- Brouwer
- Brouwer was a Dutch mathematician and philosopher (1881-1966) who fundamentally changed how mathematicians think about proof and logic. He argued that math shouldn't just accept something as true because it follows logically; instead, mathematicians should be able to actually construct or demonstrate mathematical objects to prove they exist. His ideas challenged the traditional approach to mathematics and influenced debates about the foundations of mathematical reasoning that continue today.
- Constructive witness(what intuitionistic logic requires)
- An actual, concrete example or proof that demonstrates something is true, rather than just proving that assuming it's false leads to a problem.
- Equivalence(what classical and intuitionistic logic disagree about)
- Two statements are equivalent when they mean exactly the same thing and always have the same truth value.
- Heyting(developer of intuitionistic logic)
- Arend Heyting was a Dutch logician who developed and formalized Brouwer's intuitionistic logic into a rigorous system that's easier to work with.
- classical logic(Contrasted with Hegel's dialectical approach that accepts contradictions)
- Aristotelian logic that dominated during Hegel's lifetime
- intuitionistic logic(Formal logic differing from classical logic in its treatment of structural rules and logical consequence)
- A logic that does not accept all classical structural rules, thereby invalidating certain classically valid formulas such as (p → q) ∨ p
- negation(Standard truth conditions for logical negation, used as the basis for arguments against dialetheism)
- ¬A is true if and only if A is not true
- ¬∃x[P(x) ∧ ¬D(x)](second logical formula in the statement)
- A symbolic way of saying 'there does not exist anything that has property P but lacks property D'—it's the opposite way of stating the same rule.
- ∀x[P(x) → D(x)](first logical formula in the statement)
- A symbolic way of saying 'for all things, if they have property P, then they have property D'—it's a general rule.