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

^{∁} its complement.
The last identity concerning ''A'' → ''B'' allows us to calculate the value of ¬''A'':
:$\backslash begin\; \backslash text;\; href="/html/ALL/s/neg\_A.html"\; ;"title="neg\; A">neg\; A$
With these assignments, intuitionistically valid formulas are precisely those that are assigned the value of the entire line. For example, the formula ¬(''A'' ∧ ¬''A'') is valid, because no matter what set ''X'' is chosen as the value of the formula ''A'', the value of ¬(''A'' ∧ ¬''A'') can be shown to be the entire line:
:$\backslash begin\; \backslash text;\; href="/html/ALL/s/neg(A\_\backslash land\_\backslash neg\_A).html"\; ;"title="neg(A\; \backslash land\; \backslash neg\; A)">neg(A\; \backslash land\; \backslash neg\; A)$
So the valuation of this formula is true, and indeed the formula is valid. But the law of the excluded middle, ''A'' ∨ ¬''A'', can be shown to be ''invalid'' by using a specific value of the set of positive real numbers for ''A'':
:$\backslash begin\; \backslash text;\; href="/html/ALL/s/\_\backslash lor\_\backslash neg\_A.html"\; ;"title="\; \backslash lor\; \backslash neg\; A">\; \backslash lor\; \backslash neg\; A$
The interpretation of any intuitionistically valid formula in the infinite Heyting algebra described above results in the top element, representing true, as the valuation of the formula, regardless of what values from the algebra are assigned to the variables of the formula. Conversely, for every invalid formula, there is an assignment of values to the variables that yields a valuation that differs from the top element. No finite Heyting algebra has the second of these two properties.

Written b

Joan Moschovakis

Published in ''Stanford Encyclopedia of Philosophy''.

''Logique modale propositionnelle S4 et logique intuitioniste propositionnelle''

pp. 4–5. that the translated formula is valid in the propositional modal logic S4 if and only if the original formula is valid in IPC. The above set of formulae are called the Gödel–McKinsey–Tarski translation. There is also an intuitionistic version of modal logic S4 called Constructive Modal Logic CS4.Natasha Alechina, Michael Mendler, Valeria de Paiva, and Eike Ritter

''Categorical and Kripke Semantics for Constructive S4 Modal Logic''

/ref>

"Anti-intuitionism and paraconsistency"

''Journal of Applied Logic Volume'' 3, Issue 1, Mar 2005, pp. 161–184 * Arend Heyting, 1930, "Die formalen Regeln der intuitionistischen Logik," in three parts, ''Sitzungsberichte der preussischen Akademie der Wissenschaften'': 42–71, 158–169.

Intuitionistic Logic

by Joan Moschovakis

Intuitionistic Logic

by Nick Bezhanishvili and Dick de Jongh (from the Institute for Logic, Language and Computation at the

Semantical Analysis of Intuitionistic Logic I

by Saul A. Kripke from ''Harvard University, Cambridge, Mass., USA''

Intuitionistic Logic

by '' Dirk van Dalen''

The discovery of E. W. Beth's semantics for intuitionistic logic

by A. S. Troelstra and P. van Ulsen

Expressing Database Queries with Intuitionistic Logic

by Anthony J. Bonner. L. Thorne McCarty. Kumar Vadaparty. Rutgers University, Department of Computer Science

Tableaux'method for intuitionistic logic through S4-translation

tests the intuitionistic validity of propositional formulae; provided by the Laboratoire d'Informatique de

The Oxford Handbook of Philosophy of Mathematics and Logic

'

"Intuitionism in Mathematics"

by David Charles McCarty {{Authority control Logic in computer science Non-classical logic Constructivism (mathematics) Systems of formal logic Intuitionism

constructive proof
In mathematics, a constructive proof is a method of proof that demonstrates the existence of a mathematical object by creating or providing a method for creating the object. This is in contrast to a non-constructive proof (also known as an existe ...

. In particular, systems of intuitionistic logic do not assume the law of the excluded middle and double negation elimination, which are fundamental inference rules in classical logic.
Formalized intuitionistic logic was originally developed by Arend Heyting to provide a formal basis for L. E. J. Brouwer's programme of intuitionism. From a proof-theoretic perspective, Heyting’s calculus is a restriction of classical logic in which the law of excluded middle and double negation elimination have been removed. Excluded middle and double negation elimination can still be proved for some propositions on a case by case basis, however, but do not hold universally as they do with classical logic. The standard explanation of intuitionistic logic is the BHK interpretation.
Several systems of semantics for intuitionistic logic have been studied. One of these semantics mirrors classical Boolean-valued semantics but uses Heyting algebras in place of 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 ...

