Formal Logic Question

[mharr]

Hi everyone,

I learned formal logic from the PoweScore Logical Reasoning book. PowerScore says that the "some" relationship is reversible. So, for example, A some-> B is reversible to B some-> A.

I am reviewing a couple formal logic problems in my Manhattan Preptests 41-50 book and feel a bit confused. I feel confused because Manhattan says that "some" relationships are reversible but the reversed version of a "some" relationship has one negated variable. For example, A some-> B can be reversed to B some -> ~A.

Would someone mind discussing the Manhattan explanation, please? I am having a difficult time reconciling the PowerScore and Manhattan explanations.

Thank you for your time.

[chike_eze]

Hi everyone,

I learned formal logic from the PoweScore Logical Reasoning book. PowerScore says that the "some" relationship is reversible. So, for example, A some-> B is reversible to B some-> A.

I am reviewing a couple formal logic problems in my Manhattan Preptests 41-50 book and feel a bit confused. I feel confused because Manhattan says that "some" relationships are reversible but the reversed version of a "some" relationship has one negated variable. For example, A some-> B can be reversed to B some -> ~A.

Would someone mind discussing the Manhattan explanation, please? I am having a difficult time reconciling the PowerScore and Manhattan explanations.

Thank you for your time.

To decrease your confusion, I would advise you not use arrows with "some" relationships.

"A some B" is equivalent to "B some A" because "some" is bidirectional.

Not so with "A --> B" or "A -most-> B."
"all" and "most" point in one direction. You can infer "some" relationships in the other direction, however. See below.

A --> B infers B some A
A -most-> B infers B some A

Try using Venn diagrams to visually represent these relationships.

[mharr]

Thank you for the quick response, Chike_eze! What you said makes perfect sense. I am still a bit confused though. I understand that A some B is equivalent to B some A, but I do not understand how A some B is equivalent to B some ~A? Any thoughts?

[chike_eze]

Thank you for the quick response, Chike_eze! What you said makes perfect sense. I am still a bit confused though. I understand that A some B is equivalent to B some A, but I do not understand how A some B is equivalent to B some ~A? Any thoughts?

I think that would make sense if "some" meant "some but not all of B."

Technically "some" could be between 1% - 100%. But colloquially, "some" means something more than nothing but less than all.

If "some" means "some but not all of", then you can infer B some ~A from B some A.

Translation: Some (but not all of) B is A, therefore some B is not A.

[mharr]

Thank you again for the helpful response, Chike_Eze. I appreciate it! Your explanations make perfect sense. I know where my confusion came from. I was reviewing question 22 in preptest 42, section 4. I orginally diagrammed the first statement as a "some" statement but Manhattan diagrammed it as a "most are not" statement. My "some" statement had no negations but Manhattan's "most" statement had a negation. I took the inherent inference of "most are not" which is "some are not" but for some reason equated that inference with my initial statement of "some".

I have one last question-I hope that is okay. For statements that say a "minority" or "less than half", etc. should I diagram them as "most are not" or "some" statements? I thought that diagramming those statements as "some" was accurate but now I'm not so sure...

Thank you very much for your time.

[mharr]

I think the answer to my question is a typical law school answer which is, "it depends".

For example, in this statement, "Only a minority of those who engage in political action do so out of a sense of social justice", a "most are not" statement is accurate. The inherent inference being "some are not".

However, in this statement, "Only a minority of people go to law school", can be diagrammed as a "some" statement.

[chike_eze]

Minority means less than half.
Majority means more than half.

If a town of 1000 people has only two types of people: blue and green people, and nothing else. And we know that the town has minority blue people, then it must follow that the town has majority green people.

But if we say the town contains some blue people, can we say it contains majority green people? In everyday language, perhaps. Because, generally speaking, when people say "some," they mean "some percentage greater than 0% but less than 50%," otherwise, they would have just said most. However, technically speaking (or LSAT speaking) we cannot necessarily infer "greater than 0% but less than 50%" from "some" because some includes most and all. Some could be 1%, 30%, 51%, 100%. Do you see?

You can infer "some" from a "minority," assuming minority contains at least one element in the set. But you cannot necessarily infer a "minority" from "some" because some may be between 51% and 100% (inclusive).

[mharr]

That was a great explanation!! It cleared up my misunderstanding. I did not understand why PowerScore provided a numerical range for each formal logic word (all, none, most, some, most are not, some are not) but your examples made complete sense out of it all.

I now understand how "some" could be a majority or all and also how "some are not" could still be a majority or a minority.

Thank you very much for your time!

[blackbee045]

I am still a bit confused though. I understand that A some B is equivalent to B some A, but I do not understand how A some B is equivalent to B some ~A? Any thoughts?




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[mharr]

Hi blackbee!

I'll provide an explanation. Anyone is welcome to correct me.

Lets start off with this example, all As are Bs. From this we know that it must be true that most As are Bs. It also must be true that some As are Bs. Some relationships are reversible so we know that it also must be true that some Bs are As.

Just a quick recap; when we have an "all" statement, "most" and "some" relationships are inherently known and must be true. So, if most As are Bs then we know it must be true that some As are Bs and that some Bs are As.

Now to your question. It could be true that some Bs are not As. We know what must be the case for some As are Bs, that some Bs are As. We don't know if some Bs are not As so it could be true, it could also not be true.

I hope that explanation helped. Please let me know if you would like more clarification!