logic Logic is the study of correct reasoning. It includes both formal and informal logic. Formal logic is the science of deductively valid inferences or of logical truths. It is a formal science investigating how conclusions follow from prem ...

, an infinite-valued logic (or real-valued logic or infinitely-many-valued logic) is a

many-valued logic Many-valued logic (also multi- or multiple-valued 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 ...

in which truth values comprise a

continuous Continuity or continuous may refer to: Mathematics * Continuity (mathematics), the opposing concept to discreteness; common examples include ** Continuous probability distribution or random variable in probability and statistics ** Continuous g ...

range. Traditionally, in Aristotle's logic, logic other than

bivalent logic In logic, the semantic principle (or law) of bivalence states that every declarative sentence expressing a proposition (of a theory under inspection) has exactly one truth value, either true or false. A logic satisfying this principle is cal ...

was abnormal, as the

law of the excluded middle In logic, the law of excluded middle (or the principle of excluded middle) states that for every proposition, either this proposition or its negation is true. It is one of the so-called three laws of thought, along with the law of noncontradi ...

precluded more than two possible values (i.e., "true" and "false") for any

proposition In logic and linguistics, a proposition is the meaning of a declarative sentence. In philosophy, " meaning" is understood to be a non-linguistic entity which is shared by all sentences with the same meaning. Equivalently, a proposition is the no ...

. Modern three-valued logic (ternary logic) allows for an additional possible truth value (i.e., "undecided") and is an example of finite-valued logic in which truth values are discrete, rather than continuous. Infinite-valued logic comprises continuous

fuzzy logic Fuzzy logic is a form of many-valued logic in which the truth value of variables may be any real number between 0 and 1. It is employed to handle the concept of partial truth, where the truth value may range between completely true and completel ...

, 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 In mathematical logic, a Boolean-valued model is a generalization of the ordinary Tarskian notion of structure from model theory. In a Boolean-valued model, the truth values of propositions are not limited to "true" and "false", but instead take v ...

ing, description logics, and

defuzzification Defuzzification is the process of producing a quantifiable result in crisp logic, given fuzzy sets and corresponding membership degrees. It is the process that maps a fuzzy set to a crisp set. It is typically needed in fuzzy control systems. Th ...

of fuzzy logic.

History

Isaac Newton Sir Isaac Newton (25 December 1642 – 20 March 1726/27) was an English mathematician, physicist, astronomer, alchemist, Theology, theologian, and author (described in his time as a "natural philosophy, natural philosopher"), widely ...

and

Gottfried Wilhelm Leibniz Gottfried Wilhelm (von) Leibniz . ( – 14 November 1716) was a German polymath active as a mathematician, philosopher, scientist and diplomat. He is one of the most prominent figures in both the history of philosophy and the history of ...

used both infinities and

infinitesimals In mathematics, an infinitesimal number is a quantity that is closer to zero than any standard real number, but that is not zero. The word ''infinitesimal'' comes from a 17th-century Modern Latin coinage ''infinitesimus'', which originally refe ...

to develop the differential and integral

calculus Calculus, originally called infinitesimal calculus or "the calculus of infinitesimals", is the mathematics, mathematical study of continuous change, in the same way that geometry is the study of shape, and algebra is the study of generalizati ...

in the late 17th century.

Richard Dedekind Julius Wilhelm Richard Dedekind (6 October 1831 – 12 February 1916) was a German mathematician who made important contributions to number theory, abstract algebra (particularly ring theory), and the axiomatic foundations of arithmetic. His ...

, who defined

real number In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every ...

s in terms of certain sets of

rational number In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ). The set of all ra ...

s in the 19th century, also developed an axiom of continuity stating that a single correct value exists at the limit of any

trial and error Trial and error is a fundamental method of problem-solving characterized by repeated, varied attempts which are continued until success, or until the practicer stops trying. According to W.H. Thorpe, the term was devised by C. Lloyd Morgan (18 ...

approximation An approximation is anything that is intentionally similar but not exactly equal to something else. Etymology and usage The word ''approximation'' is derived from Latin ''approximatus'', from ''proximus'' meaning ''very near'' and the prefix ' ...

Felix Hausdorff Felix Hausdorff ( , ; November 8, 1868 – January 26, 1942) was a German mathematician who is considered to be one of the founders of modern topology and who contributed significantly to set theory, descriptive set theory, measure theory, an ...

demonstrated the logical possibility of an

absolutely continuous In calculus, absolute continuity is a smoothness property of functions that is stronger than continuity and uniform continuity. The notion of absolute continuity allows one to obtain generalizations of the relationship between the two central ope ...

ordering of words comprising bivalent values, each word having 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 A paradox is a logically self-contradictory statement or a statement that runs contrary to one's expectation. It is a statement that, despite apparently valid reasoning from true premises, leads to a seemingly self-contradictory or a logically u ...

