Logic question for grammarians...or grammar question for logicians

Factual Questions
#1

With apologies to any Germans out there:

All of my friends are not German.

This sentence may be casually interpreted in two ways:

 A) It is not the case that all of my friends are German.
 B) None of my friends are German

I think that, strictly speaking, the original sentence entails B, but not A. However, I lack sufficient command of syntax and grammar rules to defend that statement.

Am I right, and more importantly, why or why not? This is simplification of a test problem in a logic course, if anyone is wondering.

#2

I don’t think this sentence is grammatically correct either way. You would say,

“Not all of my friends are German,”

or

“All of my friends are non-German.”

Is any one manual or text considered officially authoritative on English grammar?

#3

You’re right, the sentence means that none of your friends is German, i.e. that for each member of the set of your friends, the proposition that he or she is German is false.

The alternative, as you rightly suggest, should be “Not all of my friends are German”, i.e. that there is at least one member of the set of your friends for whom the proposition that he or she is German is false.

Bear in mind that the two propositions are not mutually contradictory: if non of your friends is German, then clearly not all of them are.

Having said that, as sea snake points out, it’s not the most elegantly phrased sentence. My local Chinese takeaway has a sign on the menu that says “All dishes do not include chips or rice”. What it means is that no dishes include chips or rice, which must be ordered separately; but it takes a few seconds’ thought to figure that out.

#4

Actually, when I read your sentence, I thought it entailed A more than B. Let us suppose:

You: “Dieter, Ute, Karl, Heinz, Ulrike… seems you have a thing for the Teutons.”
Me: “Yeah, but ALL of my friends aren’t German.”

It seems relatively grammatically sound, but I would say it “Not all of my friends are German,” as sea snake would.

Whereas for the other interpretation, I would be more inclined to say “None of my friends are German.” I would not say your sentence if my meaning is B, but I may in the case of A.

If this is a question on a logic course, I suspect the answer is simply ambiguous. It can logicially be read either way. My reading in casual speech is the opposite of yours, but a case can be made for both. In speech, a lot of information also comes from intontation and context, and I would venture to say that the sentence is not grammatically incorrect, simply vague. It depends on how your mind parses the “not.” Do hear it as “All of my friends are [not German.]” (Which to me sounds funny.) Or do you parse it with the subject? As in “ALL are not,” but “some may be.”

#5

Well, to bring in a bit of logic:

“All of my friends are not German”

would be (where F and G are the set of friends and Germans respectively)

“For all f in {F}, f is in {¬G}”, ie “f in {F} implies f not in {G}”

i.e. if they are your friend, they are not German. Interpretation B.

Logically, A would be

“f in {F} does not imply f in {G}”. This syntax cannot be interpreted from your sentence above.

But hell - since when was English or its interpretation logical?

pan

#6

kabbes

I agree with that assessment, from the strictest logical point-of-view.

But, logically, let’s say this. I have 8 German friends, 2 Slovak friends and 3 Austrian friends. Isn’t it possible to say “ALL my friends aren’t German.” I’m purposely using a contraction here to show how the “not” can be associated with “all” and “are” rather than “German.” Logically that’s true too, no? Think of how you stress it in speech.

“How dare you spend all your money!”
“Well, ALL of my money isn’t spent.”

“That’s a nice photo of your friends.”
“All of my friends are not pictured here.”

“Not” is a very tricky word to interpret here, and I honestly believe either interpretation is valid. You can’t quite look at language so logically.

#7

Gosh, I had no trouble “picking” B. Just diagram the sentence out. Do they still teach diagramming in school?

All (of my friends) are not “German”.

In the case:
All is the subject.
(of my friends) is a prepositional phrase that “modifies” the subject.
are not is the negative of the verb (the predicate).
“German” is the phtlidhgn that answers the question “are not what?”

The simplest form of this sentence is the subject-predicate

All are not.

Of coures, this does not convey the information necessary, but with the prior knowledge of that information, one can see what the sentence is truly saying.

This is very different from having the negative reverse the subject in:

Not all are

Am I making sense here?

