Chapter 20 - Automated Deduction for Many-Valued Logics

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].

Among non-monotonic systems of reasoning, non-monotonic modal logics, and autoepistemic logic in particular, have had considerable success. The presence of explicit modal operators allows flexibility in the embedding of other approaches. Also several theoretical results of interest have been established concerning these logics. In this paper we introduce non-monotonic modal logics based on many-valued logics, rather than on classical logic. This extends earlier work of ours on many-valued modal logics. Intended applications are to situations involving several reasoners, not just one as in the standard development.

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.

Full classical S5 in natural deduction with weak normalization

Some find it plausible that a sufficiently long duration of torture is worse than any duration of mild headaches. Similarly, it has been claimed that a million humans living great lives is better than any number of worm-like creatures feeling a few seconds of pleasure each. Some have related bad things to good things along the same lines. For example, one may hold that a future in which a sufficient number of beings experience a lifetime of torture is bad, regardless of what else that future contains, while minor bad things, such as slight unpleasantness, can always be counterbalanced by enough good things. Among the most common objections to such ideas are sequence arguments. But sequence arguments are usually formulated in classical logic. One might therefore wonder if they work if we instead adopt many-valued logic. I show that, in a common many-valued logical framework, the answer depends on which versions of transitivity are used as premises. We get valid sequence arguments if we grant any of several strong forms of transitivity of ‘is at least as bad as’ and a notion of completeness. Other, weaker forms of transitivity lead to invalid sequence arguments. The plausibility of the premises is largely set aside here, but I tentatively note that almost all of the forms of transitivity that lead to valid sequence arguments seem intuitively problematic. Still, a few moderately strong forms of transitivity that might be acceptable lead to valid sequence arguments, although weaker statements of the initial value claims avoid these arguments at least to some extent.

Here we shall extract those minimal functional means of many-valued logic which are sufficient for the construction of paraconsistent logic. Thereby we shall define a rather wide class of many-valued logics which contain paraconsistent logic as a fragment. Thus one may speak of paraconsistent structure in terms of many-valued logic exactly as one may speak of, for example, modal structure in terms of many-valued logic. Also it will be shown that paraconsistent logic extracted from many-valued logic in its turn contains a fragment which is isomorphous to classical two-valued logic. As will be evidently found out the difference between constructed paraconsistent logic and the specified model for classical logic consists only in the properties of the operation of negation (and in the choice of the set of designated truth values for paraconsistent logic).

We introduce Boolean-like algebras of dimension n (n{mathrm {BA}}s) having n constants {{{mathsf {e}}}}_1,ldots ,{{{mathsf {e}}}}_n, and an (n+1)-ary operation q (a “generalised if-then-else”) that induces a decomposition of the algebra into n factors through the so-called n-central elements. Varieties of n{mathrm {BA}}s share many remarkable properties with the variety of Boolean algebras and with primal varieties. The n{mathrm {BA}}s provide the algebraic framework for generalising the classical propositional calculus to the case of n–perfectly symmetric–truth-values. Every finite-valued tabular logic can be embedded into such a n-valued propositional logic, n{mathrm {CL}}, and this embedding preserves validity. We define a confluent and terminating first-order rewriting system for deciding validity in n{mathrm {CL}}, and, via the embeddings, in all the finite tabular logics.

In this paper, we study alethic modal operators in Łukasiewicz’ many-valued logics where truth values are certain rational numbers from the closed unit interval [0,1]. The alethic modal operators necessary and possible are added to Ł3 using Tarski’s idea. The result is a modal logic denoted by Ł\(_{3}^{\mathrm{mod}}\). A formula-equivalency between Ł n and Ł\(_{n}^{\mathrm{mod}}\) is stated for n ≥ 3. Truth value assignments, or valuations, in Łukasiewicz’ n-valued modal logic are considered. The laws of excluded middle and contradiction are considered and found that they are possible in Ł\(_{n}^{\mathrm{mod}}\). Normal modal many-valued systems based on axiom schemes (K), (T), (S4), (B) and (S5) are considered.

Combining knowledge with many-valued logics

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.

Dung's argumentation framework is an abstract framework based on a set of arguments and a binary attack relation defined over the set. One instantiation, among many others, of Dung's framework consists in constructing the arguments from a set of propositional logic formulas. Thus an argument is seen as a reason for or against the truth of a particular statement. Despite its advantages, the argumentation approach for inconsistency handling also has important shortcomings. More precisely, in some applications what one is interested in are not so much only the conclusions supported by the arguments but also the precise explications of such conclusions. We show that argumentation framework applied to classical logic formulas is not suitable to deal with this problem. On the other hand, intuitionistic logic appears to be a natural alternative candidate logic (instead of classical logic) to instantiate Dung's framework. We develop constructive argumentation framework. We show that intuitionistic logic offers nice and desirable properties of the arguments. We also provide a characterization of the arguments in this setting in terms of minimal inconsistent subsets when intuitionistic logic is embedded in the modal logic S4.

Multiple-valued logic (MVL) is a logic in which there are more than two truth values. MVL is used due to difficulty in interconnection problems in binary system. Carbon nanotube field-effect transistors (CNTFET) is used to design the Quaternary Half Subtractor (QHS) circuit. Sub blocks of Module (0210), Module (1021), Module (2102), Module (3210) using three supply voltage devices and Quaternary Multiplexer 2:1 using pass transistor have designed in this paper. Therefore, compared to existing designs, less no. of transistors are needed. Comparing proposed Half subtractor using 4:1 MUX modules to current one’s reveals that the 2:1 multiplexer-based method leads in lower power usage. These proposed designs have proven to work satisfactorily under a variety of operating situations, including power, delay, and power delay product (PDP).

Quantum logic, usually interpreted as non-distributive but two-valued logic, is reinterpreted as many-valued logic. This allows to explain the logical background of the wave-particle duality exhibited in the double-slit experiment in a better way than it can be done with the use of classical two-valued logic.

Let us first set up the most general structural formulation of two-valued logic. For our purposes the simplest case of only two propositions will suffice. Designate them as p and q, and also designate the two (postulated) truth values as T' and F, standing for "true" and "false". For a relation between the propositions the abbreviation R is convenient. Our problem is really to set up all the possible structures p R q in terms of the truth-values, I' and F, which are applicable to p, q and p R q, regardless of the meaning of p and q. We seek the fundamental morphology of R, its structural meaning. For the analysis of structure the most powerful procedure seems to be to set up a matrix or array, which reveals the group character and unity of all so-called alternative logics each of which glorifies a single version of R as primary. Immediately it is apparent that R can have only I6 possible interpretations, revealed as patterns of T' and F, as follows:

We discuss a scheme for defining and reasoning about partial recursive functions within a classical two-valued logic in which all terms denote. We show how a total extension of the partial function introduced by a recursive declaration may be axiomatized within a classical logic, and illustrate by an example the kind of reasoning that our scheme supports. By presenting a naive set-theoretic semantics, we show that the system we propose is logically consistent. Our work is motivated largely by the pragmatic issues arising from mechanical theorem proving – we discuss some of the practical benefits and limitations of our scheme for mechanical verification of software and hardware systems.