Jan Łukasiewicz Jan Łukasiewicz (; 21 December 1878 – 13 February 1956) was a Polish logician and philosopher who is best known for Polish notation and Łukasiewicz logic His work centred on philosophical logic, mathematical logic and history of logic. ...

developed a system of three-valued logic in 1920. He generalized the system to many-valued logics in 1922 and went on to develop

s with

\aleph_0

(infinite within a range) truth values.

Kurt Gödel Kurt Friedrich Gödel ( , ; April 28, 1906 – January 14, 1978) was a logician, mathematician, and philosopher. Considered along with Aristotle and Gottlob Frege to be one of the most significant logicians in history, Gödel had an imm ...

developed a

deductive system A formal system is an abstract structure used for inferring theorems from axioms according to a set of rules. These rules, which are used for carrying out the inference of theorems from axioms, are the logical calculus of the formal system. A for ...

, applicable for both finite- and infinite-valued

first-order logic First-order logic—also known as predicate logic, quantificational logic, and first-order predicate calculus—is a collection of formal systems used in mathematics, philosophy, linguistics, and computer science. First-order logic uses quantifie ...

(a formal logic in which a

predicate Predicate or predication may refer to: * Predicate (grammar), in linguistics * Predication (philosophy) * several closely related uses in mathematics and formal logic: **Predicate (mathematical logic) **Propositional function **Finitary relation, o ...

can refer to a single subject) as well as for

intermediate logic In mathematical logic, a superintuitionistic logic is a propositional logic extending intuitionistic logic. Classical logic is the strongest consistent superintuitionistic logic; thus, consistent superintuitionistic logics are called intermediate l ...

(a formal

intuitionistic logic Intuitionistic logic, sometimes more generally called constructive logic, refers to systems of symbolic logic that differ from the systems used for classical logic by more closely mirroring the notion of constructive proof. In particular, systems ...

usable to provide proofs such as a

consistency proof In classical deductive logic, a consistent theory is one that does not lead to a logical contradiction. The lack of contradiction can be defined in either semantic or syntactic terms. The semantic definition states that a theory is consistent ...

for

arithmetic Arithmetic () is an elementary part of mathematics that consists of the study of the properties of the traditional operations on numbers— addition, subtraction, multiplication, division, exponentiation, and extraction of roots. In the 19th ...

), and showed in 1932 that

logical intuition Logical Intuition, or mathematical intuition or rational intuition, is a series of instinctive foresight, know-how, and savviness often associated with the ability to perceive logical or mathematical truth—and the ability to solve mathematical ...

cannot be characterized by finite-valued logic. 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 In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the fo ...

s to express truth values. These truth values would have an imaginary dimension, for example between 0 and . Two- or higher-dimensional truth could potentially be useful in systems of

paraconsistent logic A paraconsistent logic is an attempt at a logical system to deal with contradictions in a discriminating way. Alternatively, paraconsistent logic is the subfield of logic that is concerned with studying and developing "inconsistency-tolerant" syst ...

. If practical applications were to arise for such systems, multidimensional infinite-valued logic could develop as a concept independent of real-valued logic. Lotfi A. Zadeh proposed a formal methodology of

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 A neural network is a network or circuit of biological neurons, or, in a modern sense, an artificial neural network, composed of artificial neurons or nodes. Thus, a neural network is either a biological neural network, made up of biological ...

s in the 1980s and to

machine learning Machine learning (ML) is a field of inquiry devoted to understanding and building methods that 'learn', that is, methods that leverage data to improve performance on some set of tasks. It is seen as a part of artificial intelligence. Machine ...

in the 1990s. The formal methodology also led to generalizations of mathematical theories in the family of t-norm fuzzy logics.

Examples

Basic

is the logic of continuous

t-norm In mathematics, a t-norm (also T-norm or, unabbreviated, triangular norm) is a kind of binary operation used in the framework of probabilistic metric spaces and in multi-valued logic, specifically in fuzzy logic. A t-norm generalizes intersection (s ...

s (

binary operation In mathematics, a binary operation or dyadic operation is a rule for combining two elements (called operands) to produce another element. More formally, a binary operation is an operation of arity two. More specifically, an internal binary op ...

s on the real unit interval

, 1 The comma is a punctuation mark that appears in several variants in different languages. It has the same shape as an apostrophe or single closing quotation mark () in many typefaces, but it differs from them in being placed on the baseline o ...

. Applications involving

include

facial recognition system A facial recognition system is a technology capable of matching a human face from a digital image or a video frame against a database of faces. Such a system is typically employed to authenticate users through ID verification services, and ...

home appliance A home appliance, also referred to as a domestic appliance, an electric appliance or a household appliance, is a machine which assists in household functions such as cooking, cleaning and food preservation. Appliances are divided into three ...

anti-lock braking system An anti-lock braking system (ABS) is a safety anti-skid braking system used on aircraft and on land vehicles, such as cars, motorcycles, trucks, and buses. ABS operates by preventing the wheels from locking up during braking, thereby maint ...

automatic transmission An automatic transmission (sometimes abbreviated to auto or AT) is a multi-speed transmission used in internal combustion engine-based motor vehicles that does not require any input from the driver to change forward gears under normal driving ...

