distributive lattice bounded lattice

Heyting algebra

idea

a Heyting algebra is a bounded lattice (with join and meet operations written ∨ and ∧ and with least element 0 and greatest element 1) equipped with a binary operation a → b of implication such that c ∧ a ≤ b is equivalent to c ≤ a → b. From a logical standpoint, A → B is by this definition the weakest proposition for which modus ponens, the inference rule A → B, A ⊢ B, is sound.


1.x ≼ x x ≼ x x ≼ xreflexiveaxiom▲ partially ordered set
2.(𝑥 ≼ 𝑦 𝑎𝑛𝑑 𝑦 ≼ 𝑥) ⇒ 𝑥=𝑦 (x ≼ y and y ≼ x) ⇒ x=y (x ≼ y and y ≼ x) ⇒ x=yantisymmetricaxiom▲ partially ordered set
3.(𝑥 ≼ 𝑦 𝑎𝑛𝑑 𝑦 ≼ 𝑧 ) ⇒ 𝑥 ≼ 𝑧 (x ≼ y and y ≼ z ) ⇒ x ≼ z (x ≼ y and y ≼ z ) ⇒ x ≼ ztransitiveaxiom▲ partially ordered set
گسترش‌ها Boolean algebra

تاربرگ‌های پیموده‌شده در این نشست: De Morgan duality ⟩ tautology ⟩ begging the question ⟩ The Mathematical Analysis of Logic