I understand that for the consequent to really follow from the antecedent, it (the consequent) must be both relevant and necessary given the antecedent.

So my question is: which types of conditional statements actually hold true under dialetheism? Obviously it isn't all of them e.g. A disletheist would not accept the general argument:

If 'P' and 'not-P' then 'Q'. Where 'P' and 'Q' are just anything, right?

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    It seems relevance logic is what you're describing" A notable feature of relevance logics is that they are paraconsistent logics: the existence of a contradiction will not cause "explosion". This follows from the fact that a conditional with a contradictory antecedent that does not share any propositional or predicate letters with the consequent cannot be true (or derivable)... Commented Jan 5, 2023 at 3:50
  • @DoubleKnot, a million thanks for your succinct and to-the-point comment.
    – Hudjefa
    Commented Jan 5, 2023 at 10:57
  • One should not conflate dialetheism with paraconsistency. Relevance logics are paraconsistent, because they do not include explosion, but relevance logicians do not typically allow that any contradictions are true. Dialetheism is the position that some contradictions are true. Graham Priest is an advocate of the Logic of Paradox, which is a trivalent logic under which a conditional is true if its antecedent is false or its consequent true, false if its antecedent is true and its consequent false, and paradoxical otherwise.
    – Bumble
    Commented Jan 5, 2023 at 11:28


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