s, controllers for

rapid transit Rapid transit or mass rapid transit (MRT), also known as heavy rail or metro, is a type of high-capacity public transport generally found in urban areas. A rapid transit system that primarily or traditionally runs below the surface may be ...

systems and

unmanned aerial vehicle An unmanned aerial vehicle (UAV), commonly known as a drone, is an aircraft without any human pilot, crew, or passengers on board. UAVs are a component of an unmanned aircraft system (UAS), which includes adding a ground-based controll ...

s, knowledge-based and engineering optimization systems,

weather forecasting Weather forecasting is the application of science and technology to predict the conditions of the atmosphere for a given location and time. People have attempted to predict the weather informally for millennia and formally since the 19th cen ...

pricing Pricing is the process whereby a business sets the price at which it will sell its products and services, and may be part of the business's marketing plan. In setting prices, the business will take into account the price at which it could acq ...

, and

risk assessment Broadly speaking, a risk assessment is the combined effort of: # identifying and analyzing potential (future) events that may negatively impact individuals, assets, and/or the environment (i.e. hazard analysis); and # making judgments "on the ...

modeling systems,

medical diagnosis Medical diagnosis (abbreviated Dx, Dx, or Ds) is the process of determining which disease or condition explains a person's symptoms and signs. It is most often referred to as diagnosis with the medical context being implicit. The information r ...

and treatment planning and

commodities In economics, a commodity is an economic good, usually a resource, that has full or substantial fungibility: that is, the market treats instances of the good as equivalent or nearly so with no regard to who produced them. The price of a co ...

trading systems, and more. Fuzzy logic is used to optimize efficiency in

thermostat A thermostat is a regulating device component which senses the temperature of a physical system and performs actions so that the system's temperature is maintained near a desired setpoint (control system), setpoint. Thermostats are used i ...

s for control of heating and cooling, for industrial

automation Automation describes a wide range of technologies that reduce human intervention in processes, namely by predetermining decision criteria, subprocess relationships, and related actions, as well as embodying those predeterminations in machines ...

and

process control An industrial process control in continuous production processes is a discipline that uses industrial control systems to achieve a production level of consistency, economy and safety which could not be achieved purely by human manual control. ...

computer animation Computer animation is the process used for digitally generating animations. The more general term computer-generated imagery (CGI) encompasses both static scenes ( still images) and dynamic images ( moving images), while computer animation re ...

signal processing Signal processing is an electrical engineering subfield that focuses on analyzing, modifying and synthesizing '' signals'', such as sound, images, and scientific measurements. Signal processing techniques are used to optimize transmissions, ...

, and

data analysis Data analysis is a process of inspecting, cleansing, transforming, and modeling data with the goal of discovering useful information, informing conclusions, and supporting decision-making. Data analysis has multiple facets and approaches, enc ...

. Fuzzy logic has made significant contributions in the fields of

and data mining. In

infinitary logic An infinitary logic is a logic that allows infinitely long statements and/or infinitely long proofs. Some infinitary logics may have different properties from those of standard first-order logic. In particular, infinitary logics may fail to be co ...

, 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. In

mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern 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 logics 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. Philosophical questions, including the

Sorites paradox The sorites paradox (; sometimes known as the paradox of the heap) is a paradox that results from vague predicates. A typical formulation involves a heap of sand, from which grains are removed individually. With the assumption that removing a sin ...

, have been considered based on an infinite-valued logic known as fuzzy

epistemicism Epistemicism is a position about vagueness in the philosophy of language or metaphysics, according to which there are facts about the boundaries of a vague predicate which we cannot possibly discover. Given a vague predicate, such as 'is thin' or ...

. 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. In the study of

itself, infinite-valued logic has served as an aid to understand the nature of the human understanding of logical concepts.

attempted to comprehend the human ability for

in terms of finite-valued logic before concluding that the ability is based on infinite-valued logic. Open questions remain regarding the handling, in

natural language In neuropsychology, linguistics, and philosophy of language, a natural language or ordinary language is any language that has evolved naturally in humans through use and repetition without conscious planning or premeditation. Natural languages ...

semantics, of indeterminate truth values."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 — at least those arising in connection with the semantic paradoxes — by the schemes of

Frege Friedrich Ludwig Gottlob Frege (; ; 8 November 1848 – 26 July 1925) was a German philosopher, logician, and mathematician. He was a mathematics professor at the University of Jena, and is understood by many to be the father of analytic p ...

Kleene Stephen Cole Kleene ( ; January 5, 1909 – January 25, 1994) was an American mathematician. One of the students of Alonzo Church, Kleene, along with Rózsa Péter, Alan Turing, Emil Post, and others, is best known as a founder of the branch of ...

van Fraassen A van is a type of road vehicle used for transporting goods or people. Depending on the type of van, it can be bigger or smaller than a pickup truck and SUV, and bigger than a common car. There is some varying in the scope of the word across th ...

, or perhaps some other."

References

{{Mathematical logic Logic Many-valued logic

History

Examples

See also

References