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 premises ...

and

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 ...

, a truth value, sometimes called a logical value, is a value indicating the relation of a

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 ...

truth Truth is the property of being in accord with fact or reality.Merriam-Webster's Online Dictionarytruth 2005 In everyday language, truth is typically ascribed to things that aim to represent reality or otherwise correspond to it, such as beliefs ...

, 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 class ...

has only two possible values (''

true True most commonly refers to truth, the state of being in congruence with fact or reality. True may also refer to: Places * True, West Virginia, an unincorporated community in the United States * True, Wisconsin, a town in the United States * Tr ...

'' or '' false'').

Computing

In some programming languages, any expression can be evaluated in a context that expects a Boolean data type. Typically (though this varies by programming language) expressions like the number

zero 0 (zero) is a number representing an empty quantity. In place-value notation Positional notation (or place-value notation, or positional numeral system) usually denotes the extension to any base of the Hindu–Arabic numeral system (or ...

, the

empty string In formal language theory, the empty string, or empty word, is the unique string of length zero. Formal theory Formally, a string is a finite, ordered sequence of characters such as letters, digits or spaces. The empty string is the special cas ...

, empty lists, and null evaluate to false, and strings with content (like "abc"), other numbers, and objects evaluate to true. Sometimes these classes of expressions are called "truthy" and "falsy" / "false".

Classical logic

, with its intended semantics, the truth values are ''

'' (denoted by ''1'' or the

verum The tee (⊤, \top in LaTeX) also called down tack (as opposed to the up tack) or verum is a symbol used to represent: * The top element in lattice theory. * The truth value of being true in logic, or a sentence (e.g., formula in propositional ca ...

⊤), and '' untrue'' or '' false'' (denoted by ''0'' or the

falsum The up tack or falsum (⊥, \bot in LaTeX, U+22A5 in Unicode) is a constant symbol used to represent: * The truth value 'false', or a logical constant denoting a proposition in logic that is always false (often called "falsum" or "absurdum"). * ...

⊥); that is, classical logic is a

two-valued 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 ...

. This set of two values is also called the Boolean domain. Corresponding semantics of

logical connective In logic, a logical connective (also called a logical operator, sentential connective, or sentential operator) is a logical constant. They can be used to connect logical formulas. For instance in the syntax of propositional logic, the binary co ...

s are

truth function In logic, a truth function is a function that accepts truth values as input and produces a unique truth value as output. In other words: The input and output of a truth function are all truth values; a truth function will always output exactly one ...

s, 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 argumen ...

s. Logical biconditional becomes the equality binary relation, and

negation In logic, negation, also called the logical complement, is an operation that takes a proposition P to another proposition "not P", written \neg P, \mathord P or \overline. It is interpreted intuitively as being true when P is false, and false ...

becomes a

bijection In mathematics, a bijection, also known as a bijective function, one-to-one correspondence, or invertible function, is a function between the elements of two sets, where each element of one set is paired with exactly one element of the other s ...

which permutes true and false. Conjunction and disjunction are

dual Dual or Duals may refer to: Paired/two things * Dual (mathematics), a notion of paired concepts that mirror one another ** Dual (category theory), a formalization of mathematical duality *** see more cases in :Duality theories * Dual (grammatical ...

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

In intuitionistic logic, and more generally, constructive mathematics, statements are assigned a truth value only if they can be given a constructive proof. It starts with a set of axioms, and a statement is true if one can build a proof of the statement from those axioms. A statement is false if one can deduce a contradiction from it. This leaves open the possibility of statements that have not yet been assigned a truth value. Unproven statements in intuitionistic logic are not given an intermediate truth value (as is sometimes mistakenly asserted). Indeed, one can prove that they have no third truth value, a result dating back to Glivenko in 1928.Proof that intuitionistic logic has no third truth value, Glivenko 1928
/ref> Instead, statements simply remain of unknown truth value, until they are either proven or disproven. There are various ways of interpreting intuitionistic logic, including the Brouwer–Heyting–Kolmogorov interpretation. See also .

Multi-valued logic

Multi-valued logics (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 In mathematics, the unit interval is the closed interval , that is, the set of all real numbers that are greater than or equal to 0 and less than or equal to 1. It is often denoted ' (capital letter ). In addition to its role in real analysis, ...

such structure is a

total order In mathematics, a total or linear order is a partial order in which any two elements are comparable. That is, a total order is a binary relation \leq on some set X, which satisfies the following for all a, b and c in X: # a \leq a ( reflexive) ...

; this may be expressed as the existence of various

degrees of truth In classical logic, propositions are typically unambiguously considered as being true or false. For instance, the proposition ''one is both equal and not equal to itself'' is regarded as simply false, being contrary to the Law of Noncontradiction; ...

Algebraic semantics

Not all

logical 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 form ...

s 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 semantics of classical propositional calculus.

In other theories

Intuitionistic type theory uses types in the place of truth values. Topos theory uses truth values in a special sense: the truth values of a topos are the

global element In category theory, a global element of an object ''A'' from a category is a morphism :h\colon 1 \to A, where is a terminal object of the category.. Roughly speaking, global elements are a generalization of the notion of "elements" from the categor ...

s of the subobject classifier. Having truth values in this sense does not make a logic truth valuational.

References

External links

* {{Logical truth Concepts in logic Propositions Value Value (ethics) Epistemology