Classical Logic with n Truth Values as a Symmetric Many-Valued Logic

Chapter 20 - Automated Deduction for Many-Valued Logics

According to a result of Lee, varieties of pseudocomplemented distributive lattices form an ω+1 chain in which is the trivial variety and is the variety of Boolean algebras. In the present paper it is shown that the variety contains an almost universal subcategory B in which the members of Hom(B,B') associated with minimal prime ideals of B form a countably infinite set for any B,B' ∈ B. In particular, B3contains arbitrarily large algebras whose nontrivial endomorphisms form the countably infinite right zero semigroup. Our earlier results concerning categorical properties of varieties of pseudocomplemented distributive lattices show that no further reduction of the right zero count is possible.

(1978). Orthomodular Lattices and Quantum Physics. Mathematics Magazine: Vol. 51, No. 5, pp. 299-303.

Soft computing is used to find solutions of real life problems where exact computation is not possible. Soft computation models are usually approximate in nature. In this chapter we will discuss an approximate reasoning model. The basic reasoning tools in logic are Modus Ponens and Modus Tollens where a statement is either true or false. However our practical experience tells us that often we refer to a statement as quite true or somewhat false etc. Reasoning with such statements could not be done with basic tools of logic. We have tried to find the degree of truth or falsity of these reasoning tools for statements having such linguistic truth values. We have assumed that the propositions have linguistic truth values which are not completely ordered but may be represented by a lattice (Fig. 1 or Fig. 2). It was found that Modus Ponens and Modus Tollens rules with such propositions were not tautologies as in classical logic but have linguistic truth values. These truth values have been computed for all cases in this chapter. The results may be applied to physical, biological, economic or social phenomena. We have showed an application of decision making with linguistic truth-valued propositions in healthcare.

Many-valued logic based interval reasoning plays an increasingly important role in fuzzy and other many-valued extensions of two-valued (crisp) logic. It is often overlooked that many-valued logic (MVL) reasoning has a richer inference rule base than the crisp (i.e. classical two-valued) logic. Indeed some rules that are not possible in the crisp logic (which is point-based) do come to existence when we accept intervals as the basic elements of the semantic space ([0, 1] or a more general lattice) into which the logic expressions are valuated [11, 8, 37].

Encoding deductive argumentation in quantified Boolean formulae

Intuitionistic logic encompasses the principles of reasoning which were used by L. Brouwer in developing his intuitionistic mathematics, beginning in [Bro07]. Intuitionistic logic can be succinctly described as classical logic without the law ϕ ∨ ¬ϕ of excluded middle. Brouwer observed that this law was abstracted from finite situations and its application to statements about infinite collections is not justified. One of the consequences of the rejection of the law of excluded middle is that every intuitionistic proof of an existential sentence can be effectively transformed into an intuitionistic proof of an instance of that sentence. More precisely, if a formula of the form ∃xϕ(x) without free variables is provable in the intuitionistic predicate logic, then there is a term t without free variables such that ϕ(t) is provable. In particular, if ϕ ∨ ψ is provable, then either ϕ or ψ is provable. In that sense intuitionistic logic may provide a logical basis for constructive reasoning. A formal system of intuitionistic logic was proposed in [Hey30]. A relationship between the classical propositional logic PC and intuitionistic propositional logic INT was proved by Glivenko in [Gli29], namely a PC-formula ϕ is PC-provable if and only if ¬ ¬ϕ is INT-provable. Kripke semantics for intuitionistic logic was developed in [Kri65].

Substitution Frege and extended Frege proof systems in non-classical logics

The past few decades have seen a resurgence of reasoning techniques in artificial intelligence involving both classical and non-classical logics. In his paper, ``Multi-valued Logics: A Uniform Approach to Reasoning in Artificial Intelligence'', Ginsberg has shown that through the use of bilattices, several reasoning techniques can be unified under a single framework. A bilattice is a structure that can be viewed as a class of truth values that can accommodate incomplete and inconsistent information and in certain cases default information. In bilattice theory, knowledge is ordered along two dimensions: truth/falsity and certainty/uncertainty. By defining the corresponding bilattices as truth spaces, Ginsberg has shown that the same theorem prover can be used to simulate reasoning in first order logic, default logic, prioritized default logic and assumption truth maintenance system. Although this is a significant contribution, Ginsberg's paper was lengthy and involved. This paper summarizes some of the essential concepts and foundations of bilattice theory. Furthermore, it discusses the connections of bilattice theory and several other existing multi-valued logics such as the various three-valued logics and Belnap's four-valued logic. It is noted that the set of four truth values in Belnap's logic form a lattice structure that is isomorphic to the simplest bilattice. Subsequently, Fitting proposed a conflation operation that can be used to select sub-sets of truth values from this and other bilattices. This method of selecting sub-sets of truth values provides a means for identifying sub-logic in a bilattice.

