A two-valued logic for reasoning about different types of consequence in Kleene's three-valued logic

We propose a logic for partial reasoning over all general logic programs in a certain first-order language L. We need such a logic to express statements like it is impossible that a ground atom is both true and false in any program. To construct this logic we start with Kleene's three-valued logic. Then we extend L to a metalanguage L * by including a new system of connectives. Basing upon Kripke's approach we introduce the notion of truth for formulas of L *. Comparison with classical logic and intuitionistic one is given. We prove that every intuitionistic theorem is true in our logic. We also show how classical tautologies are mapped onto true formulas.KeywordsLogic ProgrammingKleene's LogicKripke Semantics

Propositional and predicate logics of incomplete information

This paper proposes a conceptually simple but expressive framework for handling propositional information stemming from several sources, namely a two-tiered propositional logic augmented with classical modal axioms (BC-logic), a fragment of the non-normal modal logic EMN, whose semantics is expressed in terms of two-valued monotonic set-functions called Boolean capacities. We present a theorem-preserving translation of Belnap logic in this setting. As special cases, we can recover previous translations of three-valued logics such as Kleene and Priest logics. Our translation bridges the gap between Belnap logic, epistemic logic, and theories of uncertainty like possibility theory or belief functions, and paves the way to a unified approach to various inconsistency handling methods.

to reject (3) both (1) and (2) are available to us. But since (2) is not a serious alternative to (1), it cannot be a serious alternative to (3). As these remarks indicate, I am not questioning Putnam's account of the "core"~ meanings of 'true' and 'false.' Nor can I legitimately claim to have proven that three-valued logic has no future as a replacement for two-valued logic within the framework of Putnam's account of these core meanings. However, if I am correct in maintaining that the truth value concepts of a language embedded in three-valued logic must be translatable into a language with a two-valued logic and still meet Putnam's requirements, a consideration of the kinds of translations available would indicate a dim future for three-valued logic.

It is argued that two-valued logic and fuzzy logic both lack some representational and reasoning capability in modeling directly a social or natural world with a combination of positive, negative, and neutral relationships. This is due to the fact that the logic values in both lie in the interval (0, 1). An NPN (negative-positive-neutral) crisp logic and an NPN fuzzy logic are presented that assume, respectively, crisp or fuzzy values in the interval (-1, 1). The notions of NPN relations and NPN heuristic transitive closures are introduced. It is suggested that NPN logics can be embedded into NPN relations for cognitive process modeling. An application is given to illustrate the theory. >

Recently there has been an effort to solve the problems caused by the infamous NULL in relational databases, by systematically applying Kleene's three-valued logic to SQL. The third truth-value is unknown. In this paper we show that by using a fourth truth-value inconsistent, all the advantages of the three-valued approach can be retained, and that negation can be given a constructive, intuitionistic meaning that allows negative knowledge to be specified in the logic explicitly, without having to resort to extra-logical notions of stratification or to non-monotonic reasoning. The four-valued approach also allows for a computationally efficient treatment of query answering in the presence of inconsistencies. This is in contrast to the computationally intractable repair approach to inconsistency management. From a practical view-point we show that the Cylindric Star Algebra, developed by the authors, is particularly well suited for evaluating First Order queries on four-valued databases, and that the framework of data exchange can smoothly adapted to the four truth-values.

A distinctive feature of Korner’s work is the novelty and depth of the application of mathematical logic to philosophical problems. His main achievement in philosophial logic is the theory and application — particularly to empirical continua — of a three-valued logic of inexactness. The purpose of this essay is to describe the origin and mathematical development of Korner’s logic and its applications. A unifying concept is the notion of quasi-Boolean algebra, which is related to the three-valued logic and the calculus of inexact classes much as Boolean algebra is related to the classical, two-valued logic and the traditional calculus of classes.

This paper investigates a three-valued logic L3, that has been introduced in the study of natural language semantics. A complete proof system based on a three-valued analogon of negative resolution is presented. A subclass of L3 corresponding to Horn clauses in two-valued logic is defined. Its model theoretic properties are studied and it is shown to admit a PROLOG-style proof procedure.

