information:classification
Mathematical constructivism

In the semantics of classical logic, propositional formulae are assigned truth values from the two-element set { ⊤, ⊥ } {\displaystyle \{\top,\bot \}} ("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. David Hilbert considered them to be so important to the practice of mathematics that he wrote:

Intuitionistic logic has found practical use in mathematics despite the challenges presented by the inability to utilize these rules. 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 generation and verification 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.