Talk:Problem of multiple generality

Problem with Ambiguous Terms

From a TL perspective, there's a problem with the proposition "Some cat is feared by every mouse" that has to be addressed before I can accept the objection with certainty. The proposition could be logically equivalent to

"all mice are afraid of at least one cat"

or it could be equivalent to

"There exists a cat C such that every single mouse, without exception, is afraid of C."

i.e. there are ambiguous terms, which means that the proposition can't be used in a valid argument by the standards of TL. Is there any way to fix this ambiguity, and does the problem of multiple generalities remain if the ambiguity is removed? —Preceding unsigned comment added by 71.191.119.238 (talk) 20:44, 28 May 2008 (UTC)

EDIT: Never mind, that's already been dealt with by another author. Silly me for not reading all the way to the end. —Preceding unsigned comment added by 71.191.119.238 (talk) 20:47, 28 May 2008 (UTC)

Not English

However, it is not possible to express this inference in the traditional system, because the manner we
should represent the first term in the classical subject-predicate form, ensures that the second term 
our predicate will be "X is feared by every mouse"

This isn't English - what is it supposed to mean? garik 11:42, 28 January 2007 (UTC)

I will edit the article to explain to the non-logician. However, this very interesting stub badly needs references. Will the author please provide them? Rick Norwood 14:26, 28 January 2007 (UTC)

I'm not so concerned about making it clearer to the non-logician - I think this would follow to some extent from making it clearer to English speakers. The syntax is so peculiar I can only assume it wasn't written by a native speaker, or that there are words missing. But yes, I suppose it could be made easier for non-logicians too. garik 14:59, 28 January 2007 (UTC)

And I see you've done a good job of it:) garik 15:00, 28 January 2007 (UTC)

Ambiguity is equivalent to Ambiguity

This is my first time writing anything on Wikipedia. Please forgive my naivete and mistakes.

ME: The author of the problem of multiple generality begins by saying that, quote:

AUTHOR: The problem of multiple generality names a failure in traditional logic to describe certain intuitively valid inferences.

ME: So one (ie. myself) could interpret the author's meaning to be, "SOME logic is NOT 'foolproof'". The author proceeds to provide an example to demonstrate his thesis:

AUTHOR: For example, it is intuitively clear that if:

Some cat is feared by every mouse then it follows logically that:

All mice are afraid of at least one cat.

ME: Let me try to represent his intuition in TL. The author provides two propositions which are intuitively clear to him as antecedent and consequent propositions: 1. Some cat is a cause-of-fear-in-ALL-mice. (Antecedent) 2. Every mouse is affected-by-fear-of-SOME-cat? (Consequent)

Then the author proceeds to his argument which he proposes will illustrate a failure in TL.

AUTHOR: The syntax of traditional logic (TL) permits exactly four sentence types: "All As are Bs", "No As are Bs", "Some As are Bs" and "Some As are not Bs". Each type is a quantified sentence containing exactly one quantifier. Since the sentences above each contain two quantifiers ('some' and 'every' in the first sentence and 'all' and 'at least one' in the second sentence), they cannot be adequately represented in TL. The best TL can do is incorporate the second quantifier from each sentence into the second term, thus rendering the artificial sounding terms 'feared-by-every-mouse' and 'afraid-of-at-least-one-cat'. This in effect "buries" these quantifiers, which are essential to the inference's validity, within the hyphenated terms. Hence the sentence "Some cat is feared by every mouse" is alloted the same logical form as the sentence "Some cat is hungry". And so the logical form in TL is:

Some As are Bs All Cs are Ds which is clearly invalid.

ME (summing up the author's argument): In short, when an arguer attempts to use the permitted syntax of TL to express the validity of the author's hypothetical inference, the result is a logical quadruped (4 terms). And logical quadrupeds don't lead to valid inferences.

In other words, logical quadrupeds are, as the author concludes, clearly invalid --- especially in TL. But his inference is apparently "intuitively clear" and valid in modern QL. So, as his conclusion suggests, there is a PROBLEM. The same argument cannot be INTUITIVELY CLEAR and VALID in one "logic" and both obscurantist (burying quantifiers) and INVALID in an arguably different "logic".

ONE CRITICISM of the author's argument, above: Some cat (one term) is feared-by (2nd term) every mouse (3rd term) is hardly the SAME LOGICAL FORM as "SOME CAT (one term)is HUNGRY (2nd term). The author continues

AUTHOR: The first logical calculus capable of dealing with such inferences was Gottlob Frege's Begriffsschrift, the ancestor of modern predicate logic, which dealt with quantifiers by means of variable bindings. Modestly, Frege did not argue that his logic was more expressive than extant logical calculi, but commentators on Frege's logic regard this as one of his key achievements.

Using modern predicate calculus, we quickly discover that the statement is AMBIGUOUS.

