Are you a genius? Try solving this difficult Logic / Critical Reasoning problem

• 4
The following statement is NOT true: No people are not dinosaurs

Which of these logically do or do not follow?

A) Some dinosaurs are people
B) All people are dinosaurs
C) Some people are not dinosaurs
D) No dinosaurs are not people

Note: some does not exclude all

I'd really appreciate it if you could also briefly discuss your thought process as you solved it!

• 574
B isn't it?
Oh, A as well.

EDIT: I somehow missed the 'NOT'!
• 578
If the premise were True, only C would be invalid, therefore if the premise is False, A, B and D are invalid and C is valid.
• 143
No people are not dinosaurs. To simplify, cancel out the negatives and you get people are dinosaurs (but you said this is not true) so that means (some) people are not dinosaurs.

with this we can say the following

A) Some dinosaurs are people (follows)
B) All people are dinosaurs (doesn't follow)
C) Some people are not dinosaurs (follows)
D) No dinosaurs are not people (Some dinosaurs are people - follows)
• 4.2k

If I answer the question correctly, can I join Mensa?
• 5.9k
No people are not dinosaurs

I read this to mean that every person is a dinosaur.

A) Some dinosaurs are people

This is only true if there are any people because every person is a dinosaur, but you've not clarified that any people exist.

B) All people are dinosaurs

This is how I defined the sentence, so I'd say it's true.

C) Some people are not dinosaurs

This is false.

D) No dinosaurs are not people

This is false
• 108
The following statement is NOT true: No people are not dinosaurs

Translation to second-order logic:
P1: It is not the case that there exists an x, such that x is a person and x is not a dinosaur
Informally, no person you find will not also be a dinosaur
P1': Equivalently, All people are dinosaurs
Negate this.
P2: There exists an x, such that x is a person and x is not a dinosaur

C follows immediately.

A does not follow as P2 does not bind us to this. All we know is that the set of people, {P} is not a subset of the set of dinosaurs {D}. {P} and {D} could be disjoint. We don't even know if {D} has any members. The claim does not follow because we don't have support for the claim.

B is false based on the contrary case from P2. B does not follow.

D equivalent to All dinosaurs are people, does not follow as we cannot support its claim from P2. {D} maybe null, and this claim would be trivially true, but we don't have support for the claim.

A) Some dinosaurs are people (does not follow)
B) All people are dinosaurs (does not follow)
C) Some people are not dinosaurs (follows)
D) No dinosaurs are not people (does not follow)
• 4

Thank you Joseph and Fresco! :)

Special thanks to Joseph for explaining his answer!
• 578
You might find that a Venn diagram with two intersecting circles is simpler to understand than the propositional logic.
• 4
I think you're very right in saying that, Fresco :P Good idea!
• 269

You might want to work on your punctuation. I glanced at this sentence and it read like this:

"No! People are not dinosaurs."

:smile: :grin: :razz: :razz:
• 4
Lol!
• 1.9k

I'd phrase it differently:

A) Some dinosaurs are people (undetermined)
B) All people are dinosaurs (does not follow)
C) Some people are not dinosaurs (follows)
D) No dinosaurs are not people (undetermined)
• 108

When we use the phrase "does not follow", it means it cannot be justified logically from the antecedents.

While "undetermined" is fine colloquially, we need to be careful to use rigorous patterns in language to assure that we are precise in what we are communicating.

When we use 3 valued logic (e.g. SQL), undetermined might translate accurately to null. If we limit ourselves to 2 value logic, our syllogisms may only admit of follows and does not follow, in which case undetermined gives way to does not follow.
• 1.9k
While "undetermined" is fine colloquially, we need to be careful to use rigorous patterns in language to assure that we are precise in what we are communicating.

Yes, but if you look carefully at the logic, A and D are indeterminable given the premises, and B is outright false. That is to say, we can't say anything about A and D given the premises, but we can say something about B. In other words, it follows that ~B.

If you want to be "precise" about your language, you should endeavor to reflect that nuance.
• 108
I did mention it ("B is false based on the contrary case from P2").

You may have missed it in as much as there are many different ways of expressing the same thing in natural language. Nuance in natural language is one of the issues that logic was developed to deal with. Nuance can give rise to misinterpretation (inferring what was not implied).

Logic constrains our language, exchanging the power of expression in natural language for the logical validity of inference.
It was asked whether the propositions do or do not follow.

Nuance, in that respect, is not a good thing when communicating formally.

It allows misunderstanding to intrude.
• 1.9k

Actually, I did read that part. But your summary of your position was unclear, so I cleared it up for you.

A and D are neither provable nor disprovable by the premises. B is disprovable. But you're labeling all three the same way. That is how misunderstandings intrude.

