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 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 algebras. 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.
In the semantics of classical logic, propositional formulae are assigned truth values from the two-element set ("true" and "false" respectively), regardless of whether we have direct evidence 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 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 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.
The syntax of formulas of intuitionistic logic is similar to propositional logic or first-order logic. 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 (A → ⊥). In intuitionistic first-order logic both quantifiers ∃, ∀ are needed.
In propositional logic, the inference rule is modus ponens
- MP: from and infer
and the axioms are
To make this a system of first-order predicate logic, the generalization rules
- -GEN: from infer , if is not free in
- -GEN: from infer , if is not free in
are added, along with the axioms
- PRED-1: , if the term is free for substitution for the variable in (i.e., if no occurrence of any variable in becomes bound in )
- PRED-2: , with the same restriction as for PRED-1
If one wishes to include a connective for negation rather than consider it an abbreviation for , it is enough to add:
There are a number of alternatives available if one wishes to omit the connective (false). For example, one may replace the three axioms FALSE, NOT-1', and NOT-2' with the two axioms
as at Propositional calculus § Axioms. Alternatives to NOT-1 are or .
The connective for equivalence may be treated as an abbreviation, with standing for . Alternatively, one may add the axioms
IFF-1 and IFF-2 can, if desired, be combined into a single axiom using conjunction.
Relation to classical logic
The system of classical logic is obtained by adding any one of the following axioms:
- (Law of the excluded middle. May also be formulated as .)
- (Double negation elimination)
- (Peirce's law)
- (Law of contraposition)
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.
When explaining intuitionistic logic in terms of classical logic, it can be understood as a weakening thereof: 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 logic – in 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.
It does merely "not affirm" the law of the excluded middle because while it is not necessarily the case that the law is upheld in any context, no counterexample can be given either. 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. Formally, it is a simple theorem that for any two propositions. By considering any established to be false in deed shows that the double negation of the law is retained as a tautology already in minimal logic. So the propositional calculus is always compatible with classical logic. When assuming the law implies a proposition, then by applying contraposition twice and using the double-negated excluded middle, one may prove double-negated variants of various strictly classical tautologies. The situation is more intricate for predicate logic formulas, when some quantified expressions are being negated.
Akin to the above, from modus ponens in the form follows . The relation between them may always be used to obtain new formulas: A weakened premise makes for a strong implication, and vice versa. For example, note that if holds, then so does , and even weaker than the latter are the four equivalent statements , , as well as . Propositions for which double-negation elimination is possible are also called stable. Intuitionistic logic proves stability only for restricted types of propositions, such as those of negated form. A formula for which excluded middle holds can be proven stable using the disjunctive syllogism, but the converse does not hold in general, unless the excluded middle statement is stable itself.
When expresses a claim, then its double-negation merely expresses the claim that a refutation of would be inconsistent. Having proven such a mere double-negation also still aids in negating other statements through negation introduction . A double-negated existential statement does not denote existence of an entity with a property, but rather the absurdity of assumed non-existence of any such entity. Also all the principles in the next section involving quantifiers explain use of implications with hypothetical existence as premise.
Weakening statements by adding two negations before existential quantifiers (and atoms) is also the core step in the double-negation translation. It constitutes 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 be seen as a means of extending classical logic with constructive semantics.
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 with negation, 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 functions. The law of bivalence is not required to hold in intuitionistic logic. 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.
Existential vs. universal quantification
Firstly, when is not free in the proposition , then
When the domain of discourse is empty, then by the principle of explosion, an existential statement implies anything. When the domain contains at least one term, then assuming excluded middle for , the two sides become equivalent. Indeed, classically they are also equivalent to a disjunctive form discussed further below. Constructively, existence claims are however harder to come by.
If the domain of discourse is not empty and is moreover independent of , such principles are equivalent to formulas in the propositional calculus. Here, the formula then just expresses the identity . This is the curried form of modus ponens , which in the special the case with as a false proposition results in the law of non-contradiction principle . Considering a false proposition for the original implication results in the important
In words: "If there exists an entity that does not have the property , then the following is refuted: Each entity has the property ."
The quantifier formula with negations also immediately follows from the non-contradiction principle derived above, each instance of which itself already follows from the more particular . To derive a contradiction given , it suffices to establish its negation (as opposed to the stronger ) and this makes proving double-negations valuable also. By the same token, the original quantifier formula in fact still holds with weakened to . And so really
In words: "If there exists an entity that does not have the property , then the following is refuted: For each entity, one is not able to prove that it does not have the property ".
where similar considerations apply. Here the existential part is always a hypothesis and this is an equivalence. Considering the special case again,
Lastly, by using the axiom THEN-1 once, THEN-2 can be reduced to its special case and this conversion can then be used to obtain two further implications:
For simplicity, in the discussion here and below, the formulas are generally presented in weakened forms without double-negations. To name but one, a special case of the first formula here is . Note how also this formula indeed immediately implies the -direction of the equivalence bullet point listed above. The double-negations can also be placed in the antecedent.
