is fundamental to intuitionistic type theory.
These laws are not valid in intuitionistic type theory.
For further discussion, see the entry on intuitionistic logic.
The system C is a conservative extension of positive intuitionistic logic.
This dampened the enthusiasm of the mathematical community for the intuitionistic project.
Later on, Gödel (1932) tried to understand intuitionistic logic in terms of many truth degrees.
The British philosopher Michael Dummett (1973) developed a philosophical basis for Intuitionism, in particular for intuitionistic logic.
Meyer Viol (1995a, 1995b) contain further proof- and model-theoretic studies of the epsilon calculus; specifically intuitionistic epsilon calculi.
Williamson (1982) argues that Fitch’s proof is not a refutation of anti-realism, but rather a reason for the anti-realist to accept intuitionistic logic.
Intuitionistic dependence logic retains the downwards closure property: if a team satisfies an intuitionistic dependence logic formula then so do all of its subsets.
In fact, as is common in intuitionistic settings, a plethora of semantic and proof-theoretic methods are available for the study of constructive and intuitionistic set theories.
Constructive and intuitionistic Zermelo-Fraenkel set theories are axiomatic theories of sets in the style of Zermelo-Fraenkel set theory (ZF) which are based on intuitionistic logic.
A variety of interpretations for intuitionistic logic have been extended to intuitionistic and constructive set theories, such as realisability, Kripke models and Heyting-valued semantics.
Both the intuitionistic view of logic as essentially sterile, and the existence of results in intuitionistic logic that are incompatible with classical logic, depend essentially on that conception.
In intuitionistic reverse mathematics one has a similar aim, but then with respect to intuitionistic theorems: working over a weak intuitionistic theory, axioms and theorems are compared to each other.
Constructive and intuitionistic Zermelo-Fraenkel set theories are based on intuitionistic rather than classical logic, and represent a natural environment within which to codify and study mathematics based on intuitionistic logic.
Hence, if intuitionistic mathematics contains objects and principles that do not figure in classical mathematics, it may come about that intuitionistic logic, which then depends also on these non-classical elements, is no proper part of classical logic.
Double negation elimination and classical contraposition fail to be valid in intuitionistic logic (see the entry on intuitionistic logic); if one of them is added to an axiom system of intuitionistic logic, one obtains a proof system for classical logic.
logic, history of: intuitionistic logic | logic: intuitionistic | mathematics, philosophy of: intuitionism | mathematics: constructive | proof theory: development of | semantics: Montague | semantics: proof-theoretic | set theory | set theory: constructive and intuitionistic ZF | set theory: Zermelo’s axiomatization of | type theory | type theory: Church’s type theory
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