Philosophy:Manyvalued logic
Manyvalued logic (also multi or multiplevalued logic) refers to 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. Classical twovalued logic may be extended to nvalued logic for n greater than 2. Those most popular in the literature are threevalued (e.g., Łukasiewicz's and Kleene's, which accept the values "true", "false", and "unknown"), fourvalued, ninevalued, the finitevalued (finitelymany valued) with more than three values, and the infinitevalued (infinitelymanyvalued), such as fuzzy logic and probability logic.
History
It is wrong that the first known classical logician who did not 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 [twovalued] logic"^{[1]}). In fact, Aristotle did not contest the universality of the law of excluded middle, but the universality of the bivalence principle: he admitted that this principle did not all apply to future events (De Interpretatione, ch. IX),^{[2]} but he didn't create a system of multivalued 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 multivalued logic. The Polish logician and philosopher Jan Łukasiewicz began to create systems of manyvalued 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→∞. Kurt Gödel in 1932 showed that intuitionistic logic is not a finitelymany valued logic, and defined a system of Gödel logics intermediate between classical and intuitionistic logic; such logics are known as intermediate logics.
Examples
Kleene (strong) K_{3} and Priest logic P_{3}
Kleene's "(strong) logic of indeterminacy" K_{3} (sometimes [math]\displaystyle{ K_3^S }[/math]) and Priest's "logic of paradox" add a third "undefined" or "indeterminate" truth value I. The truth functions for negation (¬), conjunction (∧), disjunction (∨), implication (), and biconditional () are given by:^{[3]}





The difference between the two logics lies in how tautologies are defined. In K_{3} only T is a designated truth value, while in P_{3} 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 nor false, while in Priest's logic I can be interpreted as being "overdetermined", being both true and false. K_{3} does not have any tautologies, while P_{3} has the same tautologies as classical twovalued logic.^{[4]}
Bochvar's internal threevalued logic
Another logic is Dmitry Bochvar's "internal" threevalued logic [math]\displaystyle{ B_3^I }[/math], also called Kleene's weak threevalued logic. Except for negation and biconditional, its truth tables are all different from the above.^{[5]}



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.^{[5]}
Belnap logic (B_{4})
Belnap's logic B_{4} combines K_{3} and P_{3}. The overdetermined truth value is here denoted as B and the underdetermined truth value as N.



Gödel logics G_{k} and G_{∞}
In 1932 Gödel defined^{[6]} a family [math]\displaystyle{ G_k }[/math] of manyvalued logics, with finitely many truth values [math]\displaystyle{ 0, \tfrac{1}{k  1}, \tfrac{2}{k  1}, \ldots, \tfrac{k  2}{k  1}, 1 }[/math], for example [math]\displaystyle{ G_3 }[/math] has the truth values [math]\displaystyle{ 0, \tfrac{1}{2}, 1 }[/math] and [math]\displaystyle{ G_4 }[/math] has [math]\displaystyle{ 0, \tfrac{1}{3}, \tfrac{2}{3}, 1 }[/math]. In a similar manner he defined a logic with infinitely many truth values, [math]\displaystyle{ G_\infty }[/math], in which the truth values are all the real numbers in the interval [math]\displaystyle{ [0, 1] }[/math]. The designated truth value in these logics is 1.
The conjunction [math]\displaystyle{ \wedge }[/math] and the disjunction [math]\displaystyle{ \vee }[/math] are defined respectively as the minimum and maximum of the operands:
 [math]\displaystyle{ \begin{align} u \wedge v &:= \min\{u, v\} \\ u \vee v &:= \max\{u, v\} \end{align} }[/math]
Negation [math]\displaystyle{ \neg_G }[/math] and implication [math]\displaystyle{ \xrightarrow[G]{} }[/math] are defined as follows:
 [math]\displaystyle{ \begin{align} \neg_G u &= \begin{cases} 1, & \text{if }u = 0 \\ 0, & \text{if }u \gt 0 \end{cases} \\[3pt] u \mathrel{\xrightarrow[G]{}} v &= \begin{cases} 1, & \text{if }u \leq v \\ v, & \text{if }u \gt v \end{cases} \end{align} }[/math]
Gödel logics are completely axiomatisable, that is to say it is possible to define a logical calculus in which all tautologies are provable. The implication above is the unique Heyting implication defined by the fact that the suprema and minima operations form a complete lattice with an infinite distributive law, which defines a unique complete Heyting algebra structure on the lattice.
