View source for Infinite-valued logic

{{short description|Many-valued logic in which truth values comprise a continuous range}}
In [[logic]], an '''infinite-valued logic''' (or '''real-valued logic''' or '''infinitely-many-valued logic''') is a [[many-valued logic]] in which [[truth values]] comprise a [[continuous or discrete variable|continuous]] range. Traditionally, in [[Aristotle's logic]], logic other than [[principle of bivalence|bivalent logic]] was abnormal, as the [[law of the excluded middle]] precluded more than two possible values (i.e., "true" and "false") for any [[proposition]].<ref>{{cite web| last=Weisstein| first=Eric|authorlink = Eric Weisstein| title=Law of the Excluded Middle| publisher=[[MathWorld]]--A Wolfram Web Resource| year=2018| url=http://mathworld.wolfram.com/LawoftheExcludedMiddle.html}}</ref> Modern [[three-valued logic]] (trivalent logic) allows for an additional possible truth value (i.e., "undecided")<ref>{{cite web| last=Weisstein| first=Eric| title=Three-Valued Logic| publisher=MathWorld--A Wolfram Web Resource| year=2018| url=http://mathworld.wolfram.com/Three-ValuedLogic.html}}</ref> and is an example of [[finite-valued logic]] in which truth values are discrete, rather than continuous. Infinite-valued logic comprises continuous [[fuzzy logic]], though fuzzy logic in some of its forms can further encompass finite-valued logic. For example, finite-valued logic can be applied in [[Boolean-valued model]]ing,<ref>{{cite thesis| last=Klawltter| first=Warren A.| work=Theses and Dissertations, paper 2025| year=1976| publisher=Lehigh Preserve| title=Boolean values for fuzzy sets| url=https://preserve.lehigh.edu/cgi/viewcontent.cgi?article=3025&context=etd}}</ref><ref>{{cite web| work=4th Serbian-Hungarian Joint Symposium on Intelligent Systems| last=Perović| first=Aleksandar| title=Fuzzy Sets – a Boolean Valued Approach| publisher=Conferences and Symposia @ Óbuda University| year=2006| url=http://conf.uni-obuda.hu/sisy2006/28_Perovic.pdf}}</ref> [[description logic]]s,<ref>{{cite journal| journal=International Journal of Approximate Reasoning| volume=55| issue=9| pages=1890–1916| year=2014| last1=Cerami| first1=Marco| last2=García-Cerdaña| first2=Àngel| last3=Esteva| first3=Frances| title=On finitely-valued Fuzzy Description Logics| doi=10.1016/j.ijar.2013.09.021| hdl=10261/131932| doi-access=free}}</ref> and [[defuzzification]]<ref>{{cite journal| last1=Schockaert| first1=Steven| last2=Janssen| first2=Jeroen| last3=Vermeir| first3=Dirk| journal=[[Journal of Automated Reasoning]]| volume=49| issue=4| pages=493–550| year=2012| title=Satisfiability Checking in Łukasiewicz Logic as Finite Constraint Satisfaction| doi=10.1007/s10817-011-9227-0}}</ref><ref>{{cite web| work=Fuzzy Logic| title=1.4.4 Defuzzification| page=4| publisher=Swiss Federal Institute of Technology Zurich| year=2014| url=http://control.ee.ethz.ch/~apnoco/Script/15.pdf| access-date=2018-05-16| archive-date=2009-07-09| archive-url=https://web.archive.org/web/20090709135440/http://control.ee.ethz.ch/~apnoco/Script/15.pdf| url-status=dead}}</ref> of fuzzy logic.

== History ==

[[Isaac Newton]] and [[Gottfried Wilhelm Leibniz]] used both [[infinity|infinities]] and [[infinitesimals]] to develop the differential and integral [[calculus]] in the late 17th century. [[Richard Dedekind]], who defined [[real number]]s in terms of [[construction of the real numbers#Construction by Dedekind cuts|certain sets]] of [[rational number]]s in the 19th century,<ref>{{cite web| last=Jones| first=Roger Bishop| year=1996| title=Real Numbers - some history| url=http://www.rbjones.com/rbjpub/maths/math008.htm}}</ref> also developed an axiom of [[law of continuity|continuity]] stating that a single correct value exists at the limit of any [[trial and error]] [[approximation#Mathematics|approximation]]. [[Felix Hausdorff]] demonstrated the logical possibility of an [[absolute continuity|absolutely continuous]] ordering of words comprising bivalent values, each word having [[absolute infinite|absolutely infinite]] length, in 1938. However, the definition of a random real number, meaning a real number that has no finite description whatsoever, remains somewhat in the realm of [[paradox]].<ref>{{cite book| publisher=Princeton University Press| last=Rucker| first=Rudy| title= Infinity and the Mind| url=http://www.rudyrucker.com/infinityandthemind/#calibre_link-311 |section=sections 311 "Infinitesimals and Surreal Numbers" and 317 "Random Reals"}}</ref>