s. Another semantics uses Kripke models. These, however, are technical means for studying Heyting’s deductive system rather than formalizations of Brouwer’s original informal semantic intuitions. Semantical systems claiming to capture such intuitions, due to offering meaningful concepts of “constructive truth” (rather than merely validity or provability), are Kurt Gödel’s dialectica interpretation, Stephen Cole Kleene’s realizability, Yurii Medvedev’s logic of finite problems, or Giorgi Japaridze’s computability logic. Yet such semantics persistently induce logics properly stronger than Heyting’s logic. Some authors have argued that this might be an indication of inadequacy of Heyting’s calculus itself, deeming the latter incomplete as a constructive logic.
Mathematical constructivism

In the semantics of classical logic, propositional formulae are assignedtruth value
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'').
Computing
In some prog ...

s from the two-element set $\backslash $ ("true" and "false" respectively), regardless of whether we have direct evidence
Evidence for a proposition is what supports this proposition. It is usually understood as an indication that the supported proposition is true. What role evidence plays and how it is conceived varies from field to field.
In epistemology, evide ...

for either case. This is referred to as the 'law of excluded middle', because it excludes the possibility of any truth value besides 'true' or 'false'. In contrast, propositional formulae in intuitionistic logic are ''not'' assigned a definite truth value and are ''only'' considered "true" when we have direct evidence, hence ''proof''. (We can also say, instead of the propositional formula being "true" due to direct evidence, that it is inhabited by a proof in the Curry–Howard sense.) Operations in intuitionistic logic therefore preserve justification, with respect to evidence and provability, rather than truth-valuation.
Intuitionistic logic is a commonly-used tool in developing approaches to constructivism
Constructivism may refer to:
Art and architecture
* Constructivism (art), an early 20th-century artistic movement that extols art as a practice for social purposes
* Constructivist architecture, an architectural movement in Russia in the 1920s ...

in mathematics. The use of constructivist logics in general has been a controversial topic among mathematicians and philosophers (see, for example, the Brouwer–Hilbert controversy). A common objection to their use is the above-cited lack of two central rules of classical logic, the law of excluded middle and double negation elimination. These are considered to be so important to the practice of mathematics that David Hilbert
David Hilbert (; ; 23 January 1862 – 14 February 1943) was a German mathematician, one of the most influential mathematicians of the 19th and early 20th centuries. Hilbert discovered and developed a broad range of fundamental ideas in many a ...

wrote of them: "Taking the principle of excluded middle from the mathematician would be the same, say, as proscribing the telescope to the astronomer or to the boxer the use of his fists. To prohibit existence statements and the principle of excluded middle is tantamount to relinquishing the science of mathematics altogether."
Despite the serious challenges presented by the inability to utilize the valuable rules of excluded middle and double negation elimination, intuitionistic logic has practical use. One reason for this is that its restrictions produce proofs that have the existence property, making it also suitable for other forms of mathematical constructivism. Informally, this means that if there is a constructive proof that an object exists, that constructive proof may be used as an algorithm for generating an example of that object, a principle known as the Curry–Howard correspondence between proofs and algorithms. One reason that this particular aspect of intuitionistic logic is so valuable is that it enables practitioners to utilize a wide range of computerized tools, known as proof assistants. These tools assist their users in the verification (and generation) of large-scale proofs, whose size usually precludes the usual human-based checking that goes into publishing and reviewing a mathematical proof. As such, the use of proof assistants (such as Agda or Coq) is enabling modern mathematicians and logicians to develop and prove extremely complex systems, beyond those that are feasible to create and check solely by hand. One example of a proof that was impossible to satisfactorily verify without formal verification is the famous proof of the four color theorem. This theorem stumped mathematicians for more than a hundred years, until a proof was developed that ruled out large classes of possible counterexamples, yet still left open enough possibilities that a computer program was needed to finish the proof. That proof was controversial for some time, but, later, it was verified using Coq.
Syntax

Thesyntax
In linguistics, syntax () is the study of how words and morphemes combine to form larger units such as phrases and sentences. Central concerns of syntax include word order, grammatical relations, hierarchical sentence structure (constituency), ...

of formulas of intuitionistic logic is similar to propositional logic or first-order logic
First-order logic—also known as predicate logic, quantificational logic, and first-order predicate calculus—is a collection of formal systems used in mathematics, philosophy, linguistics, and computer science. First-order logic uses quantifie ...