One of the main features of Fuzzy Logic is its capability to deal with the concept of compatibility between two propositions, in such a way that the inference process modeled through the Compositional Rule of Inference is independent from the particular possibility distributions involved. It is in this context that the compatibility functions can be considered as fuzzy truth values, labels or qualifications, playing the same role as the values true and false play in the Classical Logic, where the meaning of propositions is nothing but its truth value. In this article we consider a restricted family of labels having the following desirable properties: (a) easy parametric representation, (b) easy semantic interpretation, (c) to allow a gradation in the family according to the modifications performed by each label, and (d) to be closed under inference processes (FR-functions), and also under some suitable and meaningful operations between them.

The purpose of this note is to show that a known and natural four-valued logic co-exists with classical two-valued logic in the familiar context of truth tables. The tool required is the power construction. The context of this paper is that of 2-valued truth tables in n propositional variables, say p1, p2, . . . , pn. The two classical truth values are 0 and 1. The truth value assignments (or valuations) are the 2 {0, 1}-vectors of length n. Propositional formulae are freely generated from the propositional variables by application of the usual classical connectives ∼ (not), & (and) and ∨ (or), each of which is defined by a given truth table. Accordingly, each propositional formula has a truth table. Formulae with the same truth table are logically equivalent, and we do not distinguish between them. The valuations form a Boolean algebra Val = (2,∧,∨, ′ ,0,1), where the Boolean operations ∧, ∨ and ′ are defined componentwise from their counterparts in the 2-element Boolean algebra 2 = {0, 1}, and 0 and 1 are respectively the vectors (0, 0, . . . , 0) and (1, 1, . . . , 1) of length n. The elements of 2 are denoted by x,y,z, . . . Up to equivalence, formulae may be regarded as truth functions from 2 to 2, and the connectives are characterised by operations in the image algebra 2. That is, if bool(A,−) : 2 → 2 is the truth function corresponding to a propositional formula A, then bool (∼A,x) = bool (A,x)′ , bool (A & B,x) = bool (A,x) ∧ bool (B,x) , (1) bool (A ∨ B,x) = bool (A,x) ∨ bool (B,x) . 1991 Mathematics Subject Classification: 03B46, 03B05. The paper is in final form and no version of it will be published elsewhere.

Chapter 1 Knowledge Representation and Classical Logic

We present many-valued disjunctive logic programs in which classical disjunctive logic program clauses are extended by a truth value that respects the material implication. Interestingly, these many-valued disjunctive logic programs have both a probabilistic semantics in probabilities over possible worlds and a truth-functional semantics. We then define minimal, perfect, and stable models and show that they have the same properties like their classical counterparts. In particular, perfect and stable models are always minimal models. Under local stratification, the perfect model semantics coincides with the stable model semantics. Finally, we show that some special cases of propositional many-valued disjunctive logic programming under minimal, perfect, and stable model semantics have the same complexity like their classical counterparts.

The purpose of this paper is to introduce a generalization of Molodtsov's approach to soft sets obtained by passing from the classical two-valued logic underlying those sets to a three-valued logic, where the third truth value can usually be interpreted as either non-determined or unknown. This extension of soft set approach allows for more intuitive and clearer representation of various decision making problems involving incomplete or uncertain information. In other words, it is a useful way to analyze soft set based multi-criteria group decision making problems under the lack of information resulting from the inability to determine the data. In this paper, we introduce the concept of three-valued soft set and its some basic operations and products. We propose the formulas to calculate all possible choice values for each object in the (weighted) three-valued soft sets, and thus calculate their respective decision values. By modifying the TOPSIS and ELECTRE methods to deal with multi-criteria group decision problems, three-valued soft set based decision making algorithms are constructed. To demonstrate the practicality of these algorithms, we address the outstanding examples adapted from the decision problems in real-life. Lastly, some aspects of the efficiency of the proposed algorithms are discussed with computational experiments.

"This book describes the aspects of mathematical logic related to computer sciences. The materials adopted in this book are intended to attend to both the peculiarities of logical systems and the requirements of computer science."