logic Logic is the study of correct reasoning. It includes both formal and informal logic. Formal logic is the study of deductively valid inferences or logical truths. It examines how conclusions follow from premises based on the structure o ...

, a finite-valued logic (also finitely many-valued logic) is a

propositional calculus The propositional calculus is a branch of logic. It is also called propositional logic, statement logic, sentential calculus, sentential logic, or sometimes zeroth-order logic. Sometimes, it is called ''first-order'' propositional logic to contra ...

in which

truth values In logic and mathematics, a truth value, sometimes called a logical value, is a value indicating the relation of a proposition to truth, which in classical logic has only two possible values (''true'' or '' false''). Truth values are used in co ...

are

discrete Discrete may refer to: *Discrete particle or quantum in physics, for example in quantum theory * Discrete device, an electronic component with just one circuit element, either passive or active, other than an integrated circuit * Discrete group, ...

. Traditionally, in Aristotle's logic, the

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 called ...

, also known as binary logic was the norm, 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 three laws of thought, along with the law of noncontradiction and th ...

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

proposition A proposition is a statement that can be either true or false. It is a central concept in the philosophy of language, semantics, logic, and related fields. Propositions are the object s denoted by declarative sentences; for example, "The sky ...

. Modern

three-valued logic In logic, a three-valued logic (also trinary logic, trivalent, ternary, or trilean, sometimes abbreviated 3VL) is any of several many-valued logic systems in which there are three truth values indicating ''true'', ''false'', and some third value ...

(ternary logic) allows for an additional possible truth value (i.e. "undecided"). The term finitely many-valued logic is typically used to describe

many-valued logic 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 Term logic, logical calculus, there were only two possible values (i.e., "true" and ...

having three or more, but not infinite, truth values. The term finite-valued logic encompasses both finitely many-valued logic and bivalent logic.

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 completely ...

s, which allow for degrees of values between "true" and "false", are typically not considered forms of finite-valued logic. However, 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 logic Description logics (DL) are a family of formal knowledge representation languages. Many DLs are more expressive than propositional logic but less expressive than first-order logic. In contrast to the latter, the core reasoning problems for DLs are ...

s, 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. The ...

of fuzzy logic. A finite-valued logic is decidable (sure to determine outcomes of the logic when it is applied to

s) if and only if it has a

computational semantics Computational semantics is the study of how to automate the process of constructing and reasoning with semantics, meaning representations of natural language expressions. It consequently plays an important role in natural language processing, nat ...

History

Aristotle Aristotle (; 384–322 BC) was an Ancient Greek philosophy, Ancient Greek philosopher and polymath. His writings cover a broad range of subjects spanning the natural sciences, philosophy, linguistics, economics, politics, psychology, a ...

's collected works regarding logic, known as the ''

Organon The ''Organon'' (, meaning "instrument, tool, organ") is the standard collection of Aristotle's six works on logical analysis and dialectic. The name ''Organon'' was given by Aristotle's followers, the Peripatetics, who maintained against the ...

'', describe bivalent logic primarily, though Aristotle's views may have allowed for propositions that are not actually true or false. The ''Organon'' influenced philosophers and mathematicians throughout the

Enlightenment Enlightenment or enlighten may refer to: Age of Enlightenment * Age of Enlightenment, period in Western intellectual history from the late 17th to late 18th century, centered in France but also encompassing (alphabetically by country or culture): ...

George Boole George Boole ( ; 2 November 1815 – 8 December 1864) was a largely self-taught English mathematician, philosopher and logician, most of whose short career was spent as the first professor of mathematics at Queen's College, Cork in Ireland. H ...

developed an

algebraic structure In mathematics, an algebraic structure or algebraic system consists of a nonempty set ''A'' (called the underlying set, carrier set or domain), a collection of operations on ''A'' (typically binary operations such as addition and multiplicatio ...

and an

algorithm In mathematics and computer science, an algorithm () is a finite sequence of Rigour#Mathematics, mathematically rigorous instructions, typically used to solve a class of specific Computational problem, problems or to perform a computation. Algo ...

probability theory Probability theory or probability calculus is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expre ...

based on bivalent logic in the 19th century.

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 logi ...

developed a system of three-valued logic in 1920.

Emil Leon Post Emil Leon Post (; February 11, 1897 – April 21, 1954) was an American mathematician and logician. He is best known for his work in the field that eventually became known as computability theory. Life Post was born in Augustów, Suwałki Gove ...

introduced further truth degrees in 1921.

Stephen Cole 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 ...

and Ulrich Blau expanded the three-valued logic system of Łukasiewicz, for

computer A computer is a machine that can be Computer programming, programmed to automatically Execution (computing), carry out sequences of arithmetic or logical operations (''computation''). Modern digital electronic computers can perform generic set ...

applications and for

natural language A natural language or ordinary language is a language that occurs naturally in a human community by a process of use, repetition, and change. It can take different forms, typically either a spoken language or a sign language. Natural languages ...

analyses, respectively.