. However, intuitionistic connectives are not definable in terms of each other in the same way as in classical logic, hence their choice matters. In intuitionistic propositional logic (IPL) it is customary to use →, ∧, ∨, ⊥ as the basic connectives, treating ¬''A'' as an abbreviation for . In intuitionistic first-order logic both quantifiers ∃, ∀ are needed.
Weaker than classical logic

Intuitionistic logic can be understood as a weakening of classical logic, meaning that it is more conservative in what it allows a reasoner to infer, while not permitting any new inferences that could not be made under classical logic. Each theorem of intuitionistic logic is a theorem in classical logic, but not conversely. Many tautologies in classical logic are not theorems in intuitionistic logicin particular, as said above, one of intuitionistic logic's chief aims is to not affirm the law of the excluded middle so as to vitiate the use of non-constructive proof by contradiction, which can be used to furnish existence claims without providing explicit examples of the objects that it proves exist. We say "not affirm" because while it is not necessarily true that the law is upheld in any context, no counterexample can be given: such a counterexample would be an inference (inferring the negation of the law for a certain proposition) disallowed under classical logic and thus is not allowed in a strict weakening like intuitionistic logic. Indeed, the double negation of the law is retained as a tautology of the system: that is, it is a theorem that $\backslash neg\backslash big(\backslash neg\; (P\; \backslash vee\; \backslash neg\; P)\backslash big)$ regardless of the proposition $P$. So the propositional calculus is always compatible with classical logic. The situation is more intricate for predicate logic formulas when quantified expressions are negated.Sequent calculus

Gerhard Gentzen discovered that a simple restriction of his system LK (his sequent calculus for classical logic) results in a system that is sound and complete with respect to intuitionistic logic. He called this system LJ. In LK any number of formulas is allowed to appear on the conclusion side of a sequent; in contrast LJ allows at most one formula in this position. Other derivatives of LK are limited to intuitionistic derivations but still allow multiple conclusions in a sequent. LJ' is one example.Hilbert-style calculus

Intuitionistic logic can be defined using the following Hilbert-style calculus. This is similar to a way of axiomatizing classical propositional logic. In propositional logic, the inference rule is modus ponens * MP: from $\backslash phi$ and $\backslash phi\; \backslash to\; \backslash psi$ infer $\backslash psi$ and the axioms are * THEN-1: $\backslash phi\; \backslash to\; (\backslash chi\; \backslash to\; \backslash phi\; )$ * THEN-2: $(\backslash phi\; \backslash to\; (\backslash chi\; \backslash to\; \backslash psi\; ))\; \backslash to\; ((\backslash phi\; \backslash to\; \backslash chi\; )\; \backslash to\; (\backslash phi\; \backslash to\; \backslash psi\; ))$ * AND-1: $\backslash phi\; \backslash land\; \backslash chi\; \backslash to\; \backslash phi$ * AND-2: $\backslash phi\; \backslash land\; \backslash chi\; \backslash to\; \backslash chi$ * AND-3: $\backslash phi\; \backslash to\; (\backslash chi\; \backslash to\; (\backslash phi\; \backslash land\; \backslash chi\; ))$ * OR-1: $\backslash phi\; \backslash to\; \backslash phi\; \backslash lor\; \backslash chi$ * OR-2: $\backslash chi\; \backslash to\; \backslash phi\; \backslash lor\; \backslash chi$ * OR-3: $(\backslash phi\; \backslash to\; \backslash psi\; )\; \backslash to\; ((\backslash chi\; \backslash to\; \backslash psi\; )\; \backslash to\; ((\backslash phi\; \backslash lor\; \backslash chi)\; \backslash to\; \backslash psi\; ))$ * FALSE: $\backslash bot\; \backslash to\; \backslash phi$ To make this a system of first-order predicate logic, the generalization rules * $\backslash forall$-GEN: from $\backslash psi\; \backslash to\; \backslash phi$ infer $\backslash psi\; \backslash to\; (\backslash forall\; x\; \backslash \; \backslash phi\; )$, if $x$ is not free in $\backslash psi$ * $\backslash exists$-GEN: from $\backslash phi\; \backslash to\; \backslash psi$ infer $(\backslash exists\; x\; \backslash \; \backslash phi\; )\; \backslash to\; \backslash psi$, if $x$ is not free in $\backslash psi$ are added, along with the axioms * PRED-1: $(\backslash forall\; x\; \backslash \; \backslash phi\; (x))\; \backslash to\; \backslash phi\; (t)$, if the term $t$ is free for substitution for the variable $x$ in $\backslash phi$ (i.e., if no occurrence of any variable in $t$ becomes bound in $\backslash phi\; (t)$) * PRED-2: $\backslash phi\; (t)\; \backslash to\; (\backslash exists\; x\; \backslash \; \backslash phi\; (x))$, with the same restriction as for PRED-1Optional connectives