#8

Actually, I still think of “ALL my friends aren’t German” as meaning that you have no German friends.

Your other two sentences are interesting.

The first sentence logically means exactly what you want it to mean. “ALL of my money” is a single object, unlike “ALL of my friends”, who are numerous (I hope). Furthermore “spent” is a completed action whereas “German” is a class. Hence the item “ALL of your money” hasn’t been “spent” - it means just what you want it to mean.

In other words, “m in {M} doesn’t imply m in {S}”. Just what we want.

The second is trickier - it comes across as being clumsier. But I think it still follows the form of the money example, not the German example. Let’s see - the two possible interpretations by analogy are[list=1][li]Some of my friends aren’t in that photo.[/li]
[li]Nobody in that photo is my friend.[/list=1]Option 1 would be “f in {F} doesn’t imply f in {P}”[/li]Option 2 would be “f in {F} implies f not in {P}”.

“All of my friends are not pictured here” is rephrased as “for each f in {F}, what?” This is where “clumsy” comes in. I don’t see where to go with the symbolic representation of the sentence. The most likely suggestion is “f in {F} does not imply f in {P}”, which actually fits best with option 1, as we would wish.

pan

#9

Sorry - Spirtle jumped in there whilst I was composing my reply to pulykamell. Obviously, my post was directed to the latter.

pan

#10

Very strictly speaking, Spritle is correct. I picked B immediately as well. The logical analysis that kabbes did backs this up.

pulykamell does, however, have a point that language, at least as used in every day speech, is not completely logical. The inflection used can change the meaning of a sentence, and the contraction “aren’t” does make for a more ambiguous interpretation, particularly when spoken. Because the sentence can be misinterpreted, I would also suggest rewriting it (picking one from Diletante and one from sea snake):

“None of my friends are German.”

or

“Not all of my friends are German.”

#11

Hmmm… this is becoming pretty interesting. I turned to two of my friends at random, and asked them what does this sentence me. They both responded that all of your friends are not German, but some may be. One was a native English language speaker, one was not. OK, a representative sample of 2 says absolutely nothing statistically significant, but at least it affirms I’m not the only one who thinks this way.

I’m gonna try once more to offer examples linguistically, and I’ll even attempt a logical explanation.

A farmer orders a crate of oranges. He gets a crate of oranges with a couple of grapefruit mixed in.

Could he say “Hey, buddy, all of these are not oranges!”
I say, sure. Would someone hearing this sentence think that his whole crate is full of grapefruit? I doubt it. Now, let’s say he indeed did get a whole crate of grapefruit. Would he say “Hey, buddy, all of these are not oranges!” To me, no. That sounds wrong. One would either say “These are not oranges” or “none of these are oranges.” Although, logically there is nothing wrong with the original sentence.

OK, now for my logical explanation. Maybe I can be a bit more coherent and mathemetical in my reasoning.

All of these fruit are not apples.

let: A=orange, B=apple, C=apple

This is where I sense a difference between the word “each” and “all.” I think most of you are treating the word “all” as “each.” In this case “each” means that:
A <> apples
B <> apples AND
C <> apples.

All three premises must be true for the statement to be true. I interpret “all” as [A and B and C] <> apples.
The difference? If any of the three are not apples, the statement is true.

Boolean:
orange = 0, apple = 1

A= orange, B= apple, C= apple
“all these fruits are not apples”
[A and B and C] <> apple (1)
[0 and 1 and 1] <> 1
0 <> 1
true

a= orange, b=orange, c=orange
“all these fruits are not apples”
[0 and 0 and 0] <> 1
0 <> 1
true

a= apple, b=apple, c=apple
“all these fruits are not apples”
[1 and 1 and 1] <> 1
1 <> 1
false

Whereas, analyzine “each” as “all” you would get for the above three examples the following equations:

a<>apples AND b<>apples AND c<>apples
0<>1 AND 1<>1 AND 1<>1
1 AND 0 AND 0
false

0<>1 AND 0<>1 AND 0<>1
1 AND 1 AND 1
true
1<>1 AND 1<>1 AND 1<>1
0 AND 0 AND 0
false

Following the exact pattern of the other board member’s reasoning.

