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In
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 ...
and
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 ...
, a truth value, sometimes called a logical value, is a value indicating the relation of a
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 ...
to
truth Truth or verity is the Property (philosophy), property of being in accord with fact or reality.Merriam-Webster's Online Dictionarytruth, 2005 In everyday language, it is typically ascribed to things that aim to represent reality or otherwise cor ...
, which in
classical logic Classical logic (or standard logic) or Frege–Russell logic is the intensively studied and most widely used class of deductive logic. Classical logic has had much influence on analytic philosophy. Characteristics Each logical system in this c ...
has only two possible values ('' true'' or '' false''). Truth values are used in
computing Computing is any goal-oriented activity requiring, benefiting from, or creating computer, computing machinery. It includes the study and experimentation of algorithmic processes, and the development of both computer hardware, hardware and softw ...
as well as various types of
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 ...
.


Computing

In some programming languages, any expression can be evaluated in a context that expects a
Boolean data type In computer science, the Boolean (sometimes shortened to Bool) is a data type that has one of two possible values (usually denoted ''true'' and ''false'') which is intended to represent the two truth values of logic and Boolean algebra. It is na ...
. Typically (though this varies by programming language) expressions like the number zero, the
empty string In formal language theory, the empty string, or empty word, is the unique String (computer science), string of length zero. Formal theory Formally, a string is a finite, ordered sequence of character (symbol), characters such as letters, digits ...
, empty lists, and
null Null may refer to: Science, technology, and mathematics Astronomy *Nuller, an optical tool using interferometry to block certain sources of light Computing *Null (SQL) (or NULL), a special marker and keyword in SQL indicating that a data value do ...
are treated as false, and strings with content (like "abc"), other numbers, and objects evaluate to true. Sometimes these classes of expressions are called falsy and truthy. For example, in
Lisp Lisp (historically LISP, an abbreviation of "list processing") is a family of programming languages with a long history and a distinctive, fully parenthesized Polish notation#Explanation, prefix notation. Originally specified in the late 1950s, ...
, nil, the empty list, is treated as false, and all other values are treated as true. In C, the number 0 or 0.0 is false, and all other values are treated as true. In
JavaScript JavaScript (), often abbreviated as JS, is a programming language and core technology of the World Wide Web, alongside HTML and CSS. Ninety-nine percent of websites use JavaScript on the client side for webpage behavior. Web browsers have ...
, the empty string (""), null, undefined, NaN, +0,
−0 Signed zero is zero with an associated Sign (mathematics), sign. In ordinary arithmetic, the number 0 does not have a sign, so that −0, +0 and 0 are equivalent. However, in computing, some number representations allow for the existence of two zer ...
and false are sometimes called ''falsy'' (of which the complement is ''truthy'') to distinguish between strictly type-checked and coerced Booleans (see also: JavaScript syntax#Type conversion). As opposed to Python, empty containers (Arrays, Maps, Sets) are considered truthy. Languages such as PHP also use this approach.


Classical logic

In
classical logic Classical logic (or standard logic) or Frege–Russell logic is the intensively studied and most widely used class of deductive logic. Classical logic has had much influence on analytic philosophy. Characteristics Each logical system in this c ...
, with its intended semantics, the truth values are '' true'' (denoted by ''1'' or the verum ⊤), and '' untrue'' or '' false'' (denoted by ''0'' or the
falsum "Up tack" is the Unicode name for a symbol (⊥, \bot in LaTeX, U+22A5 in Unicode) that is also called "bottom", "falsum", "absurdum", or "the absurdity symbol", depending on context. It is used to represent: * The truth value false (logic), 'fal ...
⊥); that is, classical logic is a two-valued logic. This set of two values is also called the
Boolean domain In mathematics and abstract algebra, a Boolean domain is a set consisting of exactly two elements whose interpretations include ''false'' and ''true''. In logic, mathematics and theoretical computer science, a Boolean domain is usually written ...
. Corresponding semantics of logical connectives are truth functions, whose values are expressed in the form of
truth table 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 arg ...
s. Logical biconditional becomes the equality binary relation, and
negation In logic, negation, also called the logical not or logical complement, is an operation (mathematics), operation that takes a Proposition (mathematics), proposition P to another proposition "not P", written \neg P, \mathord P, P^\prime or \over ...
becomes a
bijection In mathematics, a bijection, bijective function, or one-to-one correspondence is a function between two sets such that each element of the second set (the codomain) is the image of exactly one element of the first set (the domain). Equival ...
which permutes true and false. Conjunction and disjunction are dual with respect to negation, which is expressed by De Morgan's laws: : ¬( : ¬( Propositional variables become variables in the Boolean domain. Assigning values for propositional variables is referred to as valuation.


