Question

I am building a weakened version of the intuitionistic logic. It wouldn't satisfy (p∧¬p)→⊥ as a tautology, but rather, (⊤→(p∧¬p))→⊥. In plain English, contradictions admit no proof, but there might still be true contradictions anyway. (Of course, here "true" doesn't mean "tautological")

By not fully accepting the law of noncontradiction, in addition to being intuitionistic, this logic system is paraconsistent. But how should I call ⊥, if a "contradiction" is meant to indicate p∧¬p?

Answer 1

A weakened version of intuitionistic logic that satisfies (⊤→(p∧¬p))→⊥ is known as paraconsistent logic, where true contradictions may exist. A contradiction is indicated by p∧¬p.

Step-by-step explanation:

A weakened version of the intuitionistic logic that satisfies (⊤→(p∧¬p))→⊥ instead of (p∧¬p)→⊥ is known as paraconsistent logic. In this logic, contradictions do not admit a proof, but true contradictions may still exist.

In this context, a contradiction is meant to indicate p∧¬p, where p is any statement and ⊥ represents a contradiction.