# How does the ambiguity of 'All S are not P' not disturb the 4 Quantifier Negation Rules?

Source: A Concise Introduction to Logic (12 Ed, 2014) by Patrick Hurley.
Caution: In earlier editions, the Quantifier Negation Rule was called the Change of Quantifier Rule.

[p 202:] Two additional points should be noted about standard-form categorical propositions. The first is that the form [5.] “All S are not P” is not a standard form. This form is ambiguous and can be rendered as
either [6.] “No S are P”
or [7.] “Some S are not P,” depending on the content.

From p 477 (for brevity, I changed 'beautiful' to 'good'):

1.        (x) ℱx   ::   ∼(∃x) ∼ℱx
1.1 Everything is good.  =  1.2. It is not the case that something is not good.

2.                 ∼(x) ℱx   ::   (∃x) ∼ℱx
2.1. It is not the case that everything is good.  =  2.2. Something is not good.

3.        (∃x) ℱx  ::  ∼(x) ∼ℱx
3.1. Something is good.  =  3.2. It is not the case that everything is not good.

4.                 ∼(∃x) ℱx  ::  (x) ∼ℱx
4.1. It is not the case that something is good.  =  4.2. Everything is not good.

`My Problem:` Why does the ambiguity in 5 not pass or transfer into 1-4?
Why can 5 be ambiguous, but 1-4 not ambiguous?

On page 202, the author is highlighting that statement 5 is not a standard form because of the ambiguity arising in the act of rendering the statement. The quantifier negation rules only apply to standard form statements, so they do not apply to statement 5.

If we could unambiguously render statement 5 as "No S are P", then since this rendering is a statement in standard form we could formalise it as an instance of statement 4.2.

If we could unambiguously render statement 5 as "Some S are not P", then since this rendering is a statement in standard form we could formalise it as an instance of statement 2.2.

Of course, we cannot unambiguously render statement 5 as one or the other.

The ambiguity is in the rendering not in the formalisation, so as a non-standard form we cannot apply the quantifier negation rules to statement 5. That is to say, the ambiguity of 5 does not pass to 1-4 since 1-4 only apply to standard forms.

EDIT

I should have mentioned that the ambiguity in the rendering arises since it is not clear how the "scope" of the quantifier "all" is to be applied. If "all" applies just to "S" then the rendering is "No S are P", while if "all" applies to "S are not P" then the rendering is "Some S are not P".

• +1. Thanks again. Please reverse my modification if preferred; I thought to specify with the numbering of the statement in English because my question concerns the ordinary English statements, and not the Logical Statements.
– user8572
Commented Jan 4, 2016 at 19:35
• @LePressentiment I trust your edits. They're all good. I like the way you are thinking about the material presented in the text. You are clearly working hard to obtain a strong intuitive grasp of the material rather than just formally manipulating the symbols according to the rules.
– nwr
Commented Jan 4, 2016 at 19:59
• +1. Thank you again for your support; I know that I have asked many questions here and so fear that I may be seen as 'spamming' the site. So having only started in Logic, I just wish to alert users when I change something significant.
– user8572
Commented Jan 4, 2016 at 20:01

This has nothing to do with logic, and everything to do with the hybrid status of Latinate Germanic languages. Your author is trying to remove a real problem of English semantics which is contextual, with a context-free rule. Such things seldom work, and the same flaw can be pressed into other quantifications, but it is much less common.

The same single problem he is trying to avoid here infects all modal verbs, including 'is', in this kind of construction. It is always ambiguous whether a modal verb's referent, or the modal verb's sense, is complemented by a following 'not'.

(Wittgenstein in "The Blue Book" raises this as a way of indicating how deeply contextual even the most basic grammatical aspects of a language can be. Labeling the two options 'non' and 'ne', he points out some complex problems of the meaning of negation we solve constantly without realizing there ever was an ambiguity.)

Two clearer examples:

"He may not leave." can mean that he might stay or that he must stay.

