In

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

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 ...

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 ...

to 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 belie ...

, 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 clas ...

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
* ...

'' or ''false
False or falsehood may refer to:
*False (logic), the negation of truth in classical logic
*Lie or falsehood, a type of deception in the form of an untruthful statement
*false (Unix), a Unix command
* ''False'' (album), a 1992 album by Gorefest
*Ma ...

'').
Computing

In some programming languages, any expression can be evaluated in a context that expects aBoolean 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
0 (zero) is a number representing an empty quantity. In place-value notation such as the Hindu–Arabic numeral system, 0 also serves as a placeholder numerical digit, which works by multiplying digits to the left of 0 by the radix, usu ...

, 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 c ...

, 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

Inclassical 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 clas ...

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

'' (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 cal ...

⊤), and '' untrue'' or ''false
False or falsehood may refer to:
*False (logic), the negation of truth in classical logic
*Lie or falsehood, a type of deception in the form of an untruthful statement
*false (Unix), a Unix command
* ''False'' (album), a 1992 album by Gorefest
*Ma ...

'' (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").
* T ...

⊥); 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 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 ...

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 on ...

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
In logic and mathematics, the logical biconditional, sometimes known as the material biconditional, is the logical connective (\leftrightarrow) used to conjoin two statements and to form the statement " if and only if ", where is known as th ...

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 fal ...

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 ...

which permutes true and false. Conjunction and disjunction are dual with respect to negation, which is expressed by De Morgan's laws
In propositional logic and Boolean algebra, De Morgan's laws, also known as De Morgan's theorem, are a pair of transformation rules that are both valid rules of inference. They are named after Augustus De Morgan, a 19th-century British math ...

:
: ¬(
: ¬(
Propositional variable
In mathematical logic, a propositional variable (also called a sentential variable or sentential letter) is an input variable (that can either be true or false) of a truth function. Propositional variables are the basic building-blocks of proposit ...

s become variables in the Boolean domain. Assigning values for propositional variables is referred to as valuation.
Intuitionistic and constructive logic

Inintuitionistic 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, system ...

, 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./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 In mathematical logic, the Brouwer–Heyting–Kolmogorov interpretation, or BHK interpretation, of intuitionistic logic was proposed by L. E. J. Brouwer and Arend Heyting, and independently by Andrey Kolmogorov. It is also sometimes called the r ...

. See also .
Multi-valued logic

Multi-valued logics (such asfuzzy 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 complete ...

and relevance logic Relevance logic, also called relevant logic, is a kind of non-classical logic requiring the antecedent and consequent of implications to be relevantly related. They may be viewed as a family of substructural or modal logics. It is generally, but ...

) 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 analysi ...

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 ( reflexi ...

; this may be expressed as the existence of various degrees of truth.
Algebraic semantics

Not alllogical 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 fo ...

s are truth-valuational in the sense that logical connectives may be interpreted as truth functions. For example, 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, system ...

lacks a complete set of truth values because its semantics, the Brouwer–Heyting–Kolmogorov interpretation In mathematical logic, the Brouwer–Heyting–Kolmogorov interpretation, or BHK interpretation, of intuitionistic logic was proposed by L. E. J. Brouwer and Arend Heyting, and independently by Andrey Kolmogorov. It is also sometimes called the r ...

, 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 variables are the truth values ''true'' and ''false'', usually denoted 1 and 0, whereas in ...

semantics of classical propositional calculus.
In other theories

Intuitionistic type theory
Intuitionistic type theory (also known as constructive type theory, or Martin-Löf type theory) is a type theory and an alternative foundation of mathematics.
Intuitionistic type theory was created by Per Martin-Löf, a Swedish mathematician and ...

uses types
Type may refer to:
Science and technology Computing
* Typing, producing text via a keyboard, typewriter, etc.
* Data type, collection of values used for computations.
* File type
* TYPE (DOS command), a command to display contents of a file.
* ...

in the place of truth values.
Topos
In mathematics, a topos (, ; plural topoi or , or toposes) is a category that behaves like the category of sheaves of sets on a topological space (or more generally: on a site). Topoi behave much like the category of sets and possess a notion ...

theory uses truth values in a special sense: the truth values of a topos are the global elements of the subobject classifier In category theory, a subobject classifier is a special object Ω of a category such that, intuitively, the subobjects of any object ''X'' in the category correspond to the morphisms from ''X'' to Ω. In typical examples, that morphism assigns "tru ...

. Having truth values in this sense does not make a logic truth valuational.
See also

*Agnosticism
Agnosticism is the view or belief that the existence of God, of the divine or the supernatural is unknown or unknowable. (page 56 in 1967 edition) Another definition provided is the view that "human reason is incapable of providing sufficien ...

* Bayesian probability
Bayesian probability 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 quantification ...

* Circular reasoning
Circular may refer to:
* The shape of a circle
* ''Circular'' (album), a 2006 album by Spanish singer Vega
* Circular letter (disambiguation)
** Flyer (pamphlet)
A flyer (or flier) is a form of paper
Paper is a thin sheet material pro ...

* Degree of truth
* False dilemma
A false dilemma, also referred to as false dichotomy or false binary, is an informal fallacy based on a premise that erroneously limits what options are available. The source of the fallacy lies not in an invalid form of inference but in a false ...

*
* 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 ...

* Semantic theory of truth
A semantic theory of truth is a theory of truth in the philosophy of language which holds that truth is a property of sentences.
Origin
The semantic conception of truth, which is related in different ways to both the correspondence and deflat ...

* 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 closer ...

References

External links

* {{Logical truth Concepts in logic PropositionsValue
Value or values may refer to:
Ethics and social
* Value (ethics) wherein said concept may be construed as treating actions themselves as abstract objects, associating value to them
** Values (Western philosophy) expands the notion of value bey ...

Value (ethics)
Epistemology