Syntax
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 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.
Many tautologies of classical logic can no longer be proven within intuitionistic logic. Examples include not only the law of excluded middle p ∨ ¬p, but also Peirce's law ((p → q) → p) → p, and even double negation elimination. In classical logic, both p → ¬¬p and also ¬¬p → p are theorems. In intuitionistic logic, only the former is a theorem: double negation can be introduced, but it cannot be eliminated. Rejecting p ∨ ¬p may seem strange to those more familiar with classical logic, but proving this statement in constructive logic would require producing a proof for the truth or falsity of all possible statements, which is impossible for a variety of reasons.
Because many classically valid tautologies are not theorems of intuitionistic logic, but all theorems of intuitionistic logic are valid classically, intuitionistic logic can be viewed as a weakening of classical logic, albeit one with many useful properties.
Read more about this topic: Intuitionistic Logic