Feb 99/Superprep B Question

[corresponding Cor]

In "The Official LSAT Superprep" Test B's Section 2 (Logic Games), I have a question about a condition in game # 2 (dealing with maple trees, fir trees, oak trees, etc.).
One of the rules states:
"If it is not the case that the park contains BOTH laurels and oaks, then it contains firs AND spruces."

I diagram this as:
NOT L AND O -> F AND S
Contra:
NOT F OR S -> L OR O

Is that incorrect? Perhaps I'm not seeing something here, because in the explanation for question 9, it says "If there are no firs in the park, the park must contain both laurels and oaks."
Can somebody please clarify this?
Also, is it just me or this game really hard?

[Lizface killah]

In "The Official LSAT Superprep" Test B's Section 2 (Logic Games), I have a question about a condition in game # 2 (dealing maple trees, fir trees, oak trees, etc.).
One of the rules states:
"If it is not the case that the park contains BOTH laurels and oaks, then it contains firs AND spruces.

I diagram this as:
NOT L AND O -> F AND S
Contra:
NOT F OR S -> L OR O

Is that incorrect? Perhaps I'm not seeing something here, because in the explanation for question 9, it says "If there are no firs in the park, the park must contain both laurels and oaks."
Can somebody please clarify this?
Also, is it just me or this game really hard?

I think the first one should be
XL or XO --> F & S

then the contra is

XF---> L &O

[corresponding Cor]

[/quote]I think the first one should be
XL or XO --> F & S

then the contra is

XF---> L &O[/quote]

What separates this conditional from others that would be diagrammed as XL and XO? Why, exactly, is it "or" instead of "and," when it says "both?"

[Lizface killah]

I think the first one should be
XL or XO --> F & S

then the contra is

XF---> L &O

What separates this conditional from others that would be diagrammed as XL and XO? Why, exactly, is it "or" instead of "and," when it says "both?"

Because if either XL or XO or XL & XO, then it is not the case that both are there. Hard to explain, that's just how I translate that condition in my head.

[Derrex]

I diagram this as:
NOT L AND O -> F AND S
Contra:
NOT F OR S -> L OR O

Should be
Not (L and O) -> F and S ( Not (L and O) can be Not L or Not O)
Contra
Not F or Not S -> L and O

The reason yours is not correct is because of your writing Not L and O is ambiguous.
The two statements you write (NOT L AND O) and (L or O) should be the negation of each other, but it is true only if you assume (NOT L AND O) is (NOT L AND NOT O). This is not a correct interpretation of the statement "not the case the park contains BOTH laurels and oaks". It should be (NOT L OR NOT O) or (NOT (L AND O)).

[corresponding Cor]

I diagram this as:
NOT L AND O -> F AND S
Contra:
NOT F OR S -> L OR O

Should be
Not (L and O) -> F and S ( Not (L and O) can be Not L or Not O)
Contra
Not F or Not S -> L and O

The reason yours is not correct is because of your writing Not L and O is ambiguous.
The two statements you write (NOT L AND O) and (L or O) should be the negation of each other, but it is true only if you assume (NOT L AND O) is (NOT L AND NOT O). This is not a correct interpretation of the statement "not the case the park contains BOTH laurels and oaks". It should be (NOT L OR NOT O) or (NOT (L AND O)).

Ok, the only thing I'm not 100% clear is why the Contra is: "Not F or Not S -> L and O" and NOT "Not F or Not S -> L OR O." Don't all ands/ors get swapped? I apologize if I'm trying your patience here.
If Not (L and O) can be Not L or Not O, does the same ring true for all conditional relationships that contain "Not VARIABLE X and Not VARIABLE Y" - that "Not Variable X OR Not Variable Y" produces the same result?

[Derrex]

I'm sure you know this but for contra, you negate both sides and reverse the arrow.
The reason its "L and O" is because the sufficient condition in the original statement is "NOT ( L and O)", the negation of that is "L and O". Conveniently, the negation of "NOT L or NOT O" is also "L and O".

As per your second question, "NOT (X and Y) is always equivalent to "Not X or Not Y" (If you want to dig a little deeper, its an application of DeMorgan's Theorem).

