File:Logique intuitionniste Français: Logique intuitionniste – Modèle de Kripke où le tiers-exclu n’est pas satisfait. Date, 15 April. Interprétation abstraite en logique intuitionniste: extraction d’analyseurs Java certi és. Soutenue le 6 décembre devant la commission d’examen. Kleene, S. C. Review: Stanislaw Jaskowski, Recherches sur le Systeme de la Logique Intuitioniste. J. Symbolic Logic 2 (), no.
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Intuitionistic logicsometimes more generally called constructive logicrefers 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 include intuitionnishe law of the excluded middle and double negation eliminationwhich are fundamental inference rules in classical logic. Intuitionistic logic is one example of a logic in a family of non-classical logics called paracomplete logics: Formalized intuitionistic logic was originally developed by Arend Heyting to provide a formal basis for Brouwer ‘s programme of intuitionism. 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.
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. This is referred to as the ‘law of excluded middle’, because logiqhe excludes the possibility of any truth value besides ‘true’ or ‘false’. In contrast, propositional logiqje in intuitionistic logic are not assigned a definite intuitiinniste value and are only considered “true” when we have direct evidence, hence proof.
We can also say, instead of intuitionnisye propositional formula being “true” due to direct evidence, that it is inhabited by a proof in the Curry—Howard sense.
Church : Review: A. Heyting, La Conception Intuitionniste de la Logique
Operations in intuitionistic logic intuitionnisge preserve justificationwith 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 logiqeu 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 logiqque middle and double negation elimination.
These are considered to be so important to the practice of mathematics that David Hilbert wrote of them: 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 propertymaking it also suitable for other forms of mathematical constructivism.
Informally, this means that if there is a intuitionniiste 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 which are feasible to create and check solely by hand. One example of a proof which was impossible to formally verify before the advent of these tools is the famous proof of the four color theorem. This theorem stumped mathematicians for more than a hundred years, until a proof was developed which 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 it was finally 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 logichence their choice matters.
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 logic – in particular, as said above one of its chief points 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: Indeed, the double negation of the law is retained as a tautology of the system: Gentzen discovered that a simple restriction of his system LK his sequent calculus for classical logic results in a system which 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. 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. To make this a system of first-order predicate logic, the generalization rules. Alternatively, lofique may add the axioms.
Therefore, intuitionistic logic can instead be seen as a means of extending classical logic with constructive semantics. Similarly, in classical first-order logic, one of the quantifiers can logiqque defined in terms of the other and negation. These are fundamentally consequences of the law of bivalencewhich makes all such connectives merely Boolean functions.
The law of bivalence does not 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. Most of the classical identities are only theorems of intuitionistic logic in one direction, although some are theorems in both directions. They are as follows:. So, for example, “a or b” is a stronger propositional formula than “if not a, then b”, whereas these are classically interchangeable.
On the other hand, “not a or b ” is equivalent to “not a, and also not b”. If we include equivalence in the list of connectives, some of the connectives become definable from others:.
As shown by Alexander Kuznetsov, either of the following intuitionnisye — the first one ternary, the second one quinary — is by itself functionally complete: 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 Constablebut 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 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. In classical logic, we often discuss the truth values that a formula intuitionnisye take.
The values are usually chosen as the members of a Boolean algebra. Then we have the useful theorem that a formula is a valid proposition of classical logic intuitionnisge 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 an Heyting algebra, of which Boolean algebras are a special case.
A formula is intuitjonniste 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. With these assignments, intuitionistically valid formulas are precisely those that are assigned the value of the entire line.
Michel Levy LIG IMAG
So the valuation of this formula is true, and indeed the formula is valid. 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.
Building upon his work on semantics of modal logicSaul Kripke created another semantics for intuitionistic logic, known as Kripke semantics or relational semantics. It was discovered that Tarski-like semantics for intuitionistic logic were not possible to prove complete. However, Lovique Constable has shown that a weaker notion intuitionnizte 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.
Intuitionistic logic is related by duality to a paraconsistent logic known as Braziliananti-intuitionistic or dual-intuitionistic logic.
Any finite Heyting algebra which 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. Any formula of the intuitionistic propositional logic may be translated into the normal modal logic S4 as follows:. From Wikipedia, the free encyclopedia.
An International Journal for Symbolic Logicvol. Unifying Logic, Language and Philosophy. Hilbert intuitionhiste, p. The Stanford Encyclopedia of Philosophy. Lectures on the Curry-Howard Isomorphism. Studies in Logic and the Foundations of Mathematics. The Mathematics of Metamathematics. Written by Joan Moschovakis. Lgoique in Stanford Encyclopedia of Philosophy.
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Annals of Pure and Applied Logic. Notre Dame Journal of Formal Logic. Logique modale propositionnelle S4 et logique intuitioniste propositionnellepp. Studies in Logic and the Foundations of Mathematics vol. Intuitionistic logic Constructive analysis Heyting arithmetic Intuitionistic type theory Constructive set theory.
Degree of truth Fuzzy rule Fuzzy set Fuzzy finite element Fuzzy set operations. Structural rule Relevance logic Linear logic. Retrieved from ” https: Logic in computer science Non-classical logic Constructivism mathematics Systems of formal logic Intuitionism.