Hi there! I am a little confused on Example #4 under Rule #2 for Making Valid Deductions with Quantifiers. Why can you not conclude that some Xs are Ys if all Ys are Zs? Or can you not conclude that because not all Zs are Ys? I would appreciate clarification with this.
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There are two rules when we are making valid deductions with quantifiers.
First, there must be a S&N statement. We can see that this rule is satisfied here because of the premise Y - > Z.
Second, there must be a common sufficient condition. This is where the problem arises for this example.
When approaching these questions, always start by writing the contrapositive of each premise.
We know that 'some' statements are reversible:
X-some-Z Z-some-X
Then, when finding the contrapositive of S&N statements, we must always both reverse AND negate:
Y - > Z Not Z - > Not Y
So, you are correct that we cannot draw any conclusions from this because the contrapositive is Not Z - > Not Y. There must be a common sufficient condition and the arrow must be pointing away from the quantifier.
If the second premise were Z - > Y, we would be able to draw the conclusion that some Xs are Ys:
X-some-Z - > Y
We could then use the transitive property to conclude that:
X-some-Y
Hope this is helpful! Please let us know if you have any further questions.