However, be very careful. "Not (X and Y) is not the same as "Not Variable X and Not Variable Y", which I think might be the source of your confusion regarding the original relationship. That is why I said your original statement of "Not L and O" is ambiguous. "Not variable X and Not variable Y" is not an equivalent statement to "Not (X and Y)".

If you interpret "Not L and O" as "Not ( L and O)" --the correct statement--, your negation will simply be "L and O" which forms the correct necessary condition in the contra positive. If you interpret "Not L and O" as "Not L and Not O" -- incorrect--, the negation of that is "L or O", which is the wrong necessary condition and the one in your original contra positive.

Going back to your original contra positive "NOT F OR S -> L OR O", think it through for why its wrong. Lets say there is F and no S. Then this implies there's L or O, lets say there's just O and no L. This certainly works for the contra you supplied.

Then lets look at the original statement: "If it is not the case that the park contains BOTH laurels and oaks, then it contains firs AND spruces."

If there's Just O and no L, then the park does not contain both O and L, therefore it contains F and S, but we just said there is F and no S, a contradiction, hence the two statements are not equivalent.

[goansongo]

In "The Official LSAT Superprep" Test B's Section 2 (Logic Games), I have a question about a condition in game # 2 (dealing with maple trees, fir trees, oak trees, etc.).
One of the rules states:
"If it is not the case that the park contains BOTH laurels and oaks, then it contains firs AND spruces."

I diagram this as:
NOT L AND O -> F AND S
Contra:
NOT F OR S -> L OR O

Is that incorrect? Perhaps I'm not seeing something here, because in the explanation for question 9, it says "If there are no firs in the park, the park must contain both laurels and oaks."
Can somebody please clarify this?
Also, is it just me or this game really hard?

I think you should try to process the statement "If it is not the case that the park contains BOTH laurels and oaks, then it contains firs AND spruces" before diagramming it.

So if you don't have both of those things (the laurels and the oaks), then you will have firs and spruces.

What does that mean? If you don't have both things in, it means either one of those trees is out or they can both be out. To diagram that, it is ~L o ~O ----> F + S

The contrapositive is of course: ~F o ~S ----> L + O

I think you got confused by the "and" in the statement "If it is not the case that the park contains both laurels and oaks. Forget about diagramming it for a second and think about it logically instead.

And yes, I found some of the games in the Superprep book to be very very difficult. All are February tests... coincidence? Feb. is known to be very hard. And look, the LG contain circular games too.

[bluejayk]

Sometimes conditionals aren't as helpful when written out as "if --> then" statements. Especially when they follow the "~A --> B" structure, I like to state them as "A or B (or both)". For this game, you could symbolize this rule as:

(F & S) or (L & O)

You've got to have at least one of these two pairs. So anytime you don't have F, you can't have the first pair, so you must have the second pair. Thinking about it this way, to me it's much more obvious than when you don't have firs, you must have laurels and oaks.

[LawandOrder]

~L or ~O -> F & S
~F or ~S -> L & O

is my diagram.

[gmreplay]

I totally got this the first time around, but when trying it a second time the fourth rule (the one being discussed here) totally threw me. I get why it is correct to diagram it as bluejayk has, but there is one thing that seems weird:

Being that the rule states "If it is not the case that the park contains both L and O, then it contains F and S", wouldn't the contrapositive of this statement require that we change the and in both fragments to "or"?

If the rule was "If A and B, then C" we would negate it as "if ~C, then ~A or ~B". Does the fact that there is an and on both sides cancel it out, or is there another reason behind it?

[jesuis]

Sometimes conditionals aren't as helpful when written out as "if --> then" statements. Especially when they follow the "~A --> B" structure, I like to state them as "A or B (or both)". For this game, you could symbolize this rule as:

(F & S) or (L & O)

You've got to have at least one of these two pairs. So anytime you don't have F, you can't have the first pair, so you must have the second pair. Thinking about it this way, to me it's much more obvious than when you don't have firs, you must have laurels and oaks.

Thank you for putting it this way! I think your explanation and thought process for this type of conditional statement is definitely the best way to approach it!