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Intuitionistic logic is related by duality to a paraconsistent logic known as Brazilian, anti-intuitionistic or dual-intuitionistic logic. [14] The subsystem of intuitionistic logic with the FALSE (resp. NOT-2) axiom removed is known as minimal logic and some differences have been elaborated on above.
Logical Intuition, or mathematical intuition or rational intuition, is a series of instinctive foresight, know-how, and savviness often associated with the ability to perceive logical or mathematical truth—and the ability to solve mathematical challenges efficiently. [1]
In mathematical logic, the Brouwer–Heyting–Kolmogorov interpretation, or BHK interpretation, of intuitionistic logic was proposed by L. E. J. Brouwer and Arend Heyting, and independently by Andrey Kolmogorov. It is also sometimes called the realizability interpretation, because of the connection with the realizability theory of Stephen ...
The fundamental distinguishing characteristic of intuitionism is its interpretation of what it means for a mathematical statement to be true. In Brouwer's original intuitionism, the truth of a mathematical statement is a subjective claim: a mathematical statement corresponds to a mental construction, and a mathematician can assert the truth of a statement only by verifying the validity of that ...
This principle was established by Brouwer in 1928 [1] using intuitionistic principles, and can also be proven using Church's thesis. The analogous property in classical analysis is the fact that every continuous function from the continuum to {0,1} is constant.
For simplicity, the logics presented so far have been intuitionistic. Classical logic extends intuitionistic logic with an additional axiom or principle of excluded middle: For any proposition p, the proposition p ∨ ¬p is true. This statement is not obviously either an introduction or an elimination; indeed, it involves two distinct connectives.
The Dialectica interpretation has been used to build a model of Girard's refinement of intuitionistic logic known as linear logic, via the so-called Dialectica spaces. [3] Since linear logic is a refinement of intuitionistic logic, the dialectica interpretation of linear logic can also be viewed as a refinement of the dialectica interpretation ...
A sequent is said to be an intuitionistic sequent if there is at most one formula in the succedent (although multi-succedent calculi for intuitionistic logic are also possible). More precisely, the restriction of the general sequent calculus to single-succedent-formula sequents, with the same inference rules as for general sequents, constitutes ...