Translation Question: Not All X are Y

Will Dearborn

Not all X are Y. Is this translated as X some (not)Y? Also, is Not all X are Y equivalent to X some Y? For the latter question, I know in English in certain contexts, the statement "not all" of something implies "some are."

Accounts Playable

To answer your first question: for the purposes of the LSAT, "not all X are Y" means that there are some Xs that are not Ys. To see this, think of the negation of the above statement: "All Xs are Ys." That means that for every X, it's a Y--no matter what. So, if you say that isn't the case, then you are saying it's possible to have an X that is not a Y. In propositional logic, this isn't technically true because of something called the existential fallacy, but I've never seen the LSAT try to fool a test taker because of it. For that reason, just remember your negation lessons starting here: https://7sage.com/lesson/the-negation/

Second question: No. "Not all X are Y" is not equivalent to "X some Y." Say you had this full and complete world:

X
Y

It's the case that not all Xs are Ys (exactly zero Xs are Ys since there is only one X and only one Y; the two sets do not overlap). Since the "X some Y" statement is not true in this world, it cannot be equivalent to "Not all X are Y."

Lastly, it's true that some English context is ambiguous. The LSAT does its best to not be ambiguous because it doesn't want the nightmare of people challenging a lot of their questions. Thus, for the purposes of the LSAT, "not all" is simply the negation of "all." It could be the case that it implies none.

quinnxzhang

@"Accounts Playable" said:
In propositional logic, this isn't technically true because of something called the existential fallacy, but I've never seen the LSAT try to fool a test taker because of it.

This is incorrect for a couple of reasons. First, propositional logic can't even express "All X are Y" or "Some X are not Y" because propositional logic has no quantifiers. I think you have predicate logic in mind. Second, this equivalence is straightforwardly true in predicate logic and is not an existential fallacy. "Not all" is equivalent to "some not" in predicate logic -- no fallacy involved. The existential fallacy is moving from "All X are Y" to "Some X are Y", which is not what's happening here. That said, I've also never seen the LSAT abuse this.