsberg [Gin86] who was the rst to note the importance of bilattices for ....

Nuel D. Belnap. A useful four-valued logic. In J. Michael Dunn and George Epstein, editors, Modern Uses of Multiple-Valued Logic, pages 5-37. Reidel, Dordrecht, 1977.

....maps on the identity id and sw which permutes v and w. This gives rise to two different DeMorgan negations: id) sw) We w , v , v , w . Thus, the first negation is the classical one, a , and the second negation renders Belnap s four valued logic [2]. By Theorem 3, this lattice cannot have any other DeMorgan negations. 2. For a negative example, consider the finite partial order of priorities (V, where V = y, z and y and z are incomparable but have higher priority than x. This partial order is not self dual; in particular, there ....

N. Belnap. A useful four-valued logic. In J. M. Dunn, editor, Modern Uses of Many-Valued Logic, pages 8--37. Reidel, 1977.

....2. Thus, the only syntactic difference between APC and classical predicate logic is that the atomic formulas of APC are constructed from the classical atomic That is, the least upper bound, lub(a; b) is defined for every pair of elements a; b 2 BSL. Lattice with Defaults [8] 4 valued Lattice [2, 3] dt df d t f t f Fig. 1. Typical Belief Semilattices formulas by attaching annotation suffixes. For instance, if s; t; are elements of the belief semilattice, then p(X) s, q : and r(X; Y; Z) t all are atomic formulas in APC. We define only the Herbrand semantics of APC ....

N. Belnap. A Useful Four-Valued Logic. In M. Dunn and G. Epstein, editors, Modern Uses of Multi-Valued Logic, pages 8--37. Reidel Publ. Co., 1977.

....i a b and a is di erent from b. Two typical examples of BSL (which happen to be complete lattices) are shown in Figure 1. In both of them, the lattice elements are ordered upwards. The speci c BSL used in this paper is introduced later, in Figure 2. Lattice with Defaults [8] 4 valued Lattice [2, 3] dt df d t f t f Fig. 1. Typical Belief Semilattices Thus, the only syntactic di erence between APC and classical predicate logic is that the atomic formulas of APC are constructed from the classical atomic formulas by attaching annotation suxes. For instance, if s; t; are elements ....

N. Belnap. A Useful Four-Valued Logic. In M. Dunn and G. Epstein, editors, Modern Uses of Multi-Valued Logic, pages 8-37. Reidel Publ. Co., 1977.

....warranted by the underlying logic, while an unsound reasoner may infer information that is not warranted by the underlying logic. Typical examples of this approach include Boolean Constraint Propagation (BCP) McA80, McA90] a variant of unit resolution [CL73] and tautological entailment [Be177, Lev84b, Fri87], both of which are sound but incomplete. The general diiticulty with this approach is in characterizing (preferably syntactically) the class of queries that will be answered correctly, or the degree of error in the possibly incorrect answer. An extension to the incomplete unsound approach ....

....as follows: a theory is k R vivid iff i for it is identical to . 4.5.3 Comparison with Earlier Approaches We compare our family PE of tractable consequence relations with some other tractable consequence relations presented in the literature. 191 Relevance Logic and RP Entailment Belnap [Be177] presented a 4 valued model theory for PC, called relevance logic, whose entailment relation, say B, is strictly weaker than , the entailment relation for classical 2 valued model theory. Intuitively, relevance logic allows equivalences based on the properties of logical operators such as ....

N. D. Belnap. A useful four-valued logic. In G. Epstein and J. M. Dunn, editors, Modern Uses of Multiple-Valued Logics. Reidel, 1977.

.... examples of which includethelogicoffirst degree entailment (also known as system E)andthe relevant implication system (or system R) in which deductions between formulae hold only when there is some connection between the formulae (e.g. the formulae share some sentential variable) In [2] Belnap provides a semantic characterization of first degree entailment together with a sound and complete axiomatisation, emphasising its connection with the problem of how a computer should think [1] We provide a translation of Belnap s semantics into a set of first order logic formulae. ....

N. Belnap. A useful four-valued logic. In J. M. Dunn and G. Epstein, editors, Modern Uses of Multiple-valued Logic, pages 8--37. D. Reidel, 1977.

No context found.

Belnap, N. D. 1977. A useful four-valued logic. In Modern uses of multiple-valued logic, G. Epstein and J. M. Dunn, Eds. Reidel, Dordrecht, NL, 5--37.

No context found.

Nuel D. Belnap. A useful four-valued logic. In Gunnar Epstein and J. Michael Dunn, editors, Modern uses of multiple-valued logic, pages 5--37. Reidel, Dordrecht, NL, 1977.

No context found.

N. D. Belnap. A useful four-valued logic. In G. Epstein and J. Michael Dunn, editors, Modern uses of multiple-valued logic, pages 5--37. Reidel, Dordrecht, NL, 1977.

No context found.

N. D. Belnap. A useful four-valued logic. In G. Epstein and J. M. Dunn, ed., Modern uses of multiple-valued logic, pages 5--37. Reidel, Dordrecht, NL, 1977.

No context found.

Nuel D. Belnap. A useful four-valued logic. In Gunnar Epstein and J. Michael Dunn, editors, Modern uses of multiple-valued logic, pages 5--37. Reidel, Dordrecht, NL, 1977.

No context found.

BELNAP N. (1977). A useful four-valued logic. In J. DUNN & G. EPSTEIN, Eds., Modern uses of multiple-valued logic, volume 2 of Episteme, chapter ?, p. 8--37.

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Nuel D. Belnap. A useful four-valued logic. In Gunnar Epstein and J. Michael Dunn, editors, Modern uses of multiple-valuedlogic, pages 5--37. Reidel, Dordrecht, NL, 1977.