I agree with everything else you're saying, but I think it's all a better critique of your own amphibolous use of language than my insistence on precision.
• 108
A and D are neither provable nor disprovable by the premises. B is disprovable. But you're labeling all three the same way. That is how misunderstandings intrude.NKBJ

You have an issue with the expectations of logical form, not with me.

Also, look up the term precision and then compare it with accuracy.
• 1.9k
You have an issue with the expectations of logical form, not with me.

While you're at it, look up the term precision and then compare it with accuracy.

I have neither an issue with logic nor with an anonymous stranger on the interwebs (you). I'm merely trying to argue logic, and if that seems to you like something more personal, than I have to assume you're not quite as good a logician as you're attempting to paint yourself here.

And, just FYI, logicians have a long history of battling amphibolous language. They famously named a fallacy after just that.
• 108
No, you aren't. My response to your pathological tendentiousness is rather simpler, it follows thus:
• 1.9k

How charming! I hit a nerve!

Some advice: take yourself less seriously and/or toughen up a bit. The (very mild) feedback I gave you on your logic shouldn't have sent you into such a tizzy, and if that's how you routinely react to (again, very mild) feedback, then you won't last long here.
• 5.6k
The following statement is NOT true: No people are not dinosaurs

Which of these logically do or do not follow?

A) Some dinosaurs are people
B) All people are dinosaurs
C) Some people are not dinosaurs
D) No dinosaurs are not people

Note: some does not exclude all

On the square of opposition the corners are labeled A, I, E, O, from AffIrmo and NegO

A All S are P..................................................E No S are P
f.....................................................................e
f.....................................................................g
I Some S are P..............................................O Some S are not P

The question is, what kind of a statement is, "No people are not dinosaurs"?
Maybe an A: (All) no-people are not-dinosaurs?
An E seems likely: No people are not-dinosaurs.
I - I don't see how.
O - I don't see how here, either.

The relationships between the corners are well-defined:
A implies I, E implies O (called subalterns).
A-E is contrary - not contradictory. Both cannot be true, Both can be false. (Clearly one can be true and the other false).
I-O is sub-contrary (again, not contradictory). Both can be true, both cannot be false.

Let's see if it makes any difference whether "No people are not dinosaurs" is an A or an E statement.

If "No people are not dinosaurs" is an A statement and it is false, that means the O statement must be true (by contradiction). But what does the corresponding O statement look like? It must be Some no-people are not not-dinosaurs. Clearly - well maybe not clearly - this is true.

Or as an E statement: "No people are not-dinosaurs." If this is false, the corresponding I statement must be true. That is, Some people are not-dinosaurs. And this is clearly true. That is option C.

So much for the square of opposition.
• 178

May I ask (as a non-logician and a non-native speaker of English) why this does not commit the existential fallacy? I agree that A, B and D do not follow. But I do not see how C follows, either.

As I understand it, the existential fallacy is where a proposition with existential import is inferred illegitimately from a proposition with no existential import, e.g.

'All unicorns are horned'
Therefore
'Some unicorns are horned'

Where 'Some unicorns are horned' is roughly equivalent to 'There exists at least one x, such that x is a unicorn and x is horned'. The problem is that 'All unicorns are horned' does not have any existential import. It simply states that, if there is an x such that x is a unicorn, then x is horned. But there is no commitment to the truth of the antecedent. So we do not have license to infer that there really are any unicorns.

Now, if I understand your explanation, you take P2 to be the negation of P1'. That is, you take 'All people are dinosaurs', when negated, to produce 'There exists an x, such that x is a person and x is not a dinosaur'. From this, you take it that C follows.

I see that C does indeed follow from P2, but I do not see P2 as being the negation of P1'. The reason being that P1' does not seem to me to have any existential import, where P2 assuredly does. The negation should surely be 'It is not the case that all people are dinosaurs'. But, since this does not have any existential import, C would not follow.

Thoughts?
• 5.6k
Not a problem in Aristotelian logic. You have to check the rules of the game before you play. Granted, your observation has full force in modern logic. And if you wrote this post, your "non-native" English is mighty good and better than a lot of native writers'.
• 578

Correct because 'existential import' is a human value judgement outwith the formalisms of classical logic. This point is one illustration of the limitations of logic with respect to 'semantics'.
• 178

Thanks for this. So in Aristotelian logic, 'All people are dinosaurs' does have existential import? Or is it just ambiguous?

So, I agree with the consensus that 'No people are not dinosaurs' should be understood as 'All people are dinosaurs'. Negate this, and we have 'It is not the case that all people are dinosaurs', or equivalently, 'It is not the case that, if there exists an x such that x is a person, then x is a dinosaur'. If we take this as having no existential import (and it seems to me that it doesn't), then we cannot infer anything that does have existential import. This rules out A and C. Obviously we cannot infer B, since our starting proposition is the precise negation of this. D seems to be equivalent to 'All dinosaurs are people', and I don't see how we can get this from out starting proposition, either.

