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 an interdisciplinary field which studies truth and reasoning
Reason is the capacity of consciously making sense of things, applying logic
Logic (from Ancient Greek, Greek: grc, wikt:λογική, λογική, label=none, lit ...

and mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no general consensus abo ...

, 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 (linguistics), sentence. In philosophy, "Meaning (philosophy), meaning" is understood to be a non-linguistic entity which is shared by all sentences with the same mea ...

to truth
Truth is the property of being in accord with fact
A fact is something that is true
True most commonly refers to truth
Truth is the property of being in accord with fact or reality.Merriam-Webster's Online Dictionarytruth 2005 In ...

.
Computing

In some programming languages, any expression can be evaluated in a context that expects aBoolean data typeIn computer science
Computer science deals with the theoretical foundations of information, algorithms and the architectures of its computation as well as practical techniques for their application.
Computer science is the study of Algorith ...

. Typically (though this varies by programming language) expressions like the number zero
0 (zero) is a number
A number is a mathematical object used to counting, count, measurement, measure, and nominal number, label. The original examples are the natural numbers 1, 2, 3, 4, and so forth. Numbers can be represented in languag ...

, 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 o ...

, 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" / "falsey".
Classical logic

Inclassical logicClassical logic (or standard logic) is the intensively studied and most widely used class of deductive logic. Classical logic has had much influence on analytic philosophy, the type of philosophy most often found in the English-speaking world.
Chara ...

, with its intended semantics, the truth values are ''true
True most commonly refers 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 otherw ...

'' (denoted by ''1'' or the verumThe tee (⊤, \top in LaTeX
Latex is a stable dispersion (emulsion
An emulsion is a mixture of two or more liquids that are normally Miscibility, immiscible (unmixable or unblendable) owing to liquid-liquid phase separation. Emulsions are part ...

⊤), 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), ''False'' (album), a 1992 album by ...

'' (denoted by ''0'' or the falsum
The up tack or falsum (⊥, \bot in LaTeX
Latex is a stable dispersion (emulsion
An emulsion is a mixture of two or more liquids that are normally Miscibility, immiscible (unmixable or unblendable) owing to liquid-liquid phase separation. Emu ...

⊥); that is, classical logic is a two-valued logic. This set of two values is also called the Boolean domain
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

. Corresponding semantics of logical connective
In logic
Logic is an interdisciplinary field which studies truth and reasoning
Reason is the capacity of consciously making sense of things, applying logic
Logic (from Ancient Greek, Greek: grc, wikt:λογική, λογική, la ...

s are truth function
In logic
Logic (from Ancient Greek, Greek: grc, wikt:λογική, λογική, label=none, lit=possessed of reason, intellectual, dialectical, argumentative, translit=logikḗ)Also related to (''logos''), "word, thought, idea, argument, ac ...

s, whose values are expressed in the form of truth table
A truth table is a mathematical table
Mathematical tables are lists of numbers showing the results of a calculation with varying arguments. Tables of trigonometric functions were used in ancient Greece and India for applications to astronomy
...

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 the ...

becomes the equality binary relation, and negation
In logic
Logic is an interdisciplinary field which studies truth and reasoning
Reason is the capacity of consciously making sense of things, applying logic
Logic (from Ancient Greek, Greek: grc, wikt:λογική, λογική, ...

becomes a bijection
In , a bijection, bijective function, one-to-one correspondence, or invertible function, is a between the elements of two , where each element of one set is paired with exactly one element of the other set, and each element of the other set is p ...

which 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 ...

with respect to negation, which is expressed by De Morgan's laws
In propositional calculus, propositional logic and Boolean algebra, De Morgan's laws are a pair of transformation rules that are both Validity (logic), valid rule of inference, rules of inference. They are named after Augustus De Morgan, a 19th-c ...

:
: ¬(
: ¬(
Propositional variable
In mathematical logic
Mathematical logic, also called formal logic, is a subfield of mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (alge ...

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, systems of ...

, and more generally, constructive mathematics
In the philosophy of mathematics, constructivism asserts that it is necessary to find (or "construct") a mathematical object to prove that it exists. In classical mathematics, one can prove the existence of a mathematical object without "finding ...

, 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 interpretationIn mathematical logic
Mathematical logic, also called formal logic, is a subfield of mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algeb ...

