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We can formally write a contradiction as, p and (not p); these statements in classical logic will always evaluate to false. Now, a true contradiction means that actually this evaluates to true.

Now, it's just occurred to me that there is a converse to this: p or (not p) always evaluates to true. If we insist it evaluates to false, we could say it is a false co-contradiction! Is there actually a formal name for this?

They are mirror images though - if the usual laws are available since:

p or (not p) = false , then applying not gives

not (p or (not p))= not (false)

(not p) and p=true

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  • Violations of the law of excluded middle are allowed in intuitionistic logic, but there the semantics of propositions is in terms of provable/refutable instead of true/false. – David H Jun 7 '13 at 10:35
  • @DavidH: They have no truth values at all? – Mozibur Ullah Jun 7 '13 at 10:38
  • @MoziburUllah: At the very least, we don't talk about their truth-values in intuitionistic logic. That is to say, intuitionistic logic does not attempt to model truth, but rather provablility or constructibility. Philosophical motivations and interpretations vary. Some use this as a means of avoiding or even denying the concept of truth, while others take it to simply be independent of but related to truth. – Addem Jun 7 '13 at 15:38
  • @addem: ok, got you. – Mozibur Ullah Jun 7 '13 at 16:12
  • @MoziburUllah This will also vary from author to author, some are happy to speak as though truth and provability collapse into one another. – Dennis Jul 10 '13 at 22:50
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As a matter of fact there is a standard term for those propositions which are always true under any interpretation: tautologies. The mirror image as you say of a true contradiction is a false tautology.

Now, it's just occurred to me that there is a converse to this: p or (not p) always evaluates to true. If we insist it evaluates to false, we could say it is a false co-contradiction! Is there actually a formal name for this?

They are mirror images though - if the usual laws are available....

Key word being IF the usual laws are available. In saying that, you just reaffirmed the truth of the very same p∨¬p that you presently are hypothesizing to be false. By the law of non-contradiction:

⊢ ¬(ϕ ∧ ¬ϕ)

Hence, replacing ϕ with (p∨¬p), it follows that

⊢ ¬[(p ∨ ¬p) ∧ ¬(p ∨ ¬p)].

Option A) Let the whole darn thing explode by accepting that both LNC and its negation are both true. "Yes I absolutely accept the Law of Non-contradiction, and violations of it are impossible. Yes, violations of it also exist. And yes, I'm fine with that."

Option B) Our original question was what if LEM wasn't true, but we leaped straight to investing what happens if it was false. But if LEM is not true, we don't have to take that to mean it must be false any more since the middle isn't being excluded.

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