Help With A Tricky Logic Problem (multiple choice)
Hello! Can anyone help me with this? I think the answer is either D or E but don't know how to verify one way or the other. My brain be too smooth lol.
Some Ayes are Bees
All Seas are Bees
Which conclusion can be drawn?
A) Some Ayes are Seas
B) Some Seas are Ayes
C) No Seas are Ayes
D) No Ayes that are not Bees are Seas
E) None of the above conclusions can be drawn
Some Ayes are Bees
All Seas are Bees
Which conclusion can be drawn?
A) Some Ayes are Seas
B) Some Seas are Ayes
C) No Seas are Ayes
D) No Ayes that are not Bees are Seas
E) None of the above conclusions can be drawn
Comments (90)
Drawing a Venn diagram is always the right thing to do.
There are no Sees that are not Bees. Because all Sees are Bees.
Therefore nothing that is not a Bee is a See. In other words, if it's not a Bee, it is impossible that it is a See.
There are Ayes that are not Bees. That is, at least one Aye is a Bee. But there could or could not be more Ayes that are not Bees.
So if it's an Aye, and it exists inside Bees, it has a chance of being a See, but not necessarily.
But since there are no Sees outside of Bee, therefore Ayes outside of Bee are certainly not Sees.
Hence, no Ayes that are outside of all Bees are a See... precisely what D says.
---------------------
For the above, I used the concepts of "Aye" "Bee" and "See" as individuals contained necessarily or potentially or impossibly inside of all that are Ayes, Bees and Sees. Furthermore, Ayes and Bees and Sees I used as all of them, or I used them as more than zero.
Some mugs(Ayes) are copper(Bees).
All pennies(Seas) are copper(Bees).
Quoting Tobias
Nope. Can't drink mead from a penny.
Not quite. It's just that the logical statement
"All seas are bees",
on it's own doesn't allow the conclusion that the two things are equivalent. It doesn't explain their relation any further than "All seas are bees" - we don't know whether:
"All seas are bees and all bees are seas." or "All seas are bees but not all bees are seas."
i) ?x:U , A(x) ? B(x)
ii) ?x:U, S(x) ? B(x), which is equivalent to ?x:U, ¬B(x) ? ¬S(x) which immediately gives D.
Then state the theorems
A) ?x:U , A(x) ? S(x) (which can instantly be seen to not be derivable from i and ii)
B) ?x:U , A(x) ? S(x) (same as A)
C) ?x:U , S(x) ? ¬A(x) (instantly recognizable as not derivable)
The only conclusion that seems reasonable to state is some kind of categorical relationship between Ayes and Seas (Bees being the middle term).
There are 4 possibilities:
1. Some Ayes are Seas (true, but a premise)
2. Some Ayes are not Seas (undecidable)
3. All Ayes are Seas (undecidable)
4. No Ayea are Seas (false)
That's all from me. :smile:
Some can! Of all the people in the world, these are the ones that arouse my envy. My usual reaction/response is to use my finger and draw on the ground. I think this is some kind of genetic memory. Our genes "remember" a time when we used sand to draw pictures on, like Archimedes! Now there was a guy who could do things in his head! :smile:
N?l? turb?re circul?s me?s!
In modern times, universals are always interpreted as conditionals. Just translate in your head like this:
“All F’s are G” means “If anything is F, then it’s G”
“No F’s are G” means “If anything is F, then it’s not G”
The upshot of the translation is that the bits on the right are still true, even when there’s nothing that’s F.
You may also have to sit a while with this understanding of conditionals (known as “material implication”) until it feels natural.
That is not needed to see that (D) is the correct answer.
It is quite simple, without even need for Venn diagrams or symbolic logic.
(D) can be couched in two equivalent ways:
"If a thing is an Aye and not a Bee, then it is not a Sea".
"There is no thing that is an Aye and not a Bee and a Sea".
And either way you couch (D) it is entailed by the premise:
All Seas are Bees, or couched equivalently:
"If a thing is a Sea then it is a Bee."
The question is not the validity of (D) but the validity of inferring (D) from the premises.