. See also .
Multi-valued logic

Multi-valued logicIn logic
Logic (from Ancient Greek, Greek: grc, wikt:λογική, λογική, label=none, lit=possessed of reason, intellectual, dialectical, argumentative, translit=logikḗ)Also related to (''logos''), "word, thought, idea, argument, acc ...

s (such as fuzzy logic
In fuzzy mathematics, fuzzy logic is a form of many-valued logic in which the truth value
Truth is the property of being in accord with fact or reality.Merriam-Webster's Online Dictionarytruth 2005 In everyday language, truth is typically ...

and relevance logicRelevance logic, also called relevant logic, is a kind of non-classical logicClassical logic (or standard logic) is the intensively studied and most widely used class of deductive logic. Classical logic has had much influence on analytic philosophy, ...

) allow for more than two truth values, possibly containing some internal structure. For example, on the unit interval
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...

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 (mathematics), set X, which satisfies the following for all a, b and c in X:
# a \ ...

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

Not alllogical system
A formal system is 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 formal system is essentiall ...

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, systems of ...

lacks a complete set of truth values because its semantics, the Brouwer–Heyting–Kolmogorov interpretationIn mathematical logic
Mathematical logic, also called formal logic, is a subfield of mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algeb ...

, 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 algebra
__notoc__
Arend Heyting (; 9 May 1898 – 9 July 1980) was a Netherlands, Dutch mathematician and logician.
Biography
Heyting was a student of Luitzen Egbertus Jan Brouwer at the University of Amsterdam, and did much to put intuitionistic log ...

s, compared to Boolean algebra
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

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
In mathematics, logic, and computer science, a type system is a formal system in which every term has a "type" which defines its meanin ...

uses types
Type may refer to:
Science and technology Computing
* Typing
Typing is the process of writing or inputting text by pressing keys on a typewriter, computer keyboard, cell phone, or calculator. It can be distinguished from other means of text inpu ...

in the place of truth values.
Topos
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

theory uses truth values in a special sense: the truth values of a topos are the global elements of the subobject classifierIn category theory
Category theory formalizes mathematical structure and its concepts in terms of a Graph labeling, labeled directed graph called a ''Category (mathematics), category'', whose nodes are called ''objects'', and whose labelled dire ...

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

*Agnosticism
Agnosticism is the view that the existence of God, of the divine or the supernatural
The supernatural encompasses supposed phenomena that are not subject to the laws of nature.https://www.merriam-webster.com/dictionary/supernatural By de ...

* Bayesian probability
Bayesian probability is an Probability interpretations, interpretation of the concept of probability, in which, instead of frequentist probability, frequency or propensity probability, propensity of some phenomenon, probability is interpreted as rea ...

* Circular reasoning
Circular reasoning ( la, circulus in probando, "circle in proving"; also known as circular logic) is a logical fallacy in which the reasoner begins with what they are trying to end with. The components of a circular argument are often logically ...

* Degree of truth
* False dilemmaImage:Young America's dilemma - Dalrymple. LCCN2010651418.jpg, upright=1.5, Young America's dilemma: Shall I be wise and great, or rich and powerful? (1901 year poster)
A false dilemma, also referred to as false dichotomy, is an informal fallacy ba ...

*
* 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 un ...

* Semantic theory of truth
A semantic theory of truth is a theory of truth in the philosophy of language
In analytic philosophy
Analytic philosophy is a branch and tradition of philosophy
Philosophy (from , ) is the study of general and fundamental questions ...

* Slingshot argument
In philosophical logic
Understood in a narrow sense, philosophical logic is the area of philosophy that studies the application of logical methods to philosophical problems, often in the form of extended logical systems like modal logic. Some theori ...

* Supervaluationism
In philosophical logicPhilosophical logic refers to those areas of philosophy in which recognized methods of logic have Classical logic, traditionally been used to solve or advance the discussion of philosophical problems. Among these, Sybil Wolfram ...

* Truth-value semantics
* Verisimilitude
In philosophy
Philosophy (from , ) is the study of general and fundamental questions, such as those about reason, Metaphysics, existence, Epistemology, knowledge, Ethics, values, Philosophy of mind, mind, and Philosophy of language, l ...

References

External links

* {{Logical truth Concepts in logic PropositionsValue
Value or values may refer to:
* Value (ethics) it may be described as treating actions themselves as abstract objects, putting value to them
** Values (Western philosophy) expands the notion of value beyond that of ethics, but limited to Western s ...

Value (ethics)
Epistemology