= Negation

= If one wishes to include a connective $\backslash lnot$ for negation rather than consider it an abbreviation for $\backslash phi\; \backslash to\; \backslash bot$, it is enough to add: * NOT-1': $(\backslash phi\; \backslash to\; \backslash bot\; )\; \backslash to\; \backslash lnot\; \backslash phi$ * NOT-2': $\backslash lnot\; \backslash phi\; \backslash to\; (\backslash phi\; \backslash to\; \backslash bot\; )$ There are a number of alternatives available if one wishes to omit the connective $\backslash bot$ (false). For example, one may replace the three axioms FALSE, NOT-1', and NOT-2' with the two axioms * NOT-1: $(\backslash phi\; \backslash to\; \backslash chi\; )\; \backslash to\; ((\backslash phi\; \backslash to\; \backslash lnot\; \backslash chi\; )\; \backslash to\; \backslash lnot\; \backslash phi\; )$ * NOT-2: $\backslash phi\; \backslash to\; (\backslash lnot\; \backslash phi\; \backslash to\; \backslash chi\; )$ as at . Alternatives to NOT-1 are $(\backslash phi\; \backslash to\; \backslash lnot\; \backslash chi\; )\; \backslash to\; (\backslash chi\; \backslash to\; \backslash lnot\; \backslash phi\; )$ or $(\backslash phi\; \backslash to\; \backslash lnot\; \backslash phi\; )\; \backslash to\; \backslash lnot\; \backslash phi$.= Equivalence

= The connective $\backslash leftrightarrow$ for equivalence may be treated as an abbreviation, with $\backslash phi\; \backslash leftrightarrow\; \backslash chi$ standing for $(\backslash phi\; \backslash to\; \backslash chi\; )\; \backslash land\; (\backslash chi\; \backslash to\; \backslash phi\; )$. Alternatively, one may add the axioms * IFF-1: $(\backslash phi\; \backslash leftrightarrow\; \backslash chi\; )\; \backslash to\; (\backslash phi\; \backslash to\; \backslash chi\; )$ * IFF-2: $(\backslash phi\; \backslash leftrightarrow\; \backslash chi\; )\; \backslash to\; (\backslash chi\; \backslash to\; \backslash phi\; )$ * IFF-3: $(\backslash phi\; \backslash to\; \backslash chi\; )\; \backslash to\; ((\backslash chi\; \backslash to\; \backslash phi\; )\; \backslash to\; (\backslash phi\; \backslash leftrightarrow\; \backslash chi\; ))$ IFF-1 and IFF-2 can, if desired, be combined into a single axiom $(\backslash phi\; \backslash leftrightarrow\; \backslash chi\; )\; \backslash to\; ((\backslash phi\; \backslash to\; \backslash chi\; )\; \backslash land\; (\backslash chi\; \backslash to\; \backslash phi\; ))$ using conjunction.Relation to classical logic

The system of classical logic is obtained by adding any one of the following axioms: * $\backslash phi\; \backslash lor\; \backslash lnot\; \backslash phi$ (Law of the excluded middle. May also be formulated as $(\backslash phi\; \backslash to\; \backslash chi\; )\; \backslash to\; ((\backslash lnot\; \backslash phi\; \backslash to\; \backslash chi\; )\; \backslash to\; \backslash chi\; )$.) * $\backslash lnot\; \backslash lnot\; \backslash phi\; \backslash to\; \backslash phi$ (Double negation elimination) * $((\backslash phi\; \backslash to\; \backslash chi\; )\; \backslash to\; \backslash phi\; )\; \backslash to\; \backslash phi$ ( Peirce's law) * $(\backslash lnot\; \backslash phi\; \backslash to\; \backslash lnot\; \backslash chi\; )\; \backslash to\; (\backslash chi\; \backslash to\; \backslash phi\; )$ (Law of contraposition) In general, one may take as the extra axiom any classical tautology that is not valid in the two-element Kripke frame $\backslash circ\backslash circ$ (in other words, that is not included in Smetanich's logic). Another relationship is given by the Gödel–Gentzen negative translation, which provides an embedding of classical first-order logic into intuitionistic logic: a first-order formula is provable in classical logic if and only if its Gödel–Gentzen translation is provable intuitionistically. Therefore, intuitionistic logic can instead be seen as a means of extending classical logic with constructive semantics. In 1932, Kurt Gödel defined a system of logics intermediate between classical and intuitionistic logic; Gödel logics are concomitantly known asintermediate logic In mathematical logic, a superintuitionistic logic is a propositional logic extending intuitionistic logic. Classical logic is the strongest consistent superintuitionistic logic; thus, consistent superintuitionistic logics are called intermediate l ...

