All cattle ranchers dislike long winters. All ski resort owners like long winters because long winters mean increase...

Naz on June 30, 2014

Alright, let's diagram the stimulus.

P1: "All cattle ranchers dislike long winters."

We rewrite this in our "if/then" form: "If cattle rancher, then dislike long winters"

CR ==> not LLW
LLW ==> not CR

P2: "All ski resort owners like long winters because long winters mean increased profits."

Rewrite: "If ski resort owner, then like long winters." *NOTE: the rest of the sentence merely further elaborates on the Sufficient & Necessary sentence that precedes the "because."

SRO ==> LLW
not LLW ==> not SRO

P3: "Some lawyers are cattle ranchers."

L-some-CR
CR-some-L

As you can see above, quantity statements do not have contrapositives. However, you can switch the variables from one side to another while keeping the quantity word "some."

The question stem tells us that the conclusion is: "?no ski resort owners are lawyers."

Rewrite: "If ski resort owner, then not a lawyer"

SRO ==> not L
L ==> not SRO

Something to point out is that our conclusion is a Sufficient & Necessary statement with the variable "L" in it. Though P3 also has this variable, P3 is a quantifier while our conclusion is a Sufficient & Necessary. Remember, any time you combine a quantity statement with a Sufficient & Necessary statement you will be left with a quantity statement. So, in order to conclude a Sufficient & Necessary statement with the variable "L," we must have another Sufficient & Necessary statement with this variable. Therefore, the conclusion MUST include something about lawyer. Right off the bat we can eliminate answer choice (B), (D) and (E).

Just to visually help us, let's look at all the diagrammed sentences:

P1: CR ==> not LLW
LLW ==> not CR

P2: SRO ==> LLW
not LLW ==> not SRO

P3: L-some-CR
CR-some-L

C: SRO ==> not L
L ==> not SRO

Now, let's look at the correct answer, (C): "All lawyers are cattle ranchers."

Rewrite: "If lawyer, then cattle rancher."

P4: L==> CR
not CR ==> not L

With our fourth premise from answer choice (C), we can now connect P2 to the contrapositive of P1 to the contrapositive of P4 to get our conclusion like so:

SRO ==> LLW ==> not CR ==> not L

Therefore: SRO ==> not L (i.e. our conclusion).

Hope that's helpful! Let us know if you have any other questions.

Ltowns on July 6, 2014

Thanks it makes sense but I have one question. You said quantity statements do not have contrapositives. If that's case then how is that We do it with words like "all" and "none"?

Mehran on July 12, 2014

Quantity statements are SOME statements and MOST statements.

Though "all" and "none" are technically quantifiers, they are better suited for Sufficient & Necessary statements because they do have contrapositives.

Remember that "all" introduces a sufficient condition. For example, "all As are Bs" would be diagrammed as follows:

A ==> B
not B ==> not A

Remember that "no" introduces a sufficient condition and the other part of the statement negated is the necessary condition.

So for example, "no As are Bs" would be diagrammed as follows:

A ==> not B
B ==> not A

Hope that helps! Please let us know if you have any other questions.

Ltowns on July 24, 2014

Ok thanks Mehran