(D) is validly inferred from the premises. Adding an additional premise cannot vitiate an otherwise valid argument. This is the monotonic property of basic logic.
All Seas are Bees
.Quoting Agent Smith
1. Is not a premise and it is not entailed by the premises.
4. Is not made false by the premises.
What about bees swimming in all seas? Aye!
If this forum taught me something, it is to pick your bottles well. Not to win, but in order not to get caught in the quagmire of non-understanding.
So I shalt remain silent.
And picking the right battles is good too, but that's not as important.
Whether the conclusion is or not inferred according to syllogistic forms, it is inferred validly.
Quoting tim wood
All Seas are Bees
entails
No Ayes that are not Bees are Seas
whether or not there is an Aye that is not a Bee. Ayes don't even have anything to do with it. And there is no need for an existence assumption.
You've got something stuck in your head that is not the case.
Run it through your mind, or write it in symbols [where 'U' and 'E' are the universal and existential quantifiers]:
Ux(Sx -> Bx)
entails
Ux((Ax & ~Bx) -> ~Sx)
or couched equivalently:
Ux(Sx -> Bx)
entails
~Ex(Ax & ~Bx & Sx)
It's clear as day:
If there were a thing that is both not a Bee and is a Sea, then that would contradict that all Seas are Bees. And that is the case whether there is or is not anything that is an Aye or a Bee or Sea or any combination.
Conclusion: the answer is "D" and Tim Wood is a vampire no younger than 122 years old.
If x is a C, then x is a B.
Therefore, if x is not a B, then x is not a C.
So, perforce, if x is both an A and not a B, then x is not a C.
That argument involves no assumptions about whether or not there is a something that is an A, B, or C.
This argument does not prove the existence of anything.
Quoting tim wood
The problem was not what conclusion is true, but rather what conclusion can be drawn from the premises. That is inference.
Quoting tim wood
(D) follows from the premises (actually, only one of the premises is needed) no matter what does or does not exists.
I spelled out the logic of the argument explicitly. There is no existence assumption in the argument. Moreover, A's are irrelevant to the argument.
False. I expanded nothing. I made no assumption other than the premise "All Seas are Beas".
"All Seas are Beas" implies "All non-Beas are non-Seas" implies, perforce, "All things that are both non-Ayes and non-Beas are non-Seas".
And that argument holds whether or not there are Ayes or Beas or Seas.
There is no existence premise nor existence conclusion involved.
/
It is alarming that someone would fail to understand the correctness of the inference, irrespective of training in logic, but rather just as a matter of commonly acquired reasoning.
Because Aristotelian syllogisms do not exhaust even very basic reasoning.
It's not wrong. It's just that it is nowhere close to covering much of everyday deductive reasoning.
The house rules are everyday common reasoning.
What if all Seas are real numbers and Beas are complex ñumbers? Are all non-complex numbers non real numbers then?
I am afraid the image is not from an "authoritative" source.
I used Microsoft "Paint".
For a few dollars a year one can subscribe to the Forum and be able to upload images onto the Forum - well worth the money.
The picture would have been even more logical if you gave a, b, and c circular forms. Start with c in b (small circle c in circle b). The a, as the biggest circle, around them. Or outside them. Then let a move out (or in).
@TonesInDeepFreeze gets to the heart of the matter. Personally, trying to solve in words would make my head explode, so I normally have to resort to diagrams.
Meaning of "which conclusion can be drawn"
Because the question is neither "which conclusions can be drawn" nor "can any conclusion be drawn" - the question is saying that there is only one correct conclusion.
D) No Ayes that are not Bees are Seas
The statement is using abbreviated language, making life more difficult.
In full - is the proposition "there are no A's that are not B's are C's" true or false.
Potential ambiguity in "there are no A's that are not B's are C's
Potential meaning one - There are (A's) that are (not B's are C's) - is ungrammatical, therefore ignore.
Potential meaning two - There are (A's that are not B's) that are (C's) - must be what is meant.
It reminds me of the "Four colour map theorem"
Wow! That's exactly what I mean!
Quoting DavidJohnson
What's "Ayes"? I guess it means simply "anyththing" ...