More complicated variants hold. Incorporating the predicate and currying, the following generalization also entails the relation between implication and conjunction in the predicate calculus, discussed below.
If the predicate is decidedly false for all , then this equivalence is trivial. If is decidedly true for all , the schema simply reduces to the previously stated equivalence. In the language of classes, and , the special case of this equivalence with false equates two characterizations of disjointness :
Disjunction vs. conjunction
There are finite variations of the quantifier formulas, with just two propositions:
And as mentioned, stronger forms of these formulas hold. For example, . This can easily be derived from the non-contradiction principle, as all of the formulas are about deriving absurd consequences from hypotheticals, and these statements do not have to be placed in a particular order. By the same reasoning, one may obtain the mixed form of the implication and its converse
But in particular, there is no distributivity principle deriving the claim from . Negated propositions are comparably weak, in that the classically valid De Morgan's law, granting a disjunction from a single negative hypothetical, does not automatically hold constructively. See also the for a variant concerning two existentially closed decidable predicates. For an informal example of the constructive reading, consider the following: From conclusive evidence it not to be the case that both Alice and Bob showed up to their date, one cannot derive conclusive evidence, tied to either of the two persons, that this person did not show up. Instead, the intuitionistic propositional calculus and some of its extensions exhibit the disjunction property, implying the disjuncts individually would have to be derivable as well. When the fourth law is adopted, the weak excluded middle, i.e. the statement for negated propositions, immediately follows.
Conjunction vs. implication
From the general equivalence also follows the following special case, expressing incompatibility of two predicates using two different connectives:
It may be understood as a special case of currying and uncurrying. Many more considerations regarding double-negations again apply. For example, as is stronger than , from this equivalence the implication follows. However, this cannot generally be reversed, as for true the double-negation elimination principle would follow.
For either of the propositions established to be true, this again reduces to the double-negation principle for the other, and such an implication cannot generally be reversed.
Due to the symmetry of the conjunction connective, the equivalence formula for the negated conjunction again implies the already established .
Disjunction vs. implication
The two different formulas for mentioned above can be used to imply . The latter are forms of the disjunctive syllogism for negated propositions, . A strengthened form still holds in intuitionistic logic, as follows.
As above, the positions of and may be switched and may be weakened to . These principles are provable from non-contradiction and explosion. So, for example, intuitionistically "Either or " is a stronger propositional formula than "If not , then ", whereas these are classically interchangeable. The implication cannot generally be reversed, as that immediatenly implies excluded middle.
Considering in place of in the above form of the syllogism shows how excluded middle implies double-negation elimination. And in this way intuitionistic logic proves .
Of course the formulas established here may be combined to obtain yet more variations. For example, the disjunctive syllogism as presented generalizes to
If some term exists at all, the antecedent here even implies , which in turn itself also implies the conclusion here (this is again the very first formula mentioned in this section).
The bulk of the discussion in these sections applies just as well to just minimal logic. But as for the disjunctive syllogism with general , minimal logic can at most prove where denotes . The conclusion here can only be simplified to using explosion.
The above lists also contain equivalences. The equivalence involving a conjunction and a disjunction stems from actually being stronger than . Both sides of the equivalence can be understood as conjunctions of independent implications. Above, absurdity is used for . In functional interpretations, it corresponds to if-clause constructions. So e.g. "Not ( or )" is equivalent to "Not , and also not ".
An equivalence itself is generally defined as, and then equivalent to, a conjunction of implications
With it, such connectives become in turn definable from it:
In turn, and 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:
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 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 the truth values that a formula can take. The values are usually chosen as the members of a Boolean algebra. 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:
The last identity concerning A → B allows us to calculate the value of ¬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:
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:
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.
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.
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 is provable, then so is . Another example is that being provable always also means that so is . One says the system is closed under these implications as rules and they may be adopted.
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 and some differences have been elaborated on above.
Any finite Heyting algebra that is not equivalent to a Boolean algebra defines (semantically) an intermediate logic. 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.
Kurt Gödel's work involving many-valued logic 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.)
and it has been demonstrated 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.
- BHK interpretation
- Computability logic
- Constructive analysis
- Constructive proof
- Constructive set theory
- Curry–Howard correspondence
- Game semantics
- Harrop formula
- Heyting arithmetic
- Inhabited set
- Intermediate logics
- Intuitionistic type theory
- Kripke semantics
- Linear logic
- Paraconsistent logic
- Relevance theory
- Smooth infinitesimal analysis
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