# In predicate logic, does existential quantification (∃) include universal quantification (∀), i.e. can 'some' imply 'all'?

I am having a discussion whether 'some' can also imply 'all'. The definition for some, 'an unspecified number or amount of people or things' seems to leave room for this interpretation.

Discussion follows on the following statements:

1. All newspaper readers are reasonable people.

2. Some newspaper readers are criminal.

The question is whether we can validly derive this conclusion:

Not all reasonable people are criminal

• "seems to leave room for this interpretation." How so? "Is valid..." I take it you are asking whether the claims follows, rather than whether it is valid? – Acccumulation Apr 16 '19 at 21:51
• In natural language, "some" usually means "at least one, but not all". In formal language, ∃ means "at least one". if you drop the notion that ∃ means "some" and always only read ∃ as "there exists at least one", things clear up quickly. "Some" with the meaning of "at least one, but not all" is modeled not by ∃, but by ∃x ∧ ¬∀x. – Polygnome Apr 17 '19 at 11:42

"Some" does not exclude "all", but you cannot deduce "all" from "some".

Having said that, the above argument is not valid.

From premises 1 and 2 we can derive :

Some reasonable people are criminal

that is equivalent to : Not all reasonable people are not criminal.

Having said that, from "Some reasonable people are criminal" we cannot conclude by logic alone that "All reasonable people are criminal".

But we cannot exclude it either, i.e. we cannot state that "Not all reasonable people are criminal".

• I'm not sure why this answer has been upvoted and accepted, because it doesn't work. The fact that you can prove a certain statement B from the premises does not imply anything on the validity (and provability) of statement A. – Federico Poloni Apr 16 '19 at 22:04
• @FedericoPoloni - thnaks for your comment. I've highlighted the fact that the argument is not valid (and not the statement). – Mauro ALLEGRANZA Apr 17 '19 at 6:17
• @FedericoPoloni If you assume that "Some newspaper readers are criminal" conclusively means "less than all"; then there is at least one reader who is not criminal, at which point your conclusion is correct: at least one reader (and thus reasonable person) is not criminal, which means that not all reasonable people are criminal (since we know of at least one exception, the person who made you say "some" instead of "all"). However, in logic, "some" does not explicitly specify that it is definitely less than all. In everyday language it does, but not in logic. – Flater Apr 17 '19 at 15:01
• @FedericoPoloni [..] Which means that in logic, "Some newspaper readers are criminal" includes the possibility that all readers are criminal. And because this is a possibility, "Not all readers are criminal" is not provably true (it is false in the case where all readers happen to be criminals). And if that's not true, then "Not all reasonable people are criminal" can also not be provably true. – Flater Apr 17 '19 at 15:05
• @Flater I'm not sure I get your point. It looks like you are addressing another comment of mine, and you are trying to explain me how "some" is used in propositional logic vs everyday language. I am familiar with that; in my other comment I was simply pointing out that this subtlety is (in my view) the source of OP's confusion. What you write here should probably go inside an answer, then. – Federico Poloni Apr 17 '19 at 15:39

This means that at least one newspaper reader is a criminal. It can be more, it can even be all of them. We do not know. But at least one is.

All newspaper readers are reasonable people

All of the newspaper readers are reasonable people. Because at least one of the readers was a criminal, we must have one reasonable criminal.

Not all reasonable people are criminal

This is something we can't tell.

1. We do not know how many of the reasonable readers are criminals. It could be all of them, or it could be just one.
2. We only know that all newspaper readers are reasonable. But... there might also be other reasonable people who do not read newspapers. We do not have information about this.

So no, we can't tell if all reasonable people are criminals by this logic.

• Welcome to Philosophy SE! We usually like to see references, for a variety of reasons, but a well formed argument, like this one, makes everybody happy. – christo183 Apr 17 '19 at 6:52
• @christo183: Sorry, I'm new to SE in general. I also don't have any references. I had a course a few years ago which used proposition and predicate logic, so I wrote my answer purely based on what I remembered. (I also checked the other answers, which didn't actually include any references). Sorry if my answer was undesirable. – Opifex Apr 17 '19 at 6:56
• No problem, as long as your argument/answer is framed (as yours were) with obvious statements. For opinion and contentious facts we would require references. But in general references are nice for "further reading" purposes. Bottom line: your answer was quite good, a fact reflected by the relatively high number of upvotes... ;) – christo183 Apr 17 '19 at 8:46

Rewrite the phrases in a more formal-like manner as

1. For all x, N(x) implies R(x)

2. There exists x, N(x) and C(x)

And notice these do imply there are reasonable criminals, ie,

There exists x, R(x) and C(x)

Now, "Not all reasonable people are criminal" would be

Not for all x, R(x) implies C(x)

which is (classically) equivalent to

There exists x, R(x) and not C(x)

But it's easy to see one can construct a model with a single individual possessing the three predicates N, R and C, which satisfies the first three phrases, but not the last two

• If I understand correctly from OP's question, their doubts are in translating (2) into quantifiers. They have the doubt that (2), as is written, may mean "A proper subset/subclass of N(x) is R(x)", that is, there exists x, N(x) and C(x), and there exists x, N(x) and not C(x). That is the reason behind that 'some vs all' premise, and I think your answer (which is otherwise correct) should address that. – Federico Poloni Apr 17 '19 at 6:31
1. All newspaper readers are reasonable people.

2. Some newspaper readers are criminal.

... [Thus] Not all reasonable people are criminal

This syllogism is AIO in the third figure:

1. All N are R.

2. Some N are C.

Thus: Some R are not C.

The syllogism AIO-3 is invalid for two reasons. (1) A term (C) which is distributed in the conclusion is not distributed in the premise. (2) When the conclusion is negative, there must be exactly one negative premise. Here, both premises are positive.