Definitions

# Multi-valued logic

Multi-valued logics are logical calculi in which there are more than two truth values. Traditionally, logical calculi are two-valued—that is, there are only two possible truth values (i.e. truth and falsehood) for any proposition to take. An obvious extension to classical two-valued logic is an n>2-ary logic. Those most popular in the literature are three-valued (e.g. Łukasiewicz's and Kleene's) and infinite-valued (e.g. fuzzy logic) ones.

## Relation to classical logic

Logics are usually systems intended to codify rules for preserving some semantic property of propositions across transformations. In classical logic, this property is "truth." In a valid argument, the truth of the derived proposition is guaranteed if the premises are jointly true, because the application of valid steps preserves the property. However, that property doesn't have to be that of "truth"; instead, it can be some other concept.

Multi-valued logics are intended to preserve the property of designationhood (or being designated). Since there are more than two truth values, rules of inference may be intended to preserve more than just whichever corresponds (in the relevant sense) to truth. For example, in a three-valued logic, sometimes the two greatest truth-values (when they are represented as e.g. positive integers) are designated and the rules of inference preserve these values. Precisely, a valid argument will be such that the value of the premises taken jointly will always be less than or equal to the conclusion.

For example, the preserved property could be justification, the foundational concept of intuitionistic logic. Thus, a proposition is not true or false; instead, it is justified or flawed. A key difference between justification and truth, in this case, is that the law of the excluded middle doesn't hold: a proposition that is not flawed is not necessarily justified; instead, it's only not proven that it's flawed. The key difference is the determinacy of the preserved property: One may prove that P is justified, that P is flawed, or be unable to prove either. A valid argument preserves justification across transformations, so a proposition derived from justified propositions is still justified. However, there are proofs in classical logic that depend upon the law of excluded middle; since that law is not usable under this scheme, there are propositions that cannot be proven that way.

## Relation to fuzzy logic

Multi-valued logic is strictly related with Fuzzy set theory and fuzzy logic. The notion of fuzzy subset was introduced by Lotfi Zadeh as a formalization of vagueness; i.e., the phenomenon that a predicate may apply to an object not absolutely, but to a certain degree, and that there may be borderline cases. Indeed, as in multi-valued logic, fuzzy logic admits truth values different from "true" and "false". As an example, usually the set of possible truth values is the whole interval [0,1]. Nevertheless, the main difference between fuzzy logic and multi-valued logic is in the aims. In fact, in spite of its philosophical interest (it can be used to deal with the sorites paradox), fuzzy logic is devoted mainly to the applications. More precisely, there are two approaches to Fuzzy logic. The first one is very closed with multi-valued logic tradition (Hajek scool). So a set of designed values is fixed and this enables us to define an entailment relation. The deduction apparatus is defined by a suitable set of logical axioms and suitable inference rules. Another approach (Goguen, Pavelka and others) is devoted to define a deduction apparatus in which approximate reasonings are admitted. Such an apparatus is defined by a suitable fuzzy subset of logical axioms and by a suitable set of fuzzy inference rules. In the first case the logical consequence operator gives the set of logical consequence of a given set of axioms. In the latter the logical consequence operator gives the fuzzy subset of logical consequence of a given fuzzy subset of hypotheses.

Another example of an infinitely-valued logic is probability logic.

## History

The first known classical logician who didn't fully accept the law of the excluded middle was Aristotle (who, ironically, is also generally considered to be the first classical logician and the "father of logic), who admitted that his laws did not all apply to future events (De Interpretatione, ch. IX). But he didn't create a system of multi-valued logic to explain this isolated remark. The later logicians until the coming of the 20th century followed Aristotelian logic, which includes or implies the law of the excluded middle.

The 20th century brought the idea of multi-valued logic back. The Polish logician and philosopher Jan Łukasiewicz began to create systems of many-valued logic in 1920, using a third value "possible" to deal with Aristotle's paradox of the sea battle. Meanwhile, the American mathematician Emil L. Post (1921) also introduced the formulation of additional truth degrees with n>=2,where n are the truth values. Later Jan Łukasiewicz and Alfred Tarski together formulated a logic on n truth values where n>=2 and in 1932 Hans Reichenbach formulated a logic of many truth values where n→infinity. Kurt Gödel in 1932 showed that intuitionistic logic is not a finitely-many valued logic, and defined a system of Gödel logics intermediate between classical and intuitionistic logic; such logics are known as intermediate logics.

## Patents

• Takatsu, "Multiple-valued logic circuit". July 1, 1997.

## References

• Chang C.C. and Keisler H. J. 1966. Continuous Model Theory, Princeton, Princeton University Press.
• Cignoli, R. L. O., D'Ottaviano, I, M. L., Mundici, D., 2000. Algebraic Foundations of Many-valued Reasoning. Kluwer.
• Hájek P., 1998, Metamathematics of fuzzy logic. Kluwer.
• Malinowski, Gregorz, 2001, Many-Valued Logics, in Goble, Lou, ed., The Blackwell Guide to Philosophical Logic. Blackwell.
• Gerla G. 2001, Fuzzy logic: Mathematical Tools for Approximate Reasoning, Kluwer Academic Publishers, Dordrecht.
• Goguen J.A. 1968/69, The logic of inexact concepts, Synthese, 19, 325-373.
• Gottwald S. 2000, S. A Treatise on Many-Valued Logics, Research Studies Press, Baldock.
• Pavelka J. 1979, On fuzzy logic I: Many-valued rules of inference, Zeitschr. f. math. Logik und Grundlagen d. Math., 25, 45-52.