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If we do not assume the Law of Excluded Middle, we do not assume that there is no proposition $P$ for which it is false. However, if there were such a proposition, it would violate the Law of Non-contradiction. That can be formalized as such:

$$\exists P, \ \ \text{s.t.} \ \ \neg(P \lor \neg P) \iff \exists P, \ \ \text{s.t.} \ \ \neg P \land \neg \neg P $$

This would be a violation of the Law of Non-contradiction:

$$\neg(Q \land \neg Q) \iff \neg(\neg P \land \neg \neg P)$$

If we assume LNC, then LEM is implied. Why then, is LEM an axiom? How then, can constructivist logics assume LNC without implying LEM?

user110391
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    Your reasoning is incorrect. To use the technical term, it is begging the question. Also, the law of non-contradiction is not an axiom. It is a consequence of the definition of negation and, if you insist, modus ponens and the deduction theorem. – Zhen Lin Sep 23 '22 at 06:57
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    Intuitionistic Logic rejects LEM and thus, consistently, Double Negation and part of De Morgan's equivalence (that you have used above). – Mauro ALLEGRANZA Sep 23 '22 at 06:59
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    See Do De Morgan's laws hold in propositional intuitionistic logic? – Mauro ALLEGRANZA Sep 23 '22 at 07:01
  • @ZhenLin LNC is not an axiom of logic in general? Or not an axiom of constructivist logic? The latter is still strange, as I am sure I had read it was. The former however, would contradict every text I've ever read on the matter. – user110391 Sep 23 '22 at 07:10
  • It is definitely not an axiom of logic in general, because there are paraconsistent logics. But in particular it is not adopted as an axiom in constructive logic because it can be deduced from other principles. – Zhen Lin Sep 23 '22 at 07:12
  • @ZhenLin So, it isn't an axiom in most logics? Anyways, LNC does not imply LEM in constructive logics because the DeMorgan Laws do not allow that in constructive logics. That answers one part of my question. However, in classical logic, LNC and LEM are both axioms. Why is that so, when, in classical logic, the LEM is derivable from the LNC? Is it because the LEM is derivable from the LNC via DeMorgan laws, who themselves are derived from the LEM, which means the LEM is still needed as an axiom? – user110391 Sep 23 '22 at 07:33
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    Actually, I don’t think the law of non-contradiction is adopted as an axiom of classical logic either. It is difficult to go into details because you have not specified what you have in mind by classical logic. I advise you to learn at least one formal system for classical logic. Then we can talk. – Zhen Lin Sep 23 '22 at 07:44
  • LNC means that a statement and its negation cannot be both True. In propositional logic we prove the "corresponding" formula $\lnot (P \land \lnot P)$ using Ex Falso rule (or similar axiom) that is constructively sound. – Mauro ALLEGRANZA Sep 23 '22 at 10:29

1 Answers1

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Everything implies that which is a tautology (LEM in this case):

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Dan Christensen
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