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 intuitionistic logic do not include the law of the excluded middle and double negation elimination, which are fundamental inference rules in classical 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 intuitionistic logic do not include 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 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. 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 Gödel’s dialectica interpretation, Kleene’s realizability, Medvedev’s logic of finite problems, or 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 { ⊤ , ⊥ } {displaystyle { op ,ot }} ('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 which are feasible to create and check solely by hand. One example of a proof which was impossible to formally verify without algorithm 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, 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. 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: 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 ¬ [ ¬ ( P ∨ ¬ P ) ] {displaystyle eg } regardless of the proposition P {displaystyle P} . 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.