Some cat is feared by every mouse could mean (Some cat is feared) by every mouse, i.e.

For every mouse m, there exists a cat c, such that c is feared by m, xy(Mx(Cy&Fxy))

in which case the conclusion is trivial.

But it could also mean Some cat is (feared by every mouse), i.e.

There exists a cat c, such that for every mouse m, c is feared by m. yx(Mx(Cy&Fxy))

This example illustrates the importance of specifying the scope of quantifiers as for all and there exists.

2 QUESTIONS:

1. Wouldn't it be just as easy to conclude that "TL" makes it impossible to clearly express ambiguous statements, which, according to our author, are clearly shown to be ambiguous statements "using modern predicate calculus"?

2. Didn't Gottlob Frege "NOT ARGUE" that his logic was more expressive than extant logical calculi.", whereas the author of this article "ARGUES" that it is more expressive?

IN SUM: If TL makes it impossible to clearly express ambiguous "statements" in the logical syntax of TL and "modern predicate logic" clearly shows "statements" and "examples" to be ambiguous, THEN it seems that both "logics" are clearly expressing the SAME meaning:- Ambiguity is Ambiguity.

Majorsocratic 03:09, 28 August 2007 (UTC)

Your question 1 contains the phrase "to clearly express ambiguous statements". Since ambiguous = unclear, I'm not sure what that phrase (or its negation) could mean.

The answer to your second question is that logic has come a long way since Frege.

Rick Norwood 17:00, 28 August 2007 (UTC)

Criticisms of 2 Rhetorical Questions

NORWOOD:

Your question 1 contains the phrase "to clearly express ambiguous statements". Since ambiguous = unclear, I'm not sure what that phrase (or its negation) could mean. The answer to your second question is that logic has come a long way since Frege. Rick Norwood 17:00, 28 August 2007 (UTC)

THE (actually rhetorical) QUESTIONS

1. Wouldn't it be just as easy to conclude that "TL" makes it impossible to clearly express ambiguous statements, which, according to our author, are clearly shown to be ambiguous statements "using modern predicate calculus"?

2. Didn't Gottlob Frege "NOT ARGUE" that his logic was more expressive than extant logical calculi.", whereas the author of this article "ARGUES" that it is more expressive?

COMMENTS ON NORWOOD'S RESPONSES:

2. Major Logic may have "come along way" since Frege, but that may not be true of individual logicians, since it is just as easy to say that MUDDY = UNCLEAR, with respect to water, as AMBIGUOUS = UNCLEAR with respect to logic.

1. Aristotle was quite clear and unambiguous in mentioning that DEFINITIONS of terms could never be phrased as single terms. The UNCLEAR-ness of AMBIGUITY occurs as a result of the DEFINITION and meaning of AMBIGUITY as "two-(or more)-meanings", just as AMBIdextrous individuals have equal facility/dexterity in using both of their hands. So if an expression has two meanings, it is UNCLEAR to a listener of such ambiguous expressions as to which of two or more meanings the user of ambiguous expressions refers. The meanings of nonsense syllables, such as "Glub" or "Trish" are unclear, but hardly "ambiguous", since they are meaningless, rather than "double-meaninged". So UNCLEAR is hardly equivalent to or "=s" AMBIGUITY.

As to the phrase you wondered about, if we substitute UNCLEAR for AMBIGUITY, as you suggest, we get the phrase: TO CLEARLY EXPRESS UNCLEAR STATEMENTS.

Then if we put that rephrasing back into my original INTERROGATIVE/rhetorical sentence we would have:

Wouldn't it be just as easy to conclude that "TL" makes it impossible to CLEARLY express UNCLEAR (ambiguous) statements, which, according to our author, are clearly shown to be ambiguous statements "using modern predicate calculus"?

And since Mr. Norwood actually CLEARLY EXPRESSED one ambiguous statement with 2 CLEAR expressions (at least CLEAR to modern symbolic logicians), it would be silly for me to either think or say that it is IMPOSSIBLE to clearly express unclear/ambiguous statements, because that would be an "already-PROVED" false statement.

Majorsocratic 17:38, 7 September 2007 (UTC)

I think you are making too much of a simple misunderstanding. I misunderstood what you meant by "to clearly express an ambiguous statement". I agree with everything you say above. Rick Norwood (talk) 21:02, 28 May 2008 (UTC)

The Difference Between the Two Formulae Given in the Text

I believe the logical difference between the two statements given in the article...

$\forall m.\,(\,{\text{Mouse}}(m)\rightarrow \exists c.\,({\text{Cat}}(c)\land {\text{Fears}}(m,c))\,)$

$\exists c.\,(\,{\text{Cat}}(c)\land \forall m.\,({\text{Mouse}}(m)\rightarrow {\text{Fears}}(m,c))\,)$

is that the first does not explicitly state the existence of cats unless Mice exist, whereas the second states the existence of cats whether or not mice exist. The conjunction of the first sentence and...

