# What's the difference between “Not Completely True” and “Completely Not True”?

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
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

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