s.
Admissible rules

In intuitionistic logic or a fixed theory using the logic, the situation can occur that an implication always hold metatheoretically, but not in the language. For example, in the pure propositional calculus, if $(\backslash neg\; A)\backslash to(B\backslash lor\; C)$ is provable, then so is $(\backslash neg\; A\backslash to\; B)\backslash lor(\backslash neg\; A\backslash to\; C)$. Another example is that $(A\backslash to\; B)\backslash to(A\backslash lor\; C)$ being provable always also means that so is $\backslash big((A\backslash to\; B)\backslash to\; A\backslash big)\backslash lor\backslash big((A\backslash to\; B)\backslash to\; C\backslash big)$. One says the system is closed under these implications asrules
Rule or ruling may refer to:
Education
* Royal University of Law and Economics (RULE), a university in Cambodia
Human activity
* The exercise of political or personal control by someone with authority or power
* Business rule, a rule pert ...

and they may be adopted.
Non-interdefinability of operators

In classical propositional logic, it is possible to take one of conjunction, disjunction, or implication as primitive, and define the other two in terms of it together withnegation
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 fals ...

, such as in Łukasiewicz's three axioms of propositional logic. It is even possible to define all four in terms of a sole sufficient operator such as the Peirce arrow (NOR) or Sheffer stroke (NAND). Similarly, in classical first-order logic, one of the quantifiers can be defined in terms of the other and negation.
These are fundamentally consequences of the law of bivalence, which makes all such connectives merely Boolean function
In mathematics, a Boolean function is a function whose arguments and result assume values from a two-element set (usually , or ). Alternative names are switching function, used especially in older computer science literature, and truth function ...

