conditional statement and 'some'

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Zeta
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conditional statement and 'some'

Postby Zeta » Thu Nov 22, 2012 2:14 pm

Regarding the following statement:

Some X are Y

I came across this on a LR and it is easy enough to translate, but the contrapositive doesn't seem valid:

-Y then -some X

There would still be some X. Is this logically valid? What's the procedure when conditional statements involve indicators like 'some'?

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TheMostDangerousLG
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Re: conditional statement and 'some'