$\exists c.\,(\,{\text{Cat}}(c)\,)$

is logically equivalent to the second sentence.

In the context of normal speech, sentences like "Some cat is feared by every mouse" and "All mice are afraid of at least one cat" are not taken to differ in this regard as one would generally not question the existence of mice and cats when interpreting them.

If someone can verify my understanding of the logical difference, perhaps this can be added to the article. —Preceding unsigned comment added by 99.231.50.3 (talk) 04:58, 10 December 2008 (UTC)

$\exists c.\,(\,{\text{Cat}}(c)\,)\land \forall m.\,(\,{\text{Mouse}}(m)\rightarrow \exists c.\,({\text{Cat}}(c)\land {\text{Fears}}(m,c))\,)$

is not equivalent to

$\exists c.\,(\,{\text{Cat}}(c)\land \forall m.\,({\text{Mouse}}(m)\rightarrow {\text{Fears}}(m,c))\,)$

The first says that at least one cat exists, and for each mouse that exists, there exists some cat that that mouse fears. The second says that at least one cat exists and, for each mouse that exists, that mouse fears that particular cat. To illustrate the difference, consider the following model. We have two cats, Alfred and Bob, and two mice, Carl and David. Carl fears only Alfred and David fears only Bob. In this model, the first sentence will be true, as there exists two cats (Alfred and Bob) and each mouse fears at least one cat (Carl fears Alfred and David fears Bob). However, the second sentence will be false, as there is no cat that every mouse fears (Carl doesn't fear Bob and David doesn't fear Alfred). The two sentences are not equivalent. 58.160.69.15 (talk) 10:22, 27 December 2010 (UTC)

Am I going crazy or is the Cat example missing something major?

Something major being where on earth is the consideration for the state where there are no cats AT ALL in existence anymore, yet they can still exist as a fear concept within the mouse, or is this something I am simply being too obtuse about and everyone would be thinking this as well? It seems to me that this point does not appear to be addressed? Which to me in turn underpins some vital misunderstanding or miscommunication of the idea of the 'problem with multiple generality' in the sense of getting to the root of whatever supposed flaws exist? BoredextraWorkvidid (talk) 09:35, 20 September 2010 (UTC)

There is a intensional/extensional distinction here. In this example, X fearing Y is understood as a relation existing between two entities, rather than a disposition in one entity (which is probably a more physically accurate understanding). That has nothing to do with the problem of multiple generality, as 'feared' here is just an example predicate. If we were to say "Some cat is next to every mouse" and "All mice are next to at least one cat", the problem would still exist, and we could see how X can be next to Y if and only if X (and Y) exist (other than in a metaphoric sense). 58.160.69.15 (talk) 07:47, 20 December 2010 (UTC)

Redundant 'No'

The article text says:-

The syntax of traditional logic (TL) permits exactly four sentence types: "All As are Bs", "No As are Bs", "Some As are Bs" and "Some As are not Bs". Each type is a quantified sentence containing exactly one quantifier.

Looks like there are three quantifiers reading that, i.e. All, No, and Some. But "No A's are B's" can be written "All A's are not B's". So there are two quantifiers not three. Using the 'No' word is unnecessary since you have the 'not'. Why confuse the reader with redundant terms? John Middlemas (talk) 13:14, 29 July 2014 (UTC)

The number of symbolic quantifiers can be reduced to two, as you say, and in symbolic logic that is usually done. Whether that reduction will help the lay reader or not is another question. Most people think in terms of "none", "some", and "all", and even though we can do without "none", I don't think including "none" is confusing. A similar question arises in propositional logic. We can do without the binary "AND" by replacing A AND B with NOT(NOT A OR NOT B), but is that less confusing that keeping a redundant connective? Rick Norwood (talk) 13:37, 29 July 2014 (UTC)

questioning this article

I don't get the point of this article. Thats my view: The example states in both sentences the same. One time in an passive form (feared by), one time in a active form (afraid of):

I am feared by you.

You fear me.

(You are afraid of me.)

or

Some As are (by) Bs.

Bs are some As. 123qweasd (talk) 17:14, 23 January 2019 (UTC)

The point of the article is that classical syllogisms can't deal with this, and this is one of the motivations for predicate logic. 203.13.3.89 (talk) 23:35, 7 October 2024 (UTC)

Not really the point

"Using modern predicate calculus, we quickly discover that the statement is ambiguous." True, but that is not the point of the article. It's really only the 'for each mouse, there exists a cat' that demonstrates the problem of double quantification. "Toby is feared by all mice" only has a single quantifier and is not what we are attempting to talk about. 203.13.3.89 (talk) 23:33, 7 October 2024 (UTC)