I hate doing it in this esoteric boolean way, but that seems to be the only way I can explain it for the moment. I hope you can follow me. I can explain further, but I don’t want to waste the electrons at the moment. It’s the difference between “each” and “all.”

Awaiting the contestion of my theory …

#12

don’t hurt me! big apologies for the long double post. My computer gave me a “10060 – Connection timed out,” and I thought it was telling me the truth. Filthy deceiving machines. runs off to find a big hole to crawl into

[Fixed it. Trust the CGI]

[Edited by Chronos on 04-20-2001 at 10:46 AM]

#13

Aaaaaaaaaargh! Too many pulykamell posts! Make it stop - make it stop!

Ahem.

Anyway - in maths, “all” and “each” mean the same thing. Hence the fact that many proofs contain that big upsidedown capital “A”, meaning “for all”. So there.

pan

#14

This is part of a big messy subject in linguistics called “quantifier scope.” It’s not very well understood. There are lots of differences in dialects. In answer to your question, the meaning of the sentence is whatever you most naturally interpret it as being. Clearly it’s interpreted in various ways by different speakers of English. There’s no such thing as a definitive resolution of this subject by grammarians or logicians or anyone else.

#15

I think you are limitting the meaning of the phrase “All of my freinds.” This phrase has two equally possible meanings. It could mean “The group of people I consider friends.” In this case you are refering to a set. So we could say, “Memebrs of set A are not German.”

Contrarily “All of my friends” could refer to each individual who is a friend. So while some of my freinds may be German, all of my frinds are not German. The sentance could be refrased as, “Each of my friends is not German.” But since the plural form was used “All” and “are” were used instead of “Each” and “is”

The word all in this sentance can have either meaning.

On the logic front the statement “all are not” does not equal “none are.” The statements “all are not” and “some are” are not mutually exclusive. Hence, “Some of my marbles are blue but all of my marbles are not blue.” Is a perfectly acceptable sentance. It comes down to the nature of the word “all” again.

#16

I agree with Wendell 100%.

I’m not saying there is a definitive resolution, and applying mathematical analysis to grammar is plain stupid. (e.g. double negatives. They do make sense in most languages, yet mathemetically analyzed they don’t work.)

As we have demonstrated (at least I think we have) both analyses seem to work in conversation (although I still contend you will hear it in the context of “A” in real conversation more often than “B.”)And I have tried to explain as best I can what goes on in my head when I parse that sentence. Perhaps “all” and “each” mean the same in maths, but they don’t mean the same thing to me when I use them in conversation. For me “all” means “the whole,” as the OED states in its first definition for “all.” “Each” refers to individual members of the whole. That’s it. That’s how my mind parses it. Others obviously parse it differently.

Does this sound wrong to the ears of most speakers: (I mean this as an actual question. Maybe it does. To me it doesn’t)

“All of my friends are not German; some are Hungarian.”
“All of these pizzas aren’t sausage; some are pepperoni.”

#17

Think of it this way. All means 100%. So in substituting 100% in the sentences you gave:

“100% of my friends are not German; some are Hungarian.”
“100% of these pizzas aren’t sausage; some are pepperoni.”

Just because 100% of something is not German does not in any way tell you whether 5% of that same thing is German. You are making a statement about all of something not about part of something.

#18

All of the people who posted to this thread are not wrong.

#19

Thank you all for your replies. Especially Sprittle, who came up with exactly what I was looking for. Maybe I shoulda stressed the grammar part of my question a little more. Still, pulykamell’s boolean interpretation was interesting as hell.

The question regards an attempt at translating a formula in monadic predicate logic into idiomatic English. I didn’t feel like posting the entire question, so I just distilled it into what I thought was the main point of contention.

English as used in everyday conversations cannot be symbolized unambiguously, but I was wondering if in most cases the rules of grammar provide an underlying precision that we generally ignore. In this case, at least, I think they do.

#20

you have friends ??? :slight_smile: