Many-valued logic

In logic, a many-valued logic (also multi- or multiple-valued logic) is a propositional calculus in which there are more than two truth values. Traditionally, in Aristotle's logical calculus, there were only two possible values (i.e., "true" and "false") for any proposition. An obvious extension to classical two-valued logic is an n-valued logic for n greater than 2. Those most popular in the literature are three-valued (e.g., Łukasiewicz's and Kleene's, which accept the values "true", "false", and "unknown"), the finite-valued with more than three values, and the infinite-valued, such as fuzzy logic and probability logic.

History

The first known classical logician who didn't fully accept the law of excluded middle was Aristotle (who, ironically, is also generally considered to be the first classical logician and the "father of logic"). Aristotle 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. Until the coming of the 20th century, later logicians followed Aristotelian logic, which includes or assumes the law of the excluded middle.

The 20th century brought back the idea of multi-valued logic. 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. 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.

Examples

Main articles: Three-valued logic and Four-valued logic

Kleene (strong) K₃ and Priest logic P₃

Kleene's "(strong) logic of indeterminacy" K₃ (sometimes $K_{3}^{S}$ and Priest's "logic of paradox" add a third "undefined" or "indeterminate" truth value I. The truth functions for negation (¬), conjunction (∧), disjunction (∨), implication (→_K), and biconditional (↔_K) are given by:

¬ T F I I F T

∧ T I F T T I F I I I F F F F F

∨ T I F T T T T I T I I F T I F

→K T I F T T I F I T I I F T T T

↔K T I F T T I F I I I I F F I T

The difference between the two logics lies in how tautologies are defined. In K₃ only T is a designated truth value, while in P₃ both T and I are (a logical formula is considered a tautology if it evaluates to a designated truth value). In Kleene's logic I can be interpreted as being "underdetermined", being neither true not false, while in Priest's logic I can be interpreted as being "overdetermined", being both true and false. K₃ does not have any tautologies, while P₃ has the same tautologies as classical two-valued logic.

Bochvar's internal three-valued logic (also known as Kleene's weak three-valued logic)

Another logic is Bochvar's "internal" three-valued logic ($B_{3}^{I}$) also called Kleene's weak three-valued logic. Except for negation, its truth tables are all different from the above.

∧+ T I F T T I F I I I I F F I F

∨+ T I F T T I T I I I I F T I F

→+ T I F T T I F I I I I F T I T

The intermediate truth value in Bochvar's "internal" logic can be described as "contagious" because it propagates in a formula regardless of the value of any other variable.

Belnap logic (B₄)

Belnap's logic B₄ combines K₃ and P₃. The overdetermined truth value is here denoted as B and the underdetermined truth value as N.

f¬ T F B B N N F T

f∧ T B N F T T B N F B B B F F N N F N F F F F F F

f∨ T B N F T T T T T B T B T B N T T N N F T B N F

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 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 closely related to 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 closely linked with multi-valued logic tradition (Hajek school). 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 defining 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.

Applications

Applications of many-valued logic can be roughly classified into two groups. The first group uses many-valued logic domain to solve binary problems more efficiently. For example, a well-known approach to represent a multiple-output Boolean function is to treat its output part as a single many-valued variable and convert it to a single-output characteristic function. Other applications of many-valued logic include design of Programmable Logic Arrays (PLAs) with input decoders, optimization of finite state machines, testing, and verification.

The second group targets the design of electronic circuits which employ more than two discrete levels of signals, such as many-valued memories, arithmetic circuits, Field Programmable Gate Arrays (FPGA) etc. Many-valued circuits have a number of theoretical advantages over standard binary circuits. For example, the interconnect on and off chip can be reduced if signals in the circuit assume four or more levels rather than only two. In memory design, storing two instead of one bit of information per memory cell doubles the density of the memory in the same die size. Applications using arithmetic circuits often benefit from using alternatives to binary number systems. For example, residue and redundant number systems can reduce or eliminate the ripple-through carries which are involved in normal binary addition or subtraction, resulting in high-speed arithmetic operations. These number systems have a natural implementation using many-valued circuits. However, the practicality of these potential advantages heavily depends on the availability of circuit realizations, which must be compatible or competitive with present-day standard technologies.

Research venues

An IEEE International Symposium on Multiple-Valued Logic (ISMVL) has been held annually since 1970. It mostly caters to applications in digital design and verification. There is also a Journal of Multiple-Valued Logic and Soft Computing.