No context found.

N. D. Belnap. A useful four-valued logic. In G. Epstein and J. M. Dunn, editors, Modern uses of multiple-valued logic, pages 5--37. Reidel, Dordrecht, NL, 1977.

No context found.

Belnap, Jr, N. D. 1977. A useful four-valued logic. In Dunn, M., and Epstein, G., eds., Modern Uses of MultipleValued Logic. D. Reidel. 8--41.

No context found.

Nuel D. Belnap. A useful four-valued logic. In Gunnar Epstein and J. Michael Dunn, editors, Modern uses of multiple-valued logic, pages 5--37. Reidel, Dordrecht, NL, 1977.

No context found.

N. D. Belnap. A useful four-valued logic. In Modern uses of multiple-valued logic, pages 5--37. Reidel, Dordrecht, NL, 1977.

No context found.

Nuel D. Belnap. A useful four-valued logic. In Gunnar Epstein and J. Michael Dunn, editors, Modern usn of multiple-valued logic, pages 5--37. Reidel, Dordrecht, NL, 1977.

No context found.

N. D. Belnap. A useful four-valued logic. In Modern uses of multi-valued logic, pages 8--37. D. Reidel Publ. Co., 1975.

No context found.

N. D Belnap Jr.,A Useful Four-valued Logic, In: J. Michael Dunn and G. Epstein, editors, "Modern Uses of Multiple-Valued Logic," pages 8--37, 1977.

No context found.

Nuel D. Belnap. A useful four-valued logic. In Gunnar Epstein and J. Michael Dunn, editors, Modern uses of multiple-valued logic, pages 8--37. Reidel, Dordrecht, NL, 1977.

No context found.

Nuel D. Belnap. A useful four-valued logic. In Gunnar Epstein and J. Michael Dunn, editors, Modern uses of multiple-valued logic, pages 8--37. Reidel, Dordrecht, NL, 1977.

No context found.

N. D Belnap Jr. A useful four-valued logic. In J. Michael Dunn and G. Epstein, editors, Modern Uses of Multiple-Valued Logic, pages 8--37, Dordrecht, 1977. D. Reidel.

No context found.

N. D. Belnap, Jr. A useful four-valued logic, in Modern Uses of Multiple-Valued Logic, J. Michael Dunn and G. Epstein editors, D. Reidel, Boston (1977), pp. 8-37.

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N. D. Belnap, "A useful four-valued logic," pp.7--37 in Modern Uses of MultipleValued Logic, J. M. Dunn and G. Epstein (Eds.). D. Reidel Publishing Co, Boston, 1977.

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N. Belnap. A useful four-valued logic. In J. Dunn and G. Epstein, editors, Modern uses of multiple-valued logic, volume 2 of Episteme, pages 8--37. 1977.

No context found.

N. D. Belnap. A Useful Four-Valued Logic. In J. M. Dunn and G. Epstein, editors, Modern Uses of Multiple-Valued Logic, pages 8-37. D. Reidel Pub., 1975.

No context found.

N.D. Belnap, A Useful Four-Valued Logic, in \Modern Uses of Multiple-Valued Logic", ed. G. Epstein and J.M. Dunn, Boston: Reidel, 1977, pp. 8 - 37.

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N. D. Belnap. A Useful Four-Valued Logic. In J. M. Dunn and G. Epstein, editors, Modern Uses of Multiple-Valued Logic, pages 8--37. D. Reidel Pub., 1975.

No context found.

N. D. Belnap. A useful four-valued logic. In J. Michael Dunn and G. Epstein, editors, Modern Uses of Multiple-Valued Logic, pages 8-37. D. Reidel, 1977.

No context found.

Belnap, N. D., A Useful Four-Valued Logic, in G. Epstein and J. M. Dunn (eds.), Modern Uses of Multiple-Valued Logic, Reidel, 1977.

No context found.

Nuel D. Belnap. A useful four-valued logic. In J.M. Dunn and G. Epstein, editors, Modern Uses of Multiple-Valued Logic, pages 8--37. D. Reidel Publishing Co., 1977.

No context found.

N. Belnap. A useful four-valued logic. In Modern Uses of Multiple-Valued Logic, pages 5--37. Reidel, 1977.

No context found.

N Belnap. A useful four valued logic. In G Epstein, editor, Modern Uses of Multiplevalued Logic. Reidel, 1977.

No context found.

N. Belnap. "A Useful Four-Valued Logic". In Dunn and Epstein, editors, Modern Uses of Multiple-Valued Logic, pages 30--56. Reidel, 1977.

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N.D. Belnap. A useful four-valued Logics. In G. Epstein and J.M. Dunn, editors, Modern Uses of Multiple-Valued Logics, pages 7-73. Reidel Publishing Company, Boston, 1977.

No context found.

N. Belnap. A Useful Four-Valued Logic. In M. Dunn and G. Epstein, editors, Modern Uses of Multi-Valued Logic, pages 8-37. Reidel Publ. Co., 1977.

No context found.

N. D. Belnap, Jr., "A useful four-valued logic," in: J. M. Dunn and G. Epstein (eds.), Modern Uses of Multiple-valued Logic, (Reidel, 1977) pp. 8-37.

No context found.

N. D. Belnap. A useful four-valued logic. In G. Epstein and J. M. Dunn, editors, Modern Uses of Multiple-Valued Logic, pages 7--37. Reidel, 1977.

No context found.

N.D. Belnap, Jr. A useful four-valued logic. In J. Michael Dunn and G. Epstein, editors, Modern Uses of Multiple- Valued Logic, pages 8-37. Reidel, 1977.

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