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From what I understand, in second order propositional logic, ∀¬x and ¬∀x are equivalent statements. Apparently these are not equal. ¬∀x ≡ ∃¬x

However, rendered into the English language, consider the following case study:

Statement: All human beings have appendixes.  

Response 1: That's not completely true.
Response 2: That's completely not true.

In Response 1, the implication seems to be that the responder knows about appendectomy, and that some people have had their appendix removed. In Response 2, the implication seems stronger than that, and while the motivation for such an expression is likely more for emphasis than pure logic, the implication seems be that No human beings have appendixes, which is obviously not true.

So with all due haste, the question: Which one of these is the correct word choice and why? Perhaps there are logicians in the audience that can enlighten us with specific reasoning.

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In English, people almost never say "completely not true". They do say "completely untrue". –  Peter Shor Dec 11 '12 at 19:53
    
I don't agree with the assertion that Response 2 implies "No humans have appendices." Instead, it strongly asserts that the initial statement is untrue (which doesn't necessarily imply that the opposite statement is true). English isn't Boolean. –  J.R. Dec 11 '12 at 19:54
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First, it's Propositional Logic, not prepositional. Second, this is not necessarily (or even preferably) second order quantified logic; second order logic is inconsistent, whereas first order is consistent. Third, ∀¬x and ¬∀x are not equivalent; see De Morgan's Laws (p.13 on the link). –  John Lawler Dec 11 '12 at 19:55
    
@J.R. Yes, it's likely more for emphasis. However, the question is to what degree are the two statements different, and why? –  kreativitea Dec 11 '12 at 20:04
    
@JohnLawler Do you know of a resource that lists a truth table for second order logic? In effect, you're referring to this statement, correct?: ¬(∀x) φ(x) ≡ (∃x) ¬φ(x) –  kreativitea Dec 11 '12 at 20:07
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2 Answers

up vote 4 down vote accepted

Reponse 1 means that what has been said is only partially true. Response 2 means that what has been said is untrue. In practice, the thought expressed in Response 2 is more likely to occur as something like ‘That’s not true at all.’

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Response 1 suggests that there's more to add to the original statement, while response 2 suggests there's nothing else to add to the statement!

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That may be true, but it's not helpful in answering the OP's question. –  Robusto Dec 11 '12 at 20:37
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