Many-valued logic

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 [two-valued] 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 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→∞. 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.

Kleene's "(strong) logic of indeterminacy" K₃ (sometimes ${\displaystyle K_{3}^{S}}$ ) 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:^[3]

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 nor 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.^[4]

Another logic is Dmitry Bochvar's "internal" three-valued logic ${\displaystyle B_{3}^{I}}$ , also called Kleene's weak three-valued 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's logic B₄ combines K₃ and P₃. The overdetermined truth value is here denoted as B and the underdetermined truth value as N.

In 1932 Gödel defined^[6] a family ${\displaystyle G_{k}}$  of many-valued logics, with finitely many truth values ${\displaystyle 0,{\tfrac {1}{k-1}},{\tfrac {2}{k-1}},\ldots ,{\tfrac {k-2}{k-1}},1}$ , for example ${\displaystyle G_{3}}$  has the truth values ${\displaystyle 0,{\tfrac {1}{2}},1}$  and ${\displaystyle G_{4}}$  has ${\displaystyle 0,{\tfrac {1}{3}},{\tfrac {2}{3}},1}$ . In a similar manner he defined a logic with infinitely many truth values, ${\displaystyle G_{\infty }}$ , in which the truth values are all the real numbers in the interval ${\displaystyle [0,1]}$ . The designated truth value in these logics is 1.

The conjunction ${\displaystyle \wedge }$  and the disjunction ${\displaystyle \vee }$  are defined respectively as the minimum and maximum of the operands:

${\displaystyle {\begin{aligned}u\wedge v&:=\min\{u,v\}\\u\vee v&:=\max\{u,v\}\end{aligned}}}$ 

Negation ${\displaystyle \neg _{G}}$  and implication ${\displaystyle {\xrightarrow[{G}]{}}}$  are defined as follows:

${\displaystyle {\begin{aligned}\neg _{G}u&={\begin{cases}1,&{\text{if }}u=0\\0,&{\text{if }}u>0\end{cases}}\\[3pt]u\mathrel {\xrightarrow[{G}]{}} v&={\begin{cases}1,&{\text{if }}u\leq v\\v,&{\text{if }}u>v\end{cases}}\end{aligned}}}$ 

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.

Implication ${\displaystyle {\xrightarrow[{L}]{}}}$  and negation ${\displaystyle {\underset {L}{\neg }}}$  were defined by Jan Łukasiewicz through the following functions:

${\displaystyle {\begin{aligned}{\underset {L}{\neg }}u&:=1-u\\u\mathrel {\xrightarrow[{L}]{}} v&:=\min\{1,1-u+v\}\end{aligned}}}$ 

At first Łukasiewicz used these definitions in 1920 for his three-valued logic ${\displaystyle L_{3}}$ , with truth values ${\displaystyle 0,{\frac {1}{2}},1}$ . In 1922 he developed a logic with infinitely many values ${\displaystyle L_{\infty }}$ , in which the truth values spanned the real numbers in the interval ${\displaystyle [0,1]}$ . In both cases the designated truth value was 1.^[7]

By adopting truth values defined in the same way as for Gödel logics ${\displaystyle 0,{\tfrac {1}{v-1}},{\tfrac {2}{v-1}},\ldots ,{\tfrac {v-2}{v-1}},1}$ , it is possible to create a finitely-valued family of logics ${\displaystyle L_{v}}$ , the abovementioned ${\displaystyle L_{\infty }}$  and the logic ${\displaystyle L_{\aleph _{0}}}$ , in which the truth values are given by the rational numbers in the interval ${\displaystyle [0,1]}$ . The set of tautologies in ${\displaystyle L_{\infty }}$  and ${\displaystyle L_{\aleph _{0}}}$  is identical.

In product logic we have truth values in the interval ${\displaystyle [0,1]}$ , a conjunction ${\displaystyle \odot }$  and an implication ${\displaystyle {\xrightarrow[{\Pi }]{}}}$ , defined as follows^[8]

${\displaystyle {\begin{aligned}u\odot v&:=uv\\u\mathrel {\xrightarrow[{\Pi }]{}} v&:={\begin{cases}1,&{\text{if }}u\leq v\\{\frac {v}{u}},&{\text{if }}u>v\end{cases}}\end{aligned}}}$ 

Additionally there is a negative designated value ${\displaystyle {\overline {0}}}$  that denotes the concept of false. Through this value it is possible to define a negation ${\displaystyle {\underset {\Pi }{\neg }}}$  and an additional conjunction ${\displaystyle {\underset {\Pi }{\wedge }}}$  as follows:

${\displaystyle {\begin{aligned}{\underset {\Pi }{\neg }}u&:=u\mathrel {\xrightarrow[{\Pi }]{}} {\overline {0}}\\u\mathbin {\underset {\Pi }{\wedge }} v&:=u\odot \left(u\mathrel {\xrightarrow[{\Pi }]{}} v\right)\end{aligned}}}$ 

and then ${\displaystyle u\mathbin {\underset {\Pi }{\wedge }} v=\min\{u,v\}}$ .

