Some X's are Y's? Some and the Transitive Property

yeahyeah2121
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Joined: Thu Sep 24, 2009 7:39 pm

Some X's are Y's? Some and the Transitive Property

Postby yeahyeah2121 » Wed Aug 11, 2010 12:51 am

P: some a's are b's
P: all b's are c's
P: _____________
C: some d's are c's

Why is the correct missing premise "all a's are d's" instead of "all d's are a's?" It's my understanding that the if "all d's are a's" then you can create the transitive chain of d ---> a some b ---> c and come to the conclusion some d's are c's. If you insert the correct premise, the chain doesn't link up in this way. The correct answer makes sense to me, I'm just wondering why "all a's are d's" would be considered incorrect?

lawgod
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Re: Some X's are Y's? Some and the Transitive Property

Postby lawgod » Wed Aug 11, 2010 1:06 am

It cannot be "all D's are A's"
This is because we do not know which A's are B's, only that some are. Even if all D's are A's, it is possible that the "D" A's are the ones which are NOT "B", since only some A's are B.

However, If all A's are D's, we know that some of the AD group are B and that all B are C, hence some D are C. Regarding the D's which are not A, we have no information.

I don't know how to say this (or anything) in formal logic.

yeahyeah2121
Posts: 98
Joined: Thu Sep 24, 2009 7:39 pm

Re: Some X's are Y's? Some and the Transitive Property

Postby yeahyeah2121 » Wed Aug 11, 2010 1:12 am

Thanks. That was my understanding of the correct premise as well. I think this example is just another reminder to turn off auto-pilot on some formal logic questions and really think about what does and what does not logically follow in real terms. Back to studying.... whomp whomp.

lawgod
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Re: Some X's are Y's? Some and the Transitive Property

Postby lawgod » Wed Aug 11, 2010 1:17 am

Thanks for posting the question. It was a good break from reading about bad employment info and people who got into schools I was rejected from.

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Anaconda
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Re: Some X's are Y's? Some and the Transitive Property

Postby Anaconda » Wed Aug 11, 2010 2:35 am

Can someone tell me if I'm understanding this properly?


Is the reason that d -> a wouldn't work is that the chain would be:

d -> a <-> b -> c

However there are no open chains, so nothing can be inferred, so the conclusion cannot be proven.

But, if you assume a -> d

then the inference

a <-> b -> c

becomes

a <-> c

and now you can simply replace a with d to get:

d <-> c

Am I correct?

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Anaconda
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Re: Some X's are Y's? Some and the Transitive Property

Postby Anaconda » Wed Aug 11, 2010 2:36 am

Btw, is this kind of logic actually on any LR questions? I've never seen them, even though the LRB dedicates a whole chapter to it.

youknowryan
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Re: Some X's are Y's? Some and the Transitive Property

Postby youknowryan » Wed Aug 11, 2010 1:45 pm

Anaconda wrote:Btw, is this kind of logic actually on any LR questions? I've never seen them, even though the LRB dedicates a whole chapter to it.


About 1 per test, and less frequently recently than in the 30s and 40s.




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