Łukasiewicz logics L_{v} and L_{∞}
Implication [math]\displaystyle{ \xrightarrow[L]{} }[/math] and negation [math]\displaystyle{ \underset{L}{\neg} }[/math] were defined by Jan Łukasiewicz through the following functions:
 [math]\displaystyle{ \begin{align} \underset{L}{\neg} u &:= 1  u \\ u \mathrel{\xrightarrow[L]{}} v &:= \min\{1, 1  u + v\} \end{align} }[/math]
At first Łukasiewicz used these definitions in 1920 for his threevalued logic [math]\displaystyle{ L_3 }[/math], with truth values [math]\displaystyle{ 0, \frac{1}{2}, 1 }[/math]. In 1922 he developed a logic with infinitely many values [math]\displaystyle{ L_\infty }[/math], in which the truth values spanned the real numbers in the interval [math]\displaystyle{ [0, 1] }[/math]. In both cases the designated truth value was 1.^{[7]}
By adopting truth values defined in the same way as for Gödel logics [math]\displaystyle{ 0, \tfrac{1}{v1}, \tfrac{2}{v1}, \ldots, \tfrac {v2} {v1}, 1 }[/math], it is possible to create a finitelyvalued family of logics [math]\displaystyle{ L_v }[/math], the abovementioned [math]\displaystyle{ L_\infty }[/math] and the logic [math]\displaystyle{ L_{\aleph_0} }[/math], in which the truth values are given by the rational numbers in the interval [math]\displaystyle{ [0,1] }[/math]. The set of tautologies in [math]\displaystyle{ L_\infty }[/math] and [math]\displaystyle{ L_{\aleph_0} }[/math] is identical.
Product logic Π
In product logic we have truth values in the interval [math]\displaystyle{ [0,1] }[/math], a conjunction [math]\displaystyle{ \odot }[/math] and an implication [math]\displaystyle{ \xrightarrow [\Pi]{} }[/math], defined as follows^{[8]}
 [math]\displaystyle{ \begin{align} u \odot v &:= uv \\ u \mathrel{\xrightarrow[\Pi]{}} v &:= \begin{cases} 1, & \text{if } u \leq v \\ \frac{v}{u}, & \text{if } u \gt v \end{cases} \end{align} }[/math]
Additionally there is a negative designated value [math]\displaystyle{ \overline{0} }[/math] that denotes the concept of false. Through this value it is possible to define a negation [math]\displaystyle{ \underset{\Pi}{\neg} }[/math] and an additional conjunction [math]\displaystyle{ \underset{\Pi}{\wedge} }[/math] as follows:
 [math]\displaystyle{ \begin{align} \underset{\Pi}{\neg} u &:= u \mathrel{\xrightarrow[\Pi]{}} \overline{0} \\ u \mathbin{\underset{\Pi}{\wedge}} v &:= u \odot \left(u \mathrel{\xrightarrow[\Pi]{}} v\right) \end{align} }[/math]
and then [math]\displaystyle{ u \mathbin{\underset{\Pi}{\wedge}} v = \min\{u, v\} }[/math].