[[Jan Łukasiewicz]] developed a system of three-valued logic in 1920. He generalized the system to many-valued logics in 1922 and went on to develop [[Łukasiewicz logic|logic]]s with [[aleph number|<math>\aleph_0</math>]] (infinite within a range) truth values. [[Kurt Gödel]] developed a [[Gödel logic|deductive system]], applicable for both finite- and infinite-valued [[first-order logic]] (a formal logic in which a [[predicate (mathematical logic)|predicate]] can refer to a single [[proposition|subject]]) as well as for [[intermediate logic]] (a formal [[intuitionistic logic]] usable to provide proofs such as a [[dialectica interpretation|consistency proof]] for [[arithmetic]]), and showed in 1932 that [[logical intuition]] cannot be characterized by [[finite-valued logic]].<ref>{{cite book| work=The Development of Modern Logic| last1=Mancosu| first1=Paolo| last2=Zach| first2=Richard| last3=Badesa| first3=Calixto| title=9. The Development of Mathematical Logic from Russell to Tarski 1900-1935| section=7.2 Many-valued logics| publisher=Oxford University Press| pages=418–420| year=2004| url=https://books.google.com/books?id=0jXavKsArnIC| isbn=9780199722723}}</ref>

The concept of expressing truth values as real numbers in the range between 0 and 1 can bring to mind the possibility of using [[complex number]]s to express truth values. These truth values would have an [[imaginary number|imaginary]] dimension, for example between 0 and  {{math|''i''}}. Two- or higher-dimensional truth could potentially be useful in systems of [[paraconsistent logic]]. If practical applications were to arise for such systems, [[multidimensional system|multidimensional]] infinite-valued logic could develop as a concept independent of real-valued logic.<ref>{{cite web| last=Gershenson| first=Carlos| title=Multidimensional Logic: A model for Paraconsistent Logic| publisher=Cogprints Cognitive Sciences EPrint Archive| url=http://cogprints.org/1479/1/mdl.html}}</ref>

[[Lotfi A. Zadeh]] proposed a formal methodology of [[fuzzy logic]] and its applications in the early 1970s. By 1973, other researchers were applying the theory of Zadeh fuzzy controllers to various mechanical and industrial processes. The [[fuzzy modeling]] concept that evolved from this research was applied to [[neural network]]s in the 1980s and to [[machine learning]] in the 1990s. The formal methodology also led to generalizations of mathematical theories in the family of [[t-norm fuzzy logics]].<ref>{{cite web| last=Garrido| first=Angel| title=A Brief History of Fuzzy Logic| publisher=Revista EduSoft| year=2012| url=https://www.edusoft.ro/brain/index.php/brain/article/viewFile/308/390}}, Editorial</ref>

== Examples ==

Basic [[fuzzy logic]] is the logic of continuous [[t-norm]]s ([[binary operation]]s on the real unit interval [0, 1]).<ref>{{cite journal| journal=Soft Computing| volume=4| issue=2| pages=106–112| year=2000| last1=Cignoli| first1=R.| last2=Esteva| first2=F| last3=Godo| first3=L.| last4=Torrens| first4=A.| title=Basic Fuzzy Logic is the logic of continuous t-norms and their residua| doi=10.1007/s005000000044}}</ref> Applications involving [[fuzzy logic]] include [[facial recognition system]]s, [[home appliance]]s, [[anti-lock braking system]]s, [[automatic transmission]]s, controllers for [[rapid transit]] systems and [[unmanned aerial vehicle]]s, [[knowledge-based]] and [[engineering optimization]] systems, [[weather forecasting]], [[pricing]], and [[risk assessment]] [[systems modeling|modeling system]]s, [[medical diagnosis]] and treatment planning and [[commodity market|commodities]] trading systems, and more.<ref>{{cite journal| journal=Advances in Fuzzy Systems| volume=2013| pages=1–3| last1=Singh| first1=Harpreet| last2=Gupta| first2=Madan M.| last3=Meitzler| first3=Thomas| last4=Hou| first4=Zeng-Guang| last5=Garg| first5=Kum Kum| last6=Solo| first6=Ashu M. G.| title=Real-Life Applications of Fuzzy Logic| year=2013| doi=10.1155/2013/581879| doi-access=free}}</ref> Fuzzy logic is used to optimize efficiency in [[thermostat]]s for control of heating and cooling, for industrial [[automation]] and [[process control]], [[computer animation]], [[signal processing]], and [[data analysis]].<ref>{{cite web| last=Klingenberg| first=Bryan| title=Fuzzy Logic Applications| publisher=Calvin College Engineering Department| url=https://www.calvin.edu/~pribeiro/othrlnks/Fuzzy/apps.htm| access-date=2018-05-16| archive-date=2018-05-10| archive-url=https://web.archive.org/web/20180510133027/http://www.calvin.edu/~pribeiro/othrlnks/Fuzzy/apps.htm| url-status=dead}}</ref> Fuzzy logic has made significant contributions in the fields of [[machine learning]] and [[data mining]].<ref>{{cite journal| journal=Fuzzy Sets and Systems| volume=156| issue=3| year=2005| pages=387–406| last=Hüllermeier| first=Eyke| title=Fuzzy methods in machine learning and data mining: Status and prospects| url=https://pdfs.semanticscholar.org/6b96/b2b2b252bdd0502d2a2e03ae8ae009bc77a1.pdf| archive-url=https://web.archive.org/web/20180517082238/https://pdfs.semanticscholar.org/6b96/b2b2b252bdd0502d2a2e03ae8ae009bc77a1.pdf| url-status=dead| archive-date=2018-05-17| doi=10.1016/j.fss.2005.05.036| s2cid=10034299}}</ref>