See also

Mathematical logic

• Fuzzy logic
• Gödel logics
• Kleene logic (Kleene algebra)
• Łukasiewicz logic (MV-algebra)
• Post logic
• Relevance logic
• Degrees of truth
• Principle of bivalence

Philosophical logic

• False dilemma
• Mu

Digital logic

• MVCML, multiple-valued current-mode logic
• IEEE 1164 a nine-valued standard for VHDL
• IEEE 1364 a four-valued standard for Verilog
• Noise-based logic

Further reading

General

• Béziau J.-Y. (1997), What is many-valued logic ? Proceedings of the 27th International Symposium on Multiple-Valued Logic, IEEE Computer Society, Los Alamitos, pp. 117–121.
• Malinowski, Gregorz, (2001), Many-Valued Logics, in Goble, Lou, ed., The Blackwell Guide to Philosophical Logic. Blackwell.
• Bergmann, Merrie (2008), An introduction to many-valued and fuzzy logic: semantics, algebras, and derivation systems, Cambridge University Press, ISBN 978-0-521-88128-9 {{citation}}: Unknown parameter |harv= ignored (help)
• Cignoli, R. L. O., D'Ottaviano, I, M. L., Mundici, D., (2000). Algebraic Foundations of Many-valued Reasoning. Kluwer.
• Grzegorz Malinowski (1993). Many-valued logics. Clarendon Press. ISBN 978-0-19-853787-8.
• S. Gottwald, A Treatise on Many-Valued Logics. Studies in Logic and Computation, vol. 9, Research Studies Press: Baldock, Hertfordshire, England, 2001.
• Gottwald, Siegfried (2005). "Many-Valued Logics" (PDF). {{cite journal}}: Cite journal requires |journal= (help); Invalid |ref=harv (help)
• D. Michael Miller; Mitchell A. Thornton (2008). Multiple valued logic: concepts and representations. Synthesis lectures on digital circuits and systems. Vol. 12. Morgan & Claypool Publishers. ISBN 978-1-59829-190-2.
• Hájek P., (1998), Metamathematics of fuzzy logic. Kluwer. (Fuzzy logic understood as many-valued logic sui generis.)

Specific

• Alexandre Zinoviev, Philosophical Problems of Many-Valued Logic, D. Reidel Publishing Company, 169p., 1963.
• Prior A. 1957, Time and Modality. Oxford University Press, based on his 1956 John Locke lectures
• Goguen J.A. 1968/69, The logic of inexact concepts, Synthese, 19, 325–373.
• Chang C.C. and Keisler H. J. 1966. Continuous Model Theory, Princeton, Princeton University Press.
• Gerla G. 2001, Fuzzy logic: Mathematical Tools for Approximate Reasoning, Kluwer Academic Publishers, Dordrecht.
• Pavelka J. 1979, On fuzzy logic I: Many-valued rules of inference, Zeitschr. f. math. Logik und Grundlagen d. Math., 25, 45–52.
• George Metcalfe; Nicola Olivetti; Dov M. Gabbay (2008). Proof Theory for Fuzzy Logics. Springer. ISBN 978-1-4020-9408-8. Covers proof theory of many-valued logics as well, in the tradition of Hájek.
• Reiner Hähnle (1993). Automated deduction in multiple-valued logics. Clarendon Press. ISBN 978-0-19-853989-6.
• Francisco Azevedo (2003). Constraint solving over multi-valued logics: application to digital circuits. IOS Press. ISBN 978-1-58603-304-0.
• Leonard Bolc; Piotr Borowik (2003). Many-valued Logics 2: Automated reasoning and practical applications. Springer. ISBN 978-3-540-64507-8.

External links

• Gottwald, Siegfried (2009). "Many-Valued Logic". [[Stanford Encyclopedia of Philosophy]]. {{cite encyclopedia}}: Invalid |ref=harv (help); URL–wikilink conflict (help)
• Stanford Encyclopedia of Philosophy: "Truth Values"—by Yaroslav Shramko and Heinrich Wansing.
• IEEE Computer Society's Technical Committee on Multiple-Valued Logic
• Resources for Many-Valued Logic by Reiner Hähnle, Chalmers University
• Many-valued Logics W3 Server (archived)