So my answer is: neither A, B, C nor D follows. But since this differs from the answer that has already been determined correct, I am not so confident about it.
• 646
Only C) follows.

"No people are not dinosaurs" -> ~Ex[Px ^ ~Dx]
This is NOT true, so:
~~Ex[Px ^ ~Dx]
Ex[Px ^ ~Dx]

This is equivalent to saying that there is a person who isn't a dinosaur. But this is just what C) says, on the usual logical reading of 'some.'

Clearly A), B), and D) don't follow from this.
• 5.3k
As I understand it, the existential fallacy is where a proposition with existential import is inferred illegitimately from a proposition with no existential import, e.g.

'All unicorns are horned'
Therefore
'Some unicorns are horned'

Under normal circumstances you are right, but the puzzle indulges in multiple negatives. So though the truth of 'All unicorns are horned' has no existential import, its negation does. In Venn diagram terms, a universal (all or none) statement declares a region empty, but its negation declares that region populated. If not (all unicorns are horned), then there must be at least one hornless unicorn.
• 689
"No people are not dinosaurs" -> ~Ex[Px ^ ~Dx]

Agreed.

This is NOT true, so:
~~Ex[Px ^ ~Dx]

No, only this: [Edit: Yes! At least this:]

~{~Ex[Px ^ ~Dx]}

Which can't [Edit: yes it can] equate to

(~~Ex)[Px ^ ~Dx] = ~~Ex[Px ^ ~Dx] = Ex[Px ^ ~Dx]

as you hope, unless [Edit: if] it is ruled out that there are no people. Because after all, if there are no people,

~Ex[Px ^ ~Dx]

is true. So the negation of that would be false, but you want it to follow as true (option C).

[Edit: yeah, well I suppose wanting option C to follow is no good reason to say the negation is true, and that the previous line is therefore false, and that therefore there are no people. But of course the initial info is telling us the previous line is false, and that therefore there are no people. I think I kept imagining that there being no people could somehow survive the negation. Probably I was just thrown by the simple multiple negatives that unenlightened warned about above. :lol: Thanks and apologies to and ].

So,

If we take this as having no existential import (and it seems to me that it doesn't), then we cannot infer anything that does have existential import. This rules out A and C.

is quite right. [Edit: well, no.] Where you have to be a genius (as so often with puzzles) is not with the logic [Edit: although apparently that helps] but guessing which presuppositions are meant to be obvious. The testers here might be thinking it's obvious that there are people, maybe that we shouldn't take anything of the sort for granted. We don't know. Since they take the trouble to remind us that "Note: some does not exclude all", I'm guessing they have neglected to clarify (or even notice) their own assumption that there are people. So most people here are geniuses, but Virgo you fail for being a bit too clever.

In Venn diagram terms, a universal (all or none) statement declares a region empty, but its negation declares that region populated.

No, I don't think so. The region was excluded, so in the negation it is opened up again, but not necessarily populated.

[Edit: ahem]

So in Aristotelian logic, 'All people are dinosaurs' does have existential import? Or is it just ambiguous?

My question too, so I went here, https://en.wikipedia.org/wiki/Syllogism#Existential_import, and found interesting history and controversy to elaborate Tim's basic answer, i.e. yes assumed in Aristotle but not in modern systems. [Edit: beside the point after all.]
• 5.3k
No, I don't think so. The region was excluded, so in the negation it is opened up again, but not necessarily populated.

You think wrong, regions are not 'excluded' but declared empty or declared populated, the one being the negation of the other. So the negation of 'All x are y' (the region {x & not-y} is empty) is 'Some x are not y' (the region {x & not-y} is populated).

And this must be so, because if there are no unicorns, then there are no unicorns with a horn AND no unicorns without a horn. Thus they all have a horn AND they all have no horn. ({x & not-y} is empty AND {x & y} is empty.)
• 689
No, I don't think so. The region was excluded, so in the negation it is opened up again, but not necessarily populated.
— bongo fury

You think wrong,

Hey, perhaps I should have said "No I don't think so, assuming that we're using a Venn diagram to illustrate FOPL, like Snakes' - not Aristotle".

if there are no unicorns, then there are no unicorns with a horn AND no unicorns without a horn. Thus they all have a horn AND they all have no horn.

... all what have a horn and no horn??

By the way, you aren't suggesting that option C does follow on the assumption there are no people?

Or, that the assumption is incompatible with the given information? [Edit: this one :wink: ]
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