Well, I prefer to "translate" your puzzle-problem into something more meaningful:
Some animals are mammals
All cats are mammals
A) Some animals are cats: True, since mammals are animals (based on the first premise) and cats are mammals
B) Some cats are mammals: False, since we know that "All cats are mammals" (and not only some)
C) No cats are animals: "No cats" is ambiguous - In case it means "none of the cats", then it is false, based on (A)
D) No animals that are not mammals are cats: "No animals" is ambiguous - In case it means "none of the animals" then it is true, since cats are mammals
E) None of the above conclusions can be drawn: False, since there are two true statements, (A) and (D).
(If we reject (D) as ambiguous, then it remains (A) as the only true statement.)
It is true that some animals are cats. But it is not entailed by your premises.
Quoting Alkis Piskas
Wrong. In basic logic such as this, 'Some' means one or more. 'Some' does not mean 'Some but not All'.
Quoting Alkis Piskas
Wrong. 'No' means 'none of' and is unambiguous.
Quoting Alkis Piskas
(A) is true, but it is not entailed by your premises.
If by 'above' you mean the sentence you wrote before that one, then, yes, undistributed middle.
But if by 'above' you mean the discussion about Ayes, Bees, and Seas and your thought that (D) is not entailed by the premises, then you still have a severe misconception. An argument may be valid even if its validity is not within the Aristotelian syllogistic forms.
I said no such thing.
The question was "which conclusion can be drawn?" The question was not "which conclusion can be drawn by the method of Aristotelian syllogisms?".
Quoting tim wood
Undistributed middle is a fallacy. I never said otherwise. But that in no way vitiates that from
"All Seas are Bees"
we may validly conclude
"No Ayes that are not Bees are Seas".
Period.
I don't understand why you don't understand that, except that it seems you have stuck in your head that valid inferences regarding "Some, All, and No" must be within the scope of the method of Aristotelian syllogisms.
Quoting tim wood
What? It's basic everyday logic. And if that doesn't satisfy you, then one could formalize it in basic symbolic logic.
Indeed, the two salient principles used are Modus Tollens and Monotonicity of Entailment. It's pretty much that simple.
Quoting tim wood
I don't know what you think you mean by "categorically defined".
Meanwhile, it is a plain fact that "All Seas are Bees" entails "No Ayes that are not Bees are Seas".
And there are no "qualifications" needed. It is as clear as day in everyday reasoning, and it is as clear as a day on the sun with symbolic logic: Put another way:
Any circumstance in which All Seas are Bees is a circumstance in which No Ayes that are not Bees are Seas.
That is logical entailment.
From the premise "All Seas are Bees" we most certainly can validly draw the conclusion "No Ayes that are not Bees are Seas."
Period.
I already walked you through an English language demonstration of that. Or, I could do it formally in symbolic logic if anyone was captious enough to demand it.
Quoting tim wood
Of course it is a fallacy. But that is not relevant because I am not using that fallacy. I am not arguing in an Aristotelian syllogism. I am not in any way committing undistributed middle, because I'm not even inferring syllogistically.
You really still don't understand this?
Yes, so what?
Quoting tim wood
False. I already addressed that. Please read again:
Quoting TonesInDeepFreeze
Please do not elide that again.
No, you don't. A dollar is a dollar and cents are cents. Also, you cannot use some vending, gambling etc. machines if you don't have the exact amount of cents.
Quoting tim wood
You are right that you have to infer it, i.e. we don't know that directly, but it is true because its inference is valid, i.e. we can validly conclude it. (Using math sets: Mammals are a subset of animals. Cats are a subset of mammals. That is, cats are a subset of a subset of animals.)
1. Some Ayes are Bees
2. All Seas are Bees
Ergo,
3. Some Seas may be Bees.
If it is true, well, it is True! That's what I said! :smile:
Quoting TonesInDeepFreeze
I see what you say: We select one or more cats and say "these (animals) are mammals". This is true. But it refers specifically to "those" cats. Now think also about this: Saying that "some cats are mammals" suggests that there are some cats that are not mammals. Which is of course False, since we know that "All cats are mammals".