"He could not finish the course." can mean that he is still in the course and might drop out, or that has dropped out already because he cannot finish it.

The meaning is totally dependent on its context.

"All dogs are not black" can mean that a randomly selected dog might not be black or that a randomly selected dog will not be black.

The problem is with the whole class of verbs, and not with the quantifier.

But the rigidity of the other quantifiers makes the ambiguity irrelevant.

"Some dogs are not black" could theoretically mean there are dogs that might not be black or there are dogs that are non-black. But logically these have the same result: We cannot count on all dogs being black. It does not matter whether we know there are actual exceptions, or we only do not know whether-or-not there are exceptions.

"No dogs are not black" (besides just sounding stupid) could theoretically mean that there are no dogs that are not necessarily black or that there are no dogs that are non-black. Again, the more definite pronouncement swallows the less definite one logically, because the impossibility of a non-black dog rules out the impossibility of a possibly non-black dog, and vice-versus.

"All dogs are not black" does not resolve this for us. The idea of dogs in general being actually non-black, is not related in the same way to that of dogs in general being possibly non-black. The one gets us only non-black dogs, the other just some non-black dogs. These two ideas are much more independent.

In this way, all is the 'weakest' or 'most open' quantifier, which is what leads to its use as the default quantifier in open constructions. So of the common quantifiers, only 'all' really shows the weakness of the construction inside it very often.

Since it does so, however, you should take your author's advice and work around using it in any canonical form.

Beyond that, it pays off, in general, to incorporate negation into an adjectival form (non-A) or resolve it with deMorgan's laws or exclusive constructions (relying on unless, except, only, etc. for clarification) and then pull the negations back out of their embeddings at the time a representation is formulated, after any processing of the language has taken place.

• Thank you. The paragraphs under the grey horizontal line: do they describe Boolean vs Aristotelian Interpretation of Categorical Propositions (which my textbook does include)?
– user8572
Commented Jan 4, 2016 at 19:44
• I don't see how those would be relevant.
– user9166
Commented Jan 4, 2016 at 19:48

I personally feel that 5 refers to 6 ("No S are P"), but that people often misspeak and say 5 when they really mean

1. "Not all S are P",

and their saying of 5 (instead of 8) leads you to 7. I don't think 5's actually ambiguous, except for those people who misuse language. Still, it's better to avoid phrasing things like in [5], just because of this practical ambiguity, and because I think it's reminiscent of the passive voice -- it kind of just sounds awkward.

• You are right to prefer the clearer form, but the other interpretation is not wrong, because it fits into a broader pattern. Consider something like "All the executives should not leave." It is really ambiguous whether this means "The case should be avoided that the executives might all leave." or "No executive should leave." Just as "He may not go" can mean he might not go [he (may choose not) to go], or he must not go [he (may not choose) to go].
– user9166
Commented Jan 4, 2016 at 19:08
• Thanks. Please reverse my modification if preferred; I forgot to enumerate 6 and 7 too which I should have done when I first wrote the OP, and so updated your answer with them.
– user8572
Commented Jan 4, 2016 at 19:40
• @jobermark I don't think I see the ambiguity in "All the executives may not leave." It seems clear to me as "For all x such that x is an executive, x should not leave." I suppose "may not" as you explained it in your second hypothetical is ambiguous, but it might just be idiomatic. "May not" can mean "might not" or "does not have permission to." That's a genuine ambiguity. Commented Jan 5, 2016 at 5:03
• (The example does not use 'may', for exactly that reason. Its meaning has slipped through polite overuse.) Try this context "All the executives should not be allowed to leave. There would be no one to make the final call. So Jenkins will stay." Not optimal, but allowed grammar. And in that context, the meaning clearly is not the one you chose.
– user9166
Commented Jan 5, 2016 at 5:27
• I would say that, in that context, the grammar was incorrect. Commented Jan 5, 2016 at 16:13