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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?

Dannyu NDos
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    What's wrong with "bottom"? – David Gudeman Feb 25 '23 at 00:26
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    Your problem is not so much finding a name for ⊥ but giving its semantics. If your logic allows that some contradictions are true, then you cannot have the usual rules for bottom/falsum, or for negation for that matter. You might like to check out the answer to this question about minimal logic, which is also a paraconsistent sublogic of intuitionistic logic. https://math.stackexchange.com/questions/2625373/semantics-for-minimal-logic – Bumble Feb 25 '23 at 06:33
  • There's merit to this question, intuitively, ja? – Agent Smith Mar 15 '23 at 13:09
  • The ⊥ sign, otherwise known as the uptack, is an unconditional false, or contridiction. – SublimeDeck May 05 '23 at 17:43
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    ⊥ in logic is often called "[falsum](https://en.wikipedia.org/wiki/Up_tack)". However, since ⊥ is also called a contradiction, it is unwise to declare that ⊥ isn't a contradiction, and also unwise to use the word "contradiction" to describe something that can be true in your system. Perhaps "P and not P" could be called a "conjunction of opposites", for example. – kaya3 Aug 17 '23 at 11:08

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