s.
The law of bivalence is not required to hold in intuitionistic logic, only the law of non-contradiction. As a result, none of the basic connectives can be dispensed with, and the above axioms are all necessary. So most of the classical identities between connectives and quantifiers are only theorems of intuitionistic logic in one direction. Some of the theorems go in both directions, i.e. are equivalences, as subsequently discussed.
Below follows a list of implications involving negations. Combining the axioms THEN-1 and THEN-2 above, $(P\backslash to\backslash neg\; Q)\backslash leftrightarrow(Q\backslash to\backslash neg\; P)$ follows, a conversion which can be used to obtain further implications.
Existential versus universal quantification:
* $(\backslash exists\; x\; \backslash \; \backslash phi(x))\; \backslash to\; \backslash neg\; (\backslash forall\; x\; \backslash \; \backslash neg\; \backslash phi(x))$
* $(\backslash exists\; x\; \backslash \; \backslash neg\; \backslash phi(x))\; \backslash to\; \backslash neg\; (\backslash forall\; x\; \backslash \; \backslash phi(x))$
And conversely,
* $(\backslash forall\; x\; \backslash \; \backslash neg\; \backslash phi(x))\; \backslash leftrightarrow\; \backslash neg\; (\backslash exists\; x\; \backslash \; \backslash phi(x))$
* $(\backslash forall\; x\; \backslash \; \backslash phi(x))\; \backslash to\; \backslash neg\; (\backslash exists\; x\; \backslash \; \backslash neg\; \backslash phi(x))$
There are finite variations of those with just two propositions.
Disjunction vs conjunction:
* $(\backslash phi\; \backslash vee\; \backslash psi)\; \backslash to\; \backslash neg\; (\backslash neg\; \backslash phi\; \backslash wedge\; \backslash neg\; \backslash psi)$
* $(\backslash neg\; \backslash phi\; \backslash vee\; \backslash psi)\; \backslash to\; \backslash neg\; (\backslash phi\; \backslash wedge\; \backslash neg\; \backslash psi)$
* $(\backslash neg\; \backslash phi\; \backslash vee\; \backslash neg\; \backslash psi)\; \backslash to\; \backslash neg\; (\backslash phi\; \backslash wedge\; \backslash psi)$
And conversely,
* $(\backslash neg\; \backslash phi\; \backslash wedge\; \backslash neg\; \backslash psi)\; \backslash leftrightarrow\; \backslash neg\; (\backslash phi\; \backslash vee\; \backslash psi)$
* $(\backslash phi\; \backslash wedge\; \backslash neg\; \backslash psi)\; \backslash to\; \backslash neg\; (\backslash neg\; \backslash phi\; \backslash vee\; \backslash psi)$
* $(\backslash phi\; \backslash wedge\; \backslash psi)\; \backslash to\; \backslash neg\; (\backslash neg\; \backslash phi\; \backslash vee\; \backslash neg\; \backslash psi)$
So negated statements are weak antecedents, and in turn not all of the classical De Morgan's laws hold.
Going on,
Conjunction vs. implication:
* $(\backslash phi\; \backslash wedge\; \backslash psi)\; \backslash to\; \backslash neg\; (\backslash phi\; \backslash to\; \backslash neg\; \backslash psi)$
* $(\backslash phi\; \backslash wedge\; \backslash neg\; \backslash psi)\; \backslash to\; \backslash neg\; (\backslash phi\; \backslash to\; \backslash psi)$
And conversely,
* $(\backslash phi\; \backslash to\; \backslash neg\; \backslash psi)\; \backslash leftrightarrow\; \backslash neg\; (\backslash phi\; \backslash wedge\; \backslash psi)$
* $(\backslash phi\; \backslash to\; \backslash psi)\; \backslash to\; \backslash neg\; (\backslash phi\; \backslash wedge\; \backslash neg\; \backslash psi)$
The formulas for $\backslash neg\; (\backslash phi\; \backslash wedge\; \backslash psi)$ can be used to imply $(\backslash neg\; \backslash phi\; \backslash vee\; \backslash neg\; \backslash psi)\; \backslash to\; (\backslash phi\; \backslash to\; \backslash neg\; \backslash psi)$ and one also validates the version with the positions of $\backslash phi$ and $\backslash neg\backslash phi$ switched. The latter are forms of the disjunctive syllogism for negated propositions, $\backslash neg\backslash psi$. A strenghtenend form still holds in intuitionistic logic, as follows.
Disjunction versus implication:
* $(\backslash phi\; \backslash vee\; \backslash psi)\; \backslash to\; (\backslash neg\; \backslash phi\; \backslash to\; \backslash psi)$
* $(\backslash neg\; \backslash phi\; \backslash vee\; \backslash psi)\; \backslash to\; (\backslash phi\; \backslash to\; \backslash psi)$
So, for example, intuitionistically "Either $P$ or $Q$" is a stronger propositional formula than "If not $P$, then $Q$", whereas these are classically interchangeable. Such an implication for general $\backslash psi$ is not valid in minimal logic.
Equivalences:
The above lists also contain equivalences.
Firstly, the equivalence involving a conjunction and a disjunction stems from $(P\backslash lor\; Q)\backslash to\; R$ actually being stronger than $P\backslash to\; R$. Both sides of the equivalence can be understood as conjunctions of independent implications (of absurdity). In functional interpretations, it corresponds to if-clause constructions.
So e.g. "Not ($P$ or $Q$)" is equivalent to "Not $P$, and also not $Q$".
Secondly, the equivalence involving an implication and a conjunction corresponds to currying. Due to the symmetry of the conjunction connective, it again implies
:$(\backslash phi\; \backslash to\; \backslash neg\; \backslash psi)\; \backslash leftrightarrow\; (\backslash psi\; \backslash to\; \backslash neg\; \backslash phi)$
Special cases of this conversion make up the previous lists. The symmetry of the conjunction can also understood as reason for why negation of $\backslash phi$ can be moved between antecedent and consequent in the first two lists above.
The equivalences in the lists can be jointly generalized to the equivalence
:$\backslash big(\backslash forall\; x.\backslash phi(x)\backslash to\; (\backslash psi(x)\backslash to\backslash chi)\backslash big)\backslash ,\backslash ,\backslash ,\backslash leftrightarrow\backslash ,\backslash ,\backslash ,\backslash big(\backslash exists\; x.\backslash phi(x)\backslash land\; \backslash psi(x)\backslash big)\backslash to\backslash chi$
which for $\backslash chi$ as $\backslash bot$ are two characterizations of separation.
An equivalence itself is generally defined as, and then equivalent to, a conjunction of implications
* $(\backslash phi\backslash leftrightarrow\; \backslash psi)\; \backslash leftrightarrow\; \backslash big((\backslash phi\; \backslash to\; \backslash psi)\backslash land(\backslash psi\backslash to\backslash phi)\backslash big)$
With it, such connectives become in turn definable from it:
* $(\backslash phi\backslash to\backslash psi)\; \backslash leftrightarrow\; ((\backslash phi\backslash lor\backslash psi)\; \backslash leftrightarrow\; \backslash psi)$
* $(\backslash phi\backslash to\backslash psi)\; \backslash leftrightarrow\; ((\backslash phi\backslash land\backslash psi)\; \backslash leftrightarrow\; \backslash phi)$
* $(\backslash phi\backslash land\backslash psi)\; \backslash leftrightarrow\; ((\backslash phi\backslash to\backslash psi)\backslash leftrightarrow\backslash phi)$
* $(\backslash phi\backslash land\backslash psi)\; \backslash leftrightarrow\; (((\backslash phi\backslash lor\backslash psi)\backslash leftrightarrow\backslash psi)\backslash leftrightarrow\backslash phi)$
In turn, $\backslash $ and $\backslash $ are complete bases of intuitionistic connectives.
Functionally complete connectives:
As shown by Alexander V. Kuznetsov, either of the following connectives – the first one ternary, the second one quinary – is by itself functionally complete: either one can serve the role of a sole sufficient operator for intuitionistic propositional logic, thus forming an analog of the Sheffer stroke from classical propositional logic:
* $\backslash big((P\backslash lor\; Q)\backslash land\backslash neg\; R\backslash big)\backslash lor\backslash big(\backslash neg\; P\backslash land(Q\backslash leftrightarrow\; R)\backslash big)$
* $P\backslash to\backslash big(Q\backslash land\backslash neg\; R\backslash land(S\backslash lor\; T)\backslash big)$
Semantics