Post logics P_{m}
In 1921 Post defined a family of logics [math]\displaystyle{ P_m }[/math] with (as in [math]\displaystyle{ L_v }[/math] and [math]\displaystyle{ G_k }[/math]) the truth values [math]\displaystyle{ 0, \tfrac 1 {m1}, \tfrac 2 {m1}, \ldots, \tfrac {m2} {m1}, 1 }[/math]. Negation [math]\displaystyle{ \underset{P}{\neg} }[/math] and conjunction [math]\displaystyle{ \underset{P}{\wedge} }[/math] and disjunction [math]\displaystyle{ \underset{P}{\vee} }[/math] are defined as follows:
 [math]\displaystyle{ \begin{align} \underset{P}{\neg} u &:= \begin{cases} 1, & \text{if } u = 0 \\ u  \frac{1}{m  1}, & \text{if } u \not= 0 \end{cases} \\ u \mathbin{\underset{P}{\wedge}} v &:= \min\{u,v\} \\ u \mathbin{\underset{P}{\vee}} v &:= \max\{u,v\} \end{align} }[/math]
Rose logics
In 1951, Alan Rose defined another family of logics for systems whose truthvalues form lattices.^{[9]}
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.
Multivalued 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 threevalued logic, sometimes the two greatest truthvalues (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.
Suszko's thesis
Functional completeness of manyvalued logics
Functional completeness is a term used to describe a special property of finite logics and algebras. A logic's set of connectives is said to be functionally complete or adequate if and only if its set of connectives can be used to construct a formula corresponding to every possible truth function.^{[10]} An adequate algebra is one in which every finite mapping of variables can be expressed by some composition of its operations.^{[11]}
Classical logic: CL = ({0,1}, ¬, →, ∨, ∧, ↔) is functionally complete, whereas no Łukasiewicz logic or infinitely manyvalued logics has this property.^{[11]}^{[12]}
We can define a finitely manyvalued logic as being L_{n} ({1, 2, ..., n} ƒ_{1}, ..., ƒ_{m}) where n ≥ 2 is a given natural number. Post (1921) proves that assuming a logic is able to produce a function of any m^{th} order model, there is some corresponding combination of connectives in an adequate logic L_{n} that can produce a model of order m+1.^{[13]}
Applications
Known applications of manyvalued logic can be roughly classified into two groups.^{[14]} The first group uses manyvalued logic to solve binary problems more efficiently. For example, a wellknown approach to represent a multipleoutput Boolean function is to treat its output part as a single manyvalued variable and convert it to a singleoutput characteristic function (specifically, the indicator function). Other applications of manyvalued 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 that employ more than two discrete levels of signals, such as manyvalued memories, arithmetic circuits, and field programmable gate arrays (FPGAs). Manyvalued 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^{[15]} can reduce or eliminate the ripplethrough carries that are involved in normal binary addition or subtraction, resulting in highspeed arithmetic operations. These number systems have a natural implementation using manyvalued circuits. However, the practicality of these potential advantages heavily depends on the availability of circuit realizations, which must be compatible or competitive with presentday standard technologies. In addition to aiding in the design of electronic circuits, manyvalued logic is used extensively to test circuits for faults and defects. Basically all known automatic test pattern generation (ATG) algorithms used for digital circuit testing require a simulator that can resolve 5valued logic (0, 1, x, D, D').^{[16]} The additional values—x, D, and D'—represent (1) unknown/uninitialized, (2) a 0 instead of a 1, and (3) a 1 instead of a 0.
Research venues
An IEEE International Symposium on MultipleValued Logic (ISMVL) has been held annually since 1970. It mostly caters to applications in digital design and verification.^{[17]} There is also a Journal of MultipleValued Logic and Soft Computing.^{[18]}
See also
 Mathematical logic
 Degrees of truth
 Fuzzy logic
 Gödel logic
 Jaina sevenvalued logic
 Kleene logic
 Kleene algebra (with involution)
 Łukasiewicz logic
 MValgebra
 Post logic
 Principle of bivalence
 A. N. Prior
 Relevance logic
 Philosophical logic
 Digital logic
 MVCML, multiplevalued currentmode logic
 IEEE 1164 a ninevalued standard for VHDL
 IEEE 1364 a fourvalued standard for Verilog
 Threestate logic
 Noisebased logic
References
 ↑ Hurley, Patrick. A Concise Introduction to Logic, 9th edition. (2006).