Maybe this is more clear: Can we say "Some persons are humans"? It makes no sense, does it? And it is also false, based on the above reasoning.
Quoting TonesInDeepFreeze
Yes, you have already said that! :smile:
:up: Kudos! I wish I had the patience to draw all that! (I only drew it in my mind!)
What I found simpler instead was to "draw" it verbally, in my answer to @tim wood, using math sets as you did:
Mammals are a subset of animals. Cats are a subset of mammals. That is, cats are a subset of a subset of animals.
Which looks exactly what you have drawn on the right...
Now your left drawing shows that only a part of mammals are also animals. This may be correct, if we ignore the fact that we know (ourselves, not from the premises) that mammals refers to animals. That is, in a "possible world", as you say. So, to get rid of this "pitfall", we should change our premises, for example, to:
Some animals have four legs
All cats have four legs
In this case, "four legs" would not refer exclusively to animals, since tables, beds, etc. too have four legs ...
So let's see our statemets:
A) Some animals are cats: Unknown
B) Some cats have four legs: False, since we know that "All cats have four legs" (and not only some)
C) No cats are animals: "No cats" is ambiguous - At best, it's Unknown, based on (A)
D) No animals that have not four legs are cats: "No animals" is ambiguous - Assuming that it means "none of the animals" then it is True, since cats have four legs
E) None of the above conclusions can be drawn: False, if we can accept (D) as True, else True.
It all depends on (D). And this also explains the doubt of the OP, who was not sure about (D) or (E).
I agree that the left hand drawing is not correct for "our" world, where i) all mammals are animals (all B's are A's) ii) all cats are mammals (all C's are B's)
But the OP is not asking a question about "our" world. The OP is asking a question about a "possible" world, perhaps a fictional world, where i) some animals are mammals (some A's are B's) ii) all cats are mammals (all C's are B's), in which case the left hand drawing is correct.
Certainly. But I have edited my reply and gave you right after I checked the label "possible world" ... Also, I presented a more interesting and realistic scheme ...
See the update at https://thephilosophyforum.com/discussion/comment/646385
In what the OP said, I have replaced "Ayes" with "animals", "Bees" with "four legs" and "Seas" with "cats". All the rest is the same. The conclusion (D or E) is what the OP also thought (maybe for another reason though).
That is ridiculous captiousness. The example is not vitiated by quibbles about the difference between coins and bills. The point of the example is that you can have Some and also All.
Again, I pointed out to you that in the context of basic logic, 'some' means 'at least one' and doesn't mean 'some but not all'.
Quoting Alkis Piskas
To be clear, a valid inference does not ensure the truth of the conclusion. A valid inference does ensure the truth of the conclusion when either (1) the premises are true or (2) the conclusion is logically true anyway.
Quoting Alkis Piskas
No, your inference is not valid. It is true that some animals are cats, but it does not follow from your premises.
Quoting Alkis Piskas
Yes, that is valid. But that is different from your original argument.
Quoting Alkis Piskas
No, what you said is that it follows from your premises.
Quoting Alkis Piskas
In certain everyday contexts, yes, 'some' may mean or at least suggest 'not all'. But not in the study of basic logic. I'll say it again: In ordinary basic logic [also, in certain other everyday contexts]
'some' means 'at least one' and it doesn't mean 'some but not all'.
Quoting Alkis Piskas
Yet you made the same mistake in a subsequent post!
Quoting Alkis Piskas
Wrong. For the fourth time, I'm telling you that in basic logic 'some' means 'at least one' and not 'some but not all'.
Quoting Alkis Piskas
I pointed out before that that is wrong. You merely persist in claiming again what has already been explained to you to be incorrect.
"Some cats are mammals"
means
"There is a thing that is both a cat and a mammal"
"All cats are mammals"
means
"If a thing is a cat then it is a mammal"
"No cats are mammals"
means
"If a thing is a cat then it is not a mammal"
means
"All cats are not mammals"
"Some cats are not mammals"
means
"There is a thing that is a cat and is not a mammal"
There is no ambiguity.
/
Moreover:
"Some cats are mammals" does NOT imply "Some cats are not mammals".