The semantics are rather more complicated than for the classical case. A model theory can be given by Heyting algebras or, equivalently, by Kripke semantics. Recently, a Tarski-like model theory was proved complete by Bob Constable, but with a different notion of completeness than classically. Unproved statements in intuitionistic logic are not given an intermediate truth value (as is sometimes mistakenly asserted). One can prove that such statements have no third truth value, a result dating back to Glivenko in 1928. Instead they remain of unknown truth value, until they are either proved or disproved. Statements are disproved by deducing a contradiction from them. A consequence of this point of view is that intuitionistic logic has no interpretation as a two-valued logic, nor even as a finite-valued logic, in the familiar sense. Although intuitionistic logic retains the trivial propositions $\backslash $ from classical logic, each ''proof'' of a propositional formula is considered a valid propositional value, thus by Heyting's notion of propositions-as-sets, propositional formulae are (potentially non-finite) sets of their proofs.Heyting algebra semantics

In classical logic, we often discuss thetruth value
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'').
Computing
In some prog ...

s that a formula can take. The values are usually chosen as the members of 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 variables are the truth values ''true'' and ''false'', usually denoted 1 and 0, whereas in ...

. The meet and join operations in the Boolean algebra are identified with the ∧ and ∨ logical connectives, so that the value of a formula of the form ''A'' ∧ ''B'' is the meet of the value of ''A'' and the value of ''B'' in the Boolean algebra. Then we have the useful theorem that a formula is a valid proposition of classical logic if and only if its value is 1 for every valuation—that is, for any assignment of values to its variables.
A corresponding theorem is true for intuitionistic logic, but instead of assigning each formula a value from a Boolean algebra, one uses values from a Heyting algebra, of which Boolean algebras are a special case. A formula is valid in intuitionistic logic if and only if it receives the value of the top element for any valuation on any Heyting algebra.
It can be shown that to recognize valid formulas, it is sufficient to consider a single Heyting algebra whose elements are the open subsets of the real line R. In this algebra we have:
:$\backslash begin\; \backslash text;\; href="/html/ALL/s/bot.html"\; ;"title="bot">bot$
where int(''X'') is the interior of ''X'' and ''X''Kripke semantics

Building upon his work on semantics ofmodal logic
Modal logic is a collection of formal systems developed to represent statements about necessity and possibility. It plays a major role in philosophy of language, epistemology, metaphysics, and natural language semantics. Modal logics extend other ...

, Saul Kripke created another semantics for intuitionistic logic, known as Kripke semantics or relational semantics.Intuitionistic LogicWritten b

Joan Moschovakis

Published in ''Stanford Encyclopedia of Philosophy''.

Tarski-like semantics

It was discovered that Tarski-like semantics for intuitionistic logic were not possible to prove complete. However, Robert Constable has shown that a weaker notion of completeness still holds for intuitionistic logic under a Tarski-like model. In this notion of completeness we are concerned not with all of the statements that are true of every model, but with the statements that are true ''in the same way'' in every model. That is, a single proof that the model judges a formula to be true must be valid for every model. In this case, there is not only a proof of completeness, but one that is valid according to intuitionistic logic.Relation to other logics

Intuitionistic logic is related by duality to a paraconsistent logic known as ''Brazilian'', ''anti-intuitionistic'' or ''dual-intuitionistic logic''. The subsystem of intuitionistic logic with the FALSE axiom removed is known as minimal logic.Relation to many-valued logic

Kurt Gödel's work involvingmany-valued logic
Many-valued logic (also multi- or multiple-valued logic) refers to 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 ...

showed in 1932 that intuitionistic logic is not a finite-valued logic. (See the section titled Heyting algebra semantics above for an infinite-valued logic interpretation of intuitionistic logic.)
Relation to intermediate logics

Any finite Heyting algebra that is not equivalent to a Boolean algebra defines (semantically) anintermediate logic In mathematical logic, a superintuitionistic logic is a propositional logic extending intuitionistic logic. Classical logic is the strongest consistent superintuitionistic logic; thus, consistent superintuitionistic logics are called intermediate l ...