 ↑ Jules Vuillemin, Necessity or Contingency, CSLI Lecture Notes, N°56, Stanford, 1996, pp. 133167
 ↑ (Gottwald 2005)
 ↑ Humberstone, Lloyd (2011). The Connectives. Cambridge, Massachusetts: The MIT Press. pp. 201. ISBN 9780262016544. https://archive.org/details/connectives00humb.
 ↑ ^{5.0} ^{5.1} (Bergmann 2008)
 ↑ Gödel, Kurt (1932). "Zum intuitionistischen Aussagenkalkül". Anzeiger der Akademie der Wissenschaften in Wien (69): 65f.
 ↑ Kreiser, Lothar; Gottwald, Siegfried; Stelzner, Werner (1990). Nichtklassische Logik. Eine Einführung. Berlin: AkademieVerlag. pp. 41ff – 45ff. ISBN 9783050002743.
 ↑ Hajek, Petr: Fuzzy Logic. In: Edward N. Zalta: The Stanford Encyclopedia of Philosophy, Spring 2009. ([1])
 ↑ Rose, Alan (December 1951). "Systems of logic whose truthvalues form lattices". Mathematische Annalen 123: 152–165. doi:10.1007/BF02054946.
 ↑ Smith, Nicholas (2012). Logic: The Laws of Truth. Princeton University Press. pp. 124.
 ↑ ^{11.0} ^{11.1} Malinowski, Grzegorz (1993). ManyValued Logics. Clarendon Press. pp. 26–27.
 ↑ Church, Alonzo (1996) (in en). Introduction to Mathematical Logic. Princeton University Press. ISBN 9780691029061. https://books.google.com/books?id=JDLQOMKbdScC&pg=PA162.
 ↑ Post, Emil L. (1921). "Introduction to a General Theory of Elementary Propositions". American Journal of Mathematics 43 (3): 163–185. doi:10.2307/2370324. ISSN 00029327. https://www.jstor.org/stable/2370324.
 ↑ Dubrova, Elena (2002). MultipleValued Logic Synthesis and Optimization, in Hassoun S. and Sasao T., editors, Logic Synthesis and Verification, Kluwer Academic Publishers, pp. 89114
 ↑ Meher, Pramod Kumar; Valls, Javier; Juang, TsoBing; Sridharan, K.; Maharatna, Koushik (20080822). "50 Years of CORDIC: Algorithms, Architectures and Applications". IEEE Transactions on Circuits & Systems I: Regular Papers 56 (9): 1893–1907. 20090909. doi:10.1109/TCSI.2009.2025803. http://core.ac.uk/download/files/34/1509903.pdf. Retrieved 20160103.
 ↑ Abramovici, Miron; Breuer, Melvin A.; Friedman, Arthur D. (1994). Digital Systems Testing and Testable Design. New York: Computer Science Press. p. 183. ISBN 9780780310629. https://archive.org/details/digitalsystemste00abra.
 ↑ "IEEE International Symposium on MultipleValued Logic (ISMVL)". http://www.informatik.unitrier.de/~ley/db/conf/ismvl/index.html.
 ↑ "MVLSC home". http://www.oldcitypublishing.com/MVLSC/MVLSC.html.
Further reading
General
 Augusto, Luis M. (2017). Manyvalued logics: A mathematical and computational introduction. London: College Publications. 340 pages. ISBN:9781848902503. Webpage
 Béziau J.Y. (1997), What is manyvalued logic ? Proceedings of the 27th International Symposium on MultipleValued Logic, IEEE Computer Society, Los Alamitos, pp. 117–121.
 Malinowski, Gregorz, (2001), ManyValued Logics, in Goble, Lou, ed., The Blackwell Guide to Philosophical Logic. Blackwell.