Intuitionistic and constructive logic

Whereas in classical logic truth values form a
Boolean algebra In mathematics and mathematical logic, Boolean algebra is a branch of algebra. It differs from elementary algebra in two ways. First, the values of the variable (mathematics), variables are the truth values ''true'' and ''false'', usually denot ...
, in intuitionistic logic, and more generally, constructive mathematics, the truth values form a Heyting algebra. Such truth values may express various aspects of validity, including locality, temporality, or computational content. For example, one may use the open sets of a topological space as intuitionistic truth values, in which case the truth value of a formula expresses ''where'' the formula holds, not whether it holds. In realizability truth values are sets of programs, which can be understood as computational evidence of validity of a formula. For example, the truth value of the statement "for every number there is a prime larger than it" is the set of all programs that take as input a number n, and output a prime larger than n. In
category theory Category theory is a general theory of mathematical structures and their relations. It was introduced by Samuel Eilenberg and Saunders Mac Lane in the middle of the 20th century in their foundational work on algebraic topology. Category theory ...
, truth values appear as the elements of the subobject classifier. In particular, in a topos every formula of
higher-order logic In mathematics and logic, a higher-order logic (abbreviated HOL) is a form of logic that is distinguished from first-order logic by additional quantifiers and, sometimes, stronger semantics. Higher-order logics with their standard semantics are m ...
may be assigned a truth value in the subobject classifier. Even though a Heyting algebra may have many elements, this should not be understood as there being truth values that are neither true nor false, because intuitionistic logic proves \neg (p \neq \top \land p \neq \bot) ("it is not the case that p is neither true nor false").Proof that intuitionistic logic has no third truth value, Glivenko 1928
/ref> In intuitionistic type theory, the Curry-Howard correspondence exhibits an equivalence of propositions and types, according to which validity is equivalent to inhabitation of a type. For other notions of intuitionistic truth values, see the Brouwer–Heyting–Kolmogorov interpretation and .


Multi-valued logic

Multi-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 logical calculus, there were only two possible values (i.e., "true" and "false") ...
s (such as
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 ...
and relevance logic) allow for more than two truth values, possibly containing some internal structure. For example, on the unit interval such structure is a total order; this may be expressed as the existence of various degrees of truth.


Algebraic semantics

Not all logical systems are truth-valuational in the sense that logical connectives may be interpreted as truth functions. For example, intuitionistic logic lacks a complete set of truth values because its semantics, the Brouwer–Heyting–Kolmogorov interpretation, is specified in terms of provability conditions, and not directly in terms of the necessary truth of formulae. But even non-truth-valuational logics can associate values with logical formulae, as is done in algebraic semantics. The algebraic semantics of intuitionistic logic is given in terms of Heyting algebras, compared to
Boolean algebra In mathematics and mathematical logic, Boolean algebra is a branch of algebra. It differs from elementary algebra in two ways. First, the values of the variable (mathematics), variables are the truth values ''true'' and ''false'', usually denot ...
semantics of classical propositional calculus.


See also

*
Agnosticism Agnosticism is the view or belief that the existence of God, the divine, or the supernatural is either unknowable in principle or unknown in fact. (page 56 in 1967 edition) It can also mean an apathy towards such religious belief and refer t ...
*
Bayesian probability Bayesian probability ( or ) is an interpretation of the concept of probability, in which, instead of frequency or propensity of some phenomenon, probability is interpreted as reasonable expectation representing a state of knowledge or as quant ...
*
Circular reasoning Circular reasoning (, "circle in proving"; also known as circular logic) is a fallacy, logical fallacy in which the reasoner begins with what they are trying to end with. Circular reasoning is not a formal logical fallacy, but a pragmatic defect ...
* Degree of truth * False dilemma * * Paradox * Semantic theory of truth * Slingshot argument * Supervaluationism * Truth-value semantics *
Verisimilitude In philosophy, verisimilitude (or truthlikeness) is the notion that some propositions are closer to being true than other propositions. The problem of verisimilitude is the problem of articulating what it takes for one false theory to be close ...


References


External links

* {{Logical truth Concepts in logic Propositions Logical truth Concepts in epistemology