"All cats are mammals" does NOT imply 'Some cats are mammals" (because, if there are no cats, then "All cats are mammals" is vacuously true but "Some cats are mammals" is false). (Though I don't recall what, if anything, Aristotle said about that; and, while the notion of vacuous truth is basic in usual formal logic, it is not ordinarily used in everyday logic.)
"It is not the case that some cats are mammals"
is equivalent to
'No cats are mammals"
"It is not the case that all cats are mammals"
is equivalent to
"Some cats are not mammals"
"It is not the case that no cats are animals"
is equivalent to
"Some cats are mammals"
"It is not the case that some cats are not mammals"
is equivalent to
"All cats are mammals"
Yes,that matter hinges on existential import. But the problem in the first post of this thread does not. Nor does your other concern about undistributed middle.
Quoting tim wood
All of that quote seems correct to me and it in no way vitiates anything I've said, and it in no way supports your notion that the question of this thread hinges on existential import or undistributed middle.
Quoting tim wood
Of course, discussions about drawing inferences need to be in context of what principles of logic we have in mind. But the question in this thread has been answered according to everyday principles of reasoning, which also are formalized. And those particular principles do not hinge in any way on matters of existence or vacuity. I have explained exactly why that is in this case. I don't know why you continue to ignore it.
You mentioned cutting a knot with a knife. I rebutted that analogy already. But with your fixation that existential import plays a role in the particular question of this thread, you remind me of the saying that if a person has only a hammer then everything looks like a nail.
Because you never said you gave it up; and your next post seemed to still be trying to connect existential import to what we had been discussing. Granted, it is also reasonable that you were not trying to make the earlier connection, in which case I would grant my previous post would have been beating a dead horse.
Yes, "Not all Beas are Seas" is not a premise. But "All Beas are Seas" is also not a premise. So you don't get to use either in the inference.
An inference is not valid when there is an example in which the premises are true and the conclusion is false. Here's an example.
Let the set of Ayes be {Jack}
Let the set of Beas be {Jack, Lucy}
Let the set of Seas be {Lucy}
So:
"Some Ayes are Bees" True
"All Seas are Bees" True
"Some Ayes are Seas" False
So the inference is invalid.
I have redrawn my Venn Diagram, including @Raymond and @tim wood's suggestions, and using animals, etc rather than Ayes, etc. The solution is still D.
OK, D is the winner! Case closed! :sweat:
:up:
:pray: + :up:
1. Some Ayes are Bees.
2. All Seas are Bees.
No conclusion follows.
1. Some colors are white.
2. All snow are white.
Ergo,
???
Yes, it is a simple matter that (D) is the correct answer.
Quoting Agent Smith
No, many conclusions follow. And one of them is:
No Ayes that are not Bees are Seas.
Moreover, it follows from "All Seas are Beas" alone.
And one shouldn't have to rack one's brain to see that, except you still haven't racked your brain enough.
Of course. :roll:
I don't know what to make of people who still can't see it after it's been explained six ways to Sunday.
In my defense though, I wasn't looking at immediate inferences like you are; rather I was trying to see if the two statements could be used to form a classic syllogism; they can't!
Yes, we went through that with tim wood. Anyway, glad that you see now that (D) is the answer.
Good day!
Try it this way:
Some Americans are Brainy.
All Statisticians are Brainy.
We want to prove:
No American that is not Brainy is a Statistician.
But "No American that is not Brainy is a Statistician" means the same as "If something is an American and not Brainy, then it is not a Statistician."
Now, since, all Statisticians are Brainy, it follows that if something is not Brainy then it is not a Statistician. So, perforce, if something is an American and not Brainy, then it is not a Statistician. QED.
But what about the premise "Some Americans are Brainy"? Well, we never used it. We didn't need to. Which is fine. If a statement (such as "All Statisticians are Brainy") proves a conclusion, then that statement plus any other extra unneeded statement (such as "Some Americans are Brainy") still proves the conclusion (this is the principle of Monotonicity of Entailment).
Heh, Heh. :wink:
:smile: It's simply impossible!
It's impossible for you to understand?
What is the first sentence in my previous post that you don't understand?