Not All Zs are Ys - does not make sense to me - please explain.
P: Not All ax’s are Ys
X -> ~Y
~Y -some- X
P: All Xs are Zs
X -> Z
~Y -some- X -> Z
C: ~Y -some- Z
Where does the conclusion in the “correct” answer choice come from? How “Not All Zs are Ys (Z -> ~Y)” is the correct answer??? Where did that come from? It does Not follow. Or am I missing something? You can go from “All” to “some,” but you cannot go from “some” to “All”.
Replies
Create a free account to read and
take part in forum discussions.
The premise "not all Xs are Ys" can also be read as "some Xs are not Ys."
For example:
X X X X X X Y Y Y Y Q Q
We can see from this above example that not all Xs are Ys because some are Qs. Therefore, we can also say that some Xs are not Ys.
Remember that 'some' statements are reversible.
P1: X - some - not Y Contrapositive: Not Y - some X
P2: X --> Z Contrapositive: Not Z --> Not X
When connecting quantifier statements and S&N statements, the arrow must always point away from the quantifier.
How can we connect the two premises above to draw a conclusion? Notice that the sufficient condition in P2 can connect to the contrapositive of P1.
Not Y - some - X --> Z
C: Not Y - some - Z Contrapositive: Z - some - Not Y
This is the same as P1 i.e. we can conclude that "Not all Zs are Ys" because some Zs are not Ys.
Hope this helps clear things up a bit! Please let us know if you have any further questions.
LeeLarueAugust 9, 2020
Thank you, I know all that when it comes to quantifies thanks to Mehran's lesson on them. But that is precisely my question: How can we conclude from some statements to all statements? In quantifies videos Mehran specified that we can go from All statements to Some. But NOT in reverse - so why the correct answer is All when we clearly cannot go from some statements to all statements? Then The correct answer should've been: ~Y -some- Z
