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?
In a logic system that accepts true contradictions but not the law of noncontradiction, the symbol ⊥ (usually called "bottom" or "falsum") may represent something different than a standard contradiction (p∧¬p). You could refer to it as a "logical anomaly" or "logical inconsistency" to distinguish it from a standard contradiction.
So, in your system, you might say that (⊤→(p∧¬p))→⊥ represents the idea that contradictions do not necessarily lead to logical inconsistencies or anomalies. This approach aligns with the concept of paraconsistent logic, which allows for reasoning in the presence of contradictions without leading to triviality or inconsistency.