. On the other hand, validity of formulae in pure intuitionistic logic is not tied to any individual Heyting algebra but relates to any and all Heyting algebras at the same time.
Relation to modal logic

Any formula of the intuitionistic propositional logic (IPC) may be translated into the normal modal logic S4 as follows: :$\backslash begin\; \backslash bot^*\; \&=\; \backslash bot\; \backslash \backslash \; A^*\; \&=\; \backslash Box\; A\; \&\&\; \backslash text\; A\; \backslash text\; \backslash \backslash \; (A\; \backslash wedge\; B)^*\&=\; A^*\; \backslash wedge\; B^*\; \backslash \backslash \; (A\; \backslash vee\; B)^*\; \&=\; A^*\; \backslash vee\; B^*\; \backslash \backslash \; (A\; \backslash to\; B)^*\&=\; \backslash Box\; \backslash left\; (A^*\; \backslash to\; B^*\; \backslash right\; )\; \backslash \backslash \; (\backslash neg\; A)^*\&=\; \backslash Box(\backslash neg\; (A^*))\; \&\&\; \backslash neg\; A\; :=\; A\; \backslash to\; \backslash bot\; \backslash end$ and it has been demonstratedLévy, Michel (2011)''Logique modale propositionnelle S4 et logique intuitioniste propositionnelle''

pp. 4–5. that the translated formula is valid in the propositional modal logic S4 if and only if the original formula is valid in IPC. The above set of formulae are called the Gödel–McKinsey–Tarski translation. There is also an intuitionistic version of modal logic S4 called Constructive Modal Logic CS4.Natasha Alechina, Michael Mendler, Valeria de Paiva, and Eike Ritter

''Categorical and Kripke Semantics for Constructive S4 Modal Logic''

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Lambda calculus

There is an extended Curry–Howard isomorphism between IPC and simply-typed lambda calculus.See also

* BHK interpretation * Computability logic *Constructive proof
In mathematics, a constructive proof is a method of proof that demonstrates the existence of a mathematical object by creating or providing a method for creating the object. This is in contrast to a non-constructive proof (also known as an existe ...

* Curry–Howard correspondence
* Game semantics
* Inhabited set
* Intermediate logics
* 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 ...

* Kripke semantics
* Linear logic
* Paraconsistent logic
* Relevance theory
* Smooth infinitesimal analysis
Notes

References

* van Dalen, Dirk, 2001, "Intuitionistic Logic", in Goble, Lou, ed., ''The Blackwell Guide to Philosophical Logic''. Blackwell * Morten H. Sørensen, Paweł Urzyczyn, 2006, ''Lectures on the Curry-Howard Isomorphism'' (chapter 2: "Intuitionistic Logic"). Studies in Logic and the Foundations of Mathematics vol. 149, Elsevier * W. A. Carnielli (with A. B. M. Brunner"Anti-intuitionism and paraconsistency"

''Journal of Applied Logic Volume'' 3, Issue 1, Mar 2005, pp. 161–184 * Arend Heyting, 1930, "Die formalen Regeln der intuitionistischen Logik," in three parts, ''Sitzungsberichte der preussischen Akademie der Wissenschaften'': 42–71, 158–169.

External links

* ''Stanford Encyclopedia of Philosophy
The ''Stanford Encyclopedia of Philosophy'' (''SEP'') combines an online encyclopedia of philosophy with peer-reviewed publication of original papers in philosophy, freely accessible to Internet users. It is maintained by Stanford University. Eac ...

'':Intuitionistic Logic

by Joan Moschovakis

Intuitionistic Logic

by Nick Bezhanishvili and Dick de Jongh (from the Institute for Logic, Language and Computation at the

University of Amsterdam
The University of Amsterdam (abbreviated as UvA, nl, Universiteit van Amsterdam) is a public research university located in Amsterdam, Netherlands. The UvA is one of two large, publicly funded research universities in the city, the other being ...

)
Semantical Analysis of Intuitionistic Logic I

by Saul A. Kripke from ''Harvard University, Cambridge, Mass., USA''

Intuitionistic Logic

by '' Dirk van Dalen''

The discovery of E. W. Beth's semantics for intuitionistic logic

by A. S. Troelstra and P. van Ulsen

Expressing Database Queries with Intuitionistic Logic

by Anthony J. Bonner. L. Thorne McCarty. Kumar Vadaparty. Rutgers University, Department of Computer Science

Tableaux'method for intuitionistic logic through S4-translation

tests the intuitionistic validity of propositional formulae; provided by the Laboratoire d'Informatique de

Grenoble
lat, Gratianopolis
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, image = Panorama grenoble.png
, image size =
, caption = From upper left: Panorama of the city, Grenoble’s cable cars, place Saint- ...

*The Oxford Handbook of Philosophy of Mathematics and Logic

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"Intuitionism in Mathematics"

by David Charles McCarty {{Authority control Logic in computer science Non-classical logic Constructivism (mathematics) Systems of formal logic Intuitionism