Say I assume predicate A and B.
Say I show that A AND B is a contradiction.
If I then apply the law of excluded middle to say NOT A, this is a fallacy, no? Does it have a particular name?
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Anything follows from a contradiction by the law of explosion, so, as described, this is not a fallacy. (A ∧ B) ∧ ¬ (A ∧ B) → ¬ A is not very useful, but valid. Did you mean something else? I do not see where the law of excluded middle is supposed to be applied.– ConifoldAug 31 at 12:01
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In so-called Classical Logic (the "mainstream logic used in mathematics) a contradiction is a fatal wound for the "system" and produces its bankruptcy. So, there are no different levels of inconsistency...– Mauro ALLEGRANZAAug 31 at 12:25
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I don't believe it has a formal name, I propose we refer to it as a variant of the Cherry Pick Fallacy, since it involves a situation where we have ¬ A ∨ ¬ B and we concluded ( falsely) ¬ A. Also, for clarity: Predicates are always conjoined, formally. So taking A, and taking B, two "seperate" predicates, is really taking A ∧ B as a predicate. We always take the conjunction of every predicate when we do a proof.– Michael CareyAug 31 at 15:33
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1 Answer
What I think you mean is: if you start from several premises, and find that these premises together lead into a contradiction, it may be incorrect to conclude that one particular premise is the one that's wrong. All you know is that the conjunction of all premises is false, but you don't know which member of the conjunction is the problem.
I don't think there's a specific name for this. Informally, the weakest/most uncertain premise is the one we would normally reject.
It may be that there are mutually exclusive premises, eliminating any of which may remove all contradictions. So it might not even be sensible to say that any particular premise is false on its own. If we take ZFC, and then also assume the negation of the axiom of choice, we have a contradiction. But we can't say that the axiom of choice is false, and nor can we say that the negation of the axiom of choice is false. ZFC forms a consistent system, and ZF and the negation of C also does.
In Fitch-style logic, we may make an assumption, different from a normal premise. Each assumption has a limited scope where it can be used. If we find a contradiction within the scope of an assumption, then the formal conclusion, valid outside the most recent scope, is that the most recently made assumption was false.
Now, the actual problem might not have been the most recently made assumption A; it might have been an earlier premise, or an earlier assumption that was also in scope. But if some earlier assumption or premise was the false one, then it's still valid to conclude "not A," because it's valid to conclude anything from a falsehood (the earlier premise). So there is no fallacy involved in selecting the most recent assumption A as being false.
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Thanks, this is what I was looking for; I come mostly from a programming background and wasn't sure if there was a notion of scope in logic. This helps to clarify that. Is there a name for this "rule" of scope in fitch-style logic? Aug 31 at 14:38
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en.wikipedia.org/wiki/Fitch_notation has some examples. The rule of "assumption" introduces an assumption and a new scope, and the assumption can be discharged according to several possible rules, such as implication introduction (Assume A, prove B, discharge the assumption and conclude A->B) or contradiction (Assume A, prove a contradiction, discharge the assumption and conclude not A). Another thing you might look at is intrologic.stanford.edu/chapters/chapter_05.html or mathstat.dal.ca/~selinger/courses/2012W-4680/handouts/… Aug 31 at 14:43