The Nested Relational Calculus (NRC) has been an influential high-level query language, providing power and flexibility while still allowing translation to standard SQL queries. It has also been used as a basis for language-integrated query in programming languages such as F#, Scala, and Links. However, SQL’s treatment of incomplete information, using nulls and three-valued logic, is not compatible with ‘standard’ NRC based on two-valued logic. Nulls are widely used in practice for incomplete data, but the question of how to accommodate SQL-style nulls and incomplete information in NRC, or integrate such queries into a typed programming language, appears not to have been studied thoroughly. In this paper we consider two approaches: an explicit approach in which option types are used to represent (possibly) nullable primitive types, and an implicit approach in which types are treated as possibly-null by default. We give translations relating the implicit and explicit approaches, discuss handling nulls in language integration, and sketch extensions of normalization and conservativity results.

Bitwise operation is a basic operation in C program. According to the statistics, the bitwise operation accounts for 12% in all numerical operations. There are two problems for the bitwise operation when search technology is used to generate test input: one is the low efficiency for it has no corresponding interval arithmetic rule to reduce the search space; the other is the absence of a unified approach when a variable is simultaneously involved in the bitwise and non-bitwise operations. For problem one, classical two-valued logic is extended to four-valued logic to express the two states of logic variable’s value, ‘uncertain’ and ‘neither true nor false’. According to the operation rules of the four-valued logic, the search space can be reduced. For problem two, the conversion between interval representation and four-valued logic representation is given, and then provided with a unified approach based on the search algorithm. Experiments show that this method can improve the efficiency of test input generation of bitwise operation.

In this paper we present a many-valued logic programming, based on reinterpreted Belnap's 4-valued bilattice: we introduce the new semantics for a 4-valued implication by relative pseudo-complement, used for intuitionistic logics. This kind of logic programming is particularly useful for data integration with possibly incomplete and inconsistent information. We define an ontological encapsulation of the epistemic many-valued logic programs with negation, based on this bilattice, into 2-valued meta logic programs. Obtained 2-valued logic semantically reflects original epistemic many-valued logic, and can be used in order to define many-valued logic entailment and inference closure for many-valued truth assignments.

Descriptive dynamic logic and its application to reflective architectures

Knowledge representation and inference in AI have been traditionally divided between logic-based and statistical approaches. During the past decade, the rapidly developing area of Statistical Relational Learning aims to combine the two frameworks for representation and inference. In many cases, these works include probabilistic reasoning within Logic Programming frameworks. These attempts are restricted in the sense that they use only two-valued negation-as-failure semantics. However, well-founded semantics is a widely accepted three-valued-logic negation semantics scheme, which is implemented in certain Logic Programming frameworks. In this paper we introduce probabilistic inference under the well-founded semantics scheme in a single Probabilistic Logic Programming framework, where the uncertainty can be described using both statistical information (probabilities) and a third logic value.

We introduce a new approach to probabilistic logic programming in which probabilities are defined over a set of possible worlds. More precisely, classical program clauses are extended by a subinterval of [0,1] that describes a range for the conditional probability of the head of a clause given its body. We then analyze the complexity of selected probabilistic logic programming tasks. It turns out that probabilistic logic programming is computationally more complex than classical logic programming, More precisely, the tractability of special cases of classical logic programming generally does not carry over to the corresponding special cases of probabilistic logic programming. Moreover, we also draw a precise picture of the complexity of deciding and computing tight logical consequences in probabilistic reasoning with conditional constraints in general. We then present linear optimization techniques for deciding satisfiability and computing tight logical consequencesof probabilistic logic programs. These techniques are efficient in the special case in which we have little relevant purely probabilistic knowledge. We finally show that probabilistic logic programming under certain syntactic and semantic restrictions is closely related to van Emden's quantitative deduction, and thus has computational properties similar to calssical logic programming. Based on this result, we present an efficient approximation technique for probabilistic logic programming.

This paper deals with automated deduction for classical and partial logics, especially for the three-valued logic L3, which has been introduced, for example, in the study of natural language semantics. Based on ideas from a Plaisted's Gentzen style system for classical two-valued logic, we present a new tree-structured proof procedure (TMPR) together with a new completeness proof using proof transformation techniques and some improvements including the generation and use of lemmata. TMPR extends SLD-resolution with a Prolog-style backward chaining to full first-order logic by a controlled use of case analysis. This is done without having to extend negative goals needed, for example, for model elimination. A classification of TMPR, model elimination and related calculi in a common tableau framework is given. Thereafter, we present our extension of the TMPR proof procedure to L3 and show its soundness and completeness. As a side result, a TMPR proof system for the four-valued logic L4 is given. Finally, the restriction of TMPR to ‘L3-Horn clauses’ is considered, and, additionally, an idea for similarly extending model elimination and related systems to L3 (and L4) is illustrated.