Intuitionistic Logic emphasizes the idea that a mathematical statement is only considered true if there is a constructive proof or construction that verifies its truth.

Intuitionistic Logic rejects the Law of Excluded Middle, which states that for any proposition P, either P or not P is true. This rejection is a key distinction between Intuitionistic Logic and Classical Logic.

In Intuitionistic Logic, double negation (~~P) is stronger than negation (~P). This means that ~~P implies ~P, but the converse is not necessarily true.

Modus Ponens (P → Q, P ⊢ Q) is a valid inference rule in Intuitionistic Logic, allowing you to infer Q from P and P → Q.

Intuitionistic Logic is generally considered a weaker system than Classical Logic, as it rejects the Law of Excluded Middle and other principles that are valid in Classical Logic.

In Intuitionistic Logic, intuition is often seen as a guide for constructing proofs and discovering mathematical truths, but it is not considered a source of absolute knowledge.

Intuitionistic Logic has found applications in Computer Science, particularly in areas such as program verification and type theory.

L.E.J. Brouwer is widely recognized as one of the pioneers of Intuitionistic Logic, introducing and developing its fundamental principles.

The Brouwer-Heyting-Kolmogorov interpretation is a constructive interpretation of Intuitionistic Logic based on set theory, where mathematical objects are constructed step-by-step.

In Intuitionistic Logic, the disjunction connective (∨) is often interpreted as constructive disjunction, where at least one of the disjuncts can be constructively proven.

The Curry-Howard correspondence is a fundamental result in Intuitionistic Logic, establishing a deep connection between proofs and programs, where proofs can be seen as programs and vice versa.

Constructive Dilemma (P → Q, R → S, P ∨ R ⊢ Q ∨ S) is a valid inference rule in Intuitionistic Logic, allowing you to infer Q ∨ S from P → Q, R → S, and P ∨ R.

Intuitionistic Logic and Proof Theory are closely related fields, with Intuitionistic Logic providing a foundation for Proof Theory, which studies the structure and properties of proofs.

Intuitionistic Logic has found applications in Computer Science, particularly in program verification, where it is used to formally prove the correctness of computer programs.

Glivenko's Theorem establishes a fundamental connection between Intuitionistic Logic and Classical Logic, showing that a formula is provable in Intuitionistic Logic if and only if its double negation is provable in Classical Logic.