Nuel Belnap Nuel Dinsmore Belnap Jr. (; May 1, 1930 – June 12, 2024) was an American logician and philosopher who has made contributions to the philosophy of logic, temporal logic, and structural proof theory. He taught at the University of Pittsburgh ...

and J. Michael Dunn developed a four-valued logic for computer applications in 1977. Since the mid-1970s, various procedures for providing arbitrary finite-valued logics have been developed.

Examples

linguistics Linguistics is the scientific study of language. The areas of linguistic analysis are syntax (rules governing the structure of sentences), semantics (meaning), Morphology (linguistics), morphology (structure of words), phonetics (speech sounds ...

, finite-valued logic is used to treat

presuppositions In linguistics and philosophy, a presupposition is an implicit assumption about the world or background belief relating to an utterance whose truth is taken for granted in discourse. Examples of presuppositions include: * ''Jane no longer writes ...

as product systems with ordered pairs of truth degrees, or

truth tables A truth table is a mathematical table used in logic—specifically in connection with Boolean algebra, Boolean functions, and propositional calculus—which sets out the functional values of logical expressions on each of their functional ar ...

. This enables assumptions built into verbal or written statements to be associated with varying degrees of truth values in the course of natural-language processing. In the study of

formal language In logic, mathematics, computer science, and linguistics, a formal language is a set of strings whose symbols are taken from a set called "alphabet". The alphabet of a formal language consists of symbols that concatenate into strings (also c ...

s, finite-valued logic has shown that encapsulating a

truth predicate In formal theories of truth, a truth predicate is a fundamental concept based on the sentences of a formal language as interpreted logically. That is, it formalizes the concept that is normally expressed by saying that a sentence, statement or idea ...

in a language can render the language

inconsistent In deductive logic, a consistent theory is one that does not lead to a logical contradiction. A theory T is consistent if there is no formula \varphi such that both \varphi and its negation \lnot\varphi are elements of the set of consequences o ...

Saul Kripke Saul Aaron Kripke (; November 13, 1940 – September 15, 2022) was an American analytic philosophy, analytic philosopher and logician. He was Distinguished Professor of Philosophy at the Graduate Center of the City University of New York and emer ...

has built on work pioneered by

Alfred Tarski Alfred Tarski (; ; born Alfred Teitelbaum;School of Mathematics and Statistics, University of St Andrews ''School of Mathematics and Statistics, University of St Andrews''. January 14, 1901 – October 26, 1983) was a Polish-American logician ...

to demonstrate that such a truth predicate can be modeled using three-valued logic. 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 s ...

, have been considered based on a finite-valued logic known as fuzzy plurivaluationism. 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 logical model of a heap in which there are as many truth degrees as grains of sand tends to refute that suggestion. In electronics design, a logical model of the stable states of a circuit, in which there are as many truth degrees as there are states, serves as a model for finite-valued switching. Three-valued operators can be realized in

integrated circuits An integrated circuit (IC), also known as a microchip or simply chip, is a set of electronic circuits, consisting of various electronic components (such as transistors, resistors, and capacitors) and their interconnections. These components a ...

. In

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 completely ...

, typically applied for approximate reasoning, a finitely-valued logic can represent propositions that may acquire values within a finite

set Set, The Set, SET or SETS may refer to: Science, technology, and mathematics Mathematics *Set (mathematics), a collection of elements *Category of sets, the category whose objects and morphisms are sets and total functions, respectively Electro ...

. In

mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

, logical

matrices Matrix (: matrices or matrixes) or MATRIX may refer to: Science and mathematics * Matrix (mathematics), a rectangular array of numbers, symbols or expressions * Matrix (logic), part of a formula in prenex normal form * Matrix (biology), the ...

having multiple truth degrees are used to model systems of

axiom An axiom, postulate, or assumption is a statement that is taken to be true, to serve as a premise or starting point for further reasoning and arguments. The word comes from the Ancient Greek word (), meaning 'that which is thought worthy or ...

s. Biophysical indications suggest that in the

brain The brain is an organ (biology), organ that serves as the center of the nervous system in all vertebrate and most invertebrate animals. It consists of nervous tissue and is typically located in the head (cephalization), usually near organs for ...

, synaptic charge injections occur in finite steps, and that

neuron A neuron (American English), neurone (British English), or nerve cell, is an membrane potential#Cell excitability, excitable cell (biology), cell that fires electric signals called action potentials across a neural network (biology), neural net ...

arrangements can be modeled based on the

probability distribution In probability theory and statistics, a probability distribution is a Function (mathematics), function that gives the probabilities of occurrence of possible events for an Experiment (probability theory), experiment. It is a mathematical descri ...

of a finitely valued

random variable A random variable (also called random quantity, aleatory variable, or stochastic variable) is a Mathematics, mathematical formalization of a quantity or object which depends on randomness, random events. The term 'random variable' in its mathema ...

. In the study of

itself, finite-valued logic has served as an aid to understand the nature and existence of infinite-valued logic.

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 profoundly ...

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.

References

{{Mathematical logic Logic Many-valued logic

History

Examples

See also

References