In [[infinitary logic]], degrees of provability of propositions can be expressed in terms of infinite-valued logic that can be described via evaluated formulas, written as ordered pairs each consisting of a truth degree symbol and a formula.<ref>{{cite journal| journal=Many-Valued Logics| last=Gottwald| first=Siegfried| title=12. Pavelka Style Extensions| pages=40–41| publisher=philpapers.org| year=2005| doi=10.1016/B978-044451541-4/50021-X| s2cid=8412503| url=https://pdfs.semanticscholar.org/be53/749719383724aff246f656a73e8a2c1ce702.pdf| archive-url=https://web.archive.org/web/20180517082239/https://pdfs.semanticscholar.org/be53/749719383724aff246f656a73e8a2c1ce702.pdf| url-status=dead| archive-date=2018-05-17}}</ref>

In [[mathematics]], number-free semantics can express facts about classical mathematical notions and make them derivable by logical deductions in infinite-valued logic. [[T-norm fuzzy logic]]s can be applied to eliminate references to real numbers from definitions and theorems, in order to simplify certain mathematical concepts and facilitate certain generalizations. A framework employed for number-free formalization of mathematical concepts is known as fuzzy class theory.<ref>{{cite web| last=Běhounek| first=Libor| title=Number-free Mathematics Based on T-norm Fuzzy Logic| publisher=University of Ostrava| year=2009| s2cid=9991521| url=https://pdfs.semanticscholar.org/284d/5342de90884fae07eaabe4a295e745af419b.pdf| archive-url=https://web.archive.org/web/20180517082219/https://pdfs.semanticscholar.org/284d/5342de90884fae07eaabe4a295e745af419b.pdf| url-status=dead| archive-date=2018-05-17}}</ref>

Philosophical questions, including the [[Sorites paradox]], have been considered based on an infinite-valued logic known as fuzzy [[epistemicism]].<ref>{{cite book| work=Cuts and Clouds| last=MacFarlane| first=John| title=Fuzzy Epistemicism| publisher=Oxford University Press| year=2010| url=https://johnmacfarlane.net/fuzzy-epistemicism.pdf}}</ref> The Sorites paradox suggests that if adding a grain of sand to something that is not a heap cannot create a heap, then a heap of sand cannot be created. A stepwise approach toward a limit, in which truth is gradually "leaked", tends to refute that suggestion.<ref>{{cite journal| journal=[[Synthese]]| volume=134| issue=3| year=2003| pages=363–387| last=Paoli| first=Francesco| title=A Really Fuzzy Approach to the Sorites Paradox| doi=10.1023/A:1022995202767}}</ref>

In the study of [[logic]] itself, infinite-valued logic has served as an aid to understand the nature of the human understanding of logical concepts. [[Kurt Gödel]] attempted to comprehend the human ability for [[logical intuition]] in terms of finite-valued logic before concluding that the ability is based on infinite-valued logic.<ref>{{cite web| last=Burgess| first=John| title=Intuitions of Three Kinds in Gödel's Views on the Continuum| url=https://www.princeton.edu/~jburgess/Goedel.pdf}}</ref> Open questions remain regarding the handling, in [[natural language]] semantics, of indeterminate truth values.<ref>"The moral: an adequate theory must allow our statements involving the notion of truth to be risky: they risk being paradoxical if the empirical facts are extremely (and unexpectedly) unfavorable. There can be no syntactic or semantic 'sieve' that will winnow out the 'bad' cases while preserving the 'good' ones. ... I  am somewhat uncertain whether there is a definite factual question as to whether natural language handles truth-value gaps &mdash; at least those arising in connection with the semantic paradoxes &mdash; by the schemes of [[Gottlob Frege|Frege]], [[Stephen Cole Kleene|Kleene]], [[Bas van Fraassen|van Fraassen]], or perhaps some other." {{cite journal| journal=[[The Journal of Philosophy]]| volume=72| number=19| pages=690–716| year=1975| last=Kripke| first=Saul| title=Outline of a Theory of Truth| url=https://www.impan.pl/~kz/truthseminar/Kripke_Outline.pdf| doi=10.2307/2024634| jstor=2024634}}</ref>

== See also ==
*[[Many-valued logic]]
*[[Finite-valued logic]]
*[[Intuitionistic logic]]
*[[Logical intuition]]
*[[Fuzzy logic]]
*[[Fuzzy number]]
*[[Construction of t-norms]]
*[[Set theory]]

==References==
{{reflist}}

{{Mathematical logic}}

[[Category:Logic]]
[[Category:Many-valued logic]]