 Bergmann, Merrie (2008), An introduction to manyvalued and fuzzy logic: semantics, algebras, and derivation systems, Cambridge University Press, ISBN 9780521881289
 Cignoli, R. L. O., D'Ottaviano, I, M. L., Mundici, D., (2000). Algebraic Foundations of Manyvalued Reasoning. Kluwer.
 Malinowski, Grzegorz (1993). Manyvalued logics. Clarendon Press. ISBN 9780198537878.
 S. Gottwald, A Treatise on ManyValued Logics. Studies in Logic and Computation, vol. 9, Research Studies Press: Baldock, Hertfordshire, England, 2001.
 Gottwald, Siegfried (2005). ManyValued Logics. http://www.unileipzig.de/~logik/gottwald/SGforDJ.pdf.
 Miller, D. Michael; Thornton, Mitchell A. (2008). Multiple valued logic: concepts and representations. Synthesis lectures on digital circuits and systems. 12. Morgan & Claypool Publishers. ISBN 9781598291902.
 Hájek P., (1998), Metamathematics of fuzzy logic. Kluwer. (Fuzzy logic understood as manyvalued logic sui generis.)
Specific
 Alexandre Zinoviev, Philosophical Problems of ManyValued 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.
 Novák, V., Perfilieva, I., Močkoř, J., (1999), Mathematical Principles of Fuzzy Logic. Kluwer, Boston.
 Pavelka J. 1979, On fuzzy logic I: Manyvalued rules of inference, Zeitschr. f. math. Logik und Grundlagen d. Math., 25, 45–52.
 Metcalfe, George; Olivetti, Nicola; Dov M. Gabbay (2008). Proof Theory for Fuzzy Logics. Springer. ISBN 9781402094088. Covers proof theory of manyvalued logics as well, in the tradition of Hájek.
 Hähnle, Reiner (1993). Automated deduction in multiplevalued logics. Clarendon Press. ISBN 9780198539896.
 Azevedo, Francisco (2003). Constraint solving over multivalued logics: application to digital circuits. IOS Press. ISBN 9781586033040.
 Bolc, Leonard; Borowik, Piotr (2003). Manyvalued Logics 2: Automated reasoning and practical applications. Springer. ISBN 9783540645078.
 Stanković, Radomir S.; Astola, Jaakko T.; Moraga, Claudio (2012). Representation of MultipleValued Logic Functions. Morgan & Claypool Publishers. doi:10.2200/S00420ED1V01Y201205DCS037. ISBN 9781608459421.
 Abramovici, Miron; Breuer, Melvin A.; Friedman, Arthur D. (1994). Digital Systems Testing and Testable Design. New York: Computer Science Press. ISBN 9780780310629.
External links
 Gottwald, Siegfried (2022). "ManyValued Logic". in Zalta, Edward N.. The Stanford Encyclopedia of Philosophy (Summer 2022 Edition). https://plato.stanford.edu/archives/sum2022/entries/logicmanyvalued/.
 Shramko, Yaroslav; Wansing, Heinrich (2021). "Truth Values". in Zalta, Edward N.. The Stanford Encyclopedia of Philosophy (Winter 2021 Edition). https://plato.stanford.edu/archives/win2021/entries/truthvalues/.
 IEEE Computer Society's Technical Committee on MultipleValued Logic
 Resources for ManyValued Logic by Reiner Hähnle, Chalmers University
 Manyvalued Logics W3 Server (archived)
 Yaroslav Shramko and Heinrich Wansing (2020). "Suszko's Thesis". Stanford Encyclopedia of Philosophy. https://plato.stanford.edu/entries/truthvalues/suszkothesis.html.
 Carlos Caleiro, Walter Carnielli, Marcelo E. Coniglio and João Marcos, Two's company: "The humbug of many logical values" in JeanYves Beziau, ed (2007). Logica Universalis: Towards a General Theory of Logic (2nd ed.). Springer Science & Business Media. pp. 174–194. ISBN 9783764383541.
Original source: https://en.wikipedia.org/wiki/Manyvalued logic.
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