In 1921 Post defined a family of logics ${\displaystyle P_{m}}$  with (as in ${\displaystyle L_{v}}$  and ${\displaystyle G_{k}}$ ) the truth values ${\displaystyle 0,{\tfrac {1}{m-1}},{\tfrac {2}{m-1}},\ldots ,{\tfrac {m-2}{m-1}},1}$ . Negation ${\displaystyle {\underset {P}{\neg }}}$  and conjunction ${\displaystyle {\underset {P}{\wedge }}}$  and disjunction ${\displaystyle {\underset {P}{\vee }}}$  are defined as follows:

${\displaystyle {\begin{aligned}{\underset {P}{\neg }}u&:={\begin{cases}1,&{\text{if }}u=0\\u-{\frac {1}{m-1}},&{\text{if }}u\not =0\end{cases}}\\[6pt]u\mathbin {\underset {P}{\wedge }} v&:=\min\{u,v\}\\[6pt]u\mathbin {\underset {P}{\vee }} v&:=\max\{u,v\}\end{aligned}}}$ 

In 1951, Alan Rose defined another family of logics for systems whose truth-values form lattices.^[9]

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.

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 many-valued logics has this property.^[11][12]

We can define a finitely many-valued logic as being L_n ({1, 2, ..., n} ƒ₁, ..., 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]

Known applications of many-valued logic can be roughly classified into two groups.^[14] The first group uses many-valued logic 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 (specifically, the indicator 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 that employ more than two discrete levels of signals, such as many-valued memories, arithmetic circuits, and field programmable gate arrays (FPGAs). 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^[15] can reduce or eliminate the ripple-through carries that 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. In addition to aiding in the design of electronic circuits, many-valued 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 5-valued 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.

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.^[17] There is also a Journal of Multiple-Valued Logic and Soft Computing.^[18]

Mathematical logic

• Degrees of truth
• Fuzzy logic
• Gödel logic
• Jaina seven-valued logic
• Kleene logic
• Kleene algebra (with involution)
• Łukasiewicz logic
• MV-algebra
• Post logic
• Principle of bivalence
• A. N. Prior
• Relevance logic

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
• Three-state logic
• Noise-based logic

General

• Augusto, Luis M. (2017). Many-valued logics: A mathematical and computational introduction. London: College Publications. 340 pages. ISBN 978-1-84890-250-3. Webpage
• 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
• Cignoli, R. L. O., D'Ottaviano, I, M. L., Mundici, D., (2000). Algebraic Foundations of Many-valued Reasoning. Kluwer.
• Malinowski, Grzegorz (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). Archived from the original on March 3, 2016. {{cite journal}}: Cite journal requires |journal= (help)CS1 maint: bot: original URL status unknown (link)
• Miller, D. Michael; Thornton, Mitchell A. (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.
• Novák, V., Perfilieva, I., Močkoř, J., (1999), Mathematical Principles of Fuzzy Logic. Kluwer, Boston.
• Pavelka J. 1979, On fuzzy logic I: Many-valued 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 978-1-4020-9408-8. Covers proof theory of many-valued logics as well, in the tradition of Hájek.
• Hähnle, Reiner (1993). Automated deduction in multiple-valued logics. Clarendon Press. ISBN 978-0-19-853989-6.
• Azevedo, Francisco (2003). Constraint solving over multi-valued logics: application to digital circuits. IOS Press. ISBN 978-1-58603-304-0.
• Bolc, Leonard; Borowik, Piotr (2003). Many-valued Logics 2: Automated reasoning and practical applications. Springer. ISBN 978-3-540-64507-8.
• Stanković, Radomir S.; Astola, Jaakko T.; Moraga, Claudio (2012). Representation of Multiple-Valued Logic Functions. Morgan & Claypool Publishers. doi:10.2200/S00420ED1V01Y201205DCS037. ISBN 978-1-60845-942-1.
• Abramovici, Miron; Breuer, Melvin A.; Friedman, Arthur D. (1994). Digital Systems Testing and Testable Design. New York: Computer Science Press. ISBN 978-0-7803-1062-9.

• Gottwald, Siegfried (2022). "Many-Valued Logic". In Zalta, Edward N. (ed.). The Stanford Encyclopedia of Philosophy (Summer 2022 Edition).
• Shramko, Yaroslav and Wansing, Heinrich (2021). "Truth Values". In Zalta, Edward N. (ed.). The Stanford Encyclopedia of Philosophy (Winter 2021 Edition).
• 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)
• Yaroslav Shramko; Heinrich Wansing (2020). "Suszko's Thesis". Stanford Encyclopedia of Philosophy.
• Carlos Caleiro, Walter Carnielli, Marcelo E. Coniglio and João Marcos, Two's company: "The humbug of many logical values" in Jean-Yves Beziau, ed. (2007). Logica Universalis: Towards a General Theory of Logic (2nd ed.). Springer Science & Business Media. pp. 174–194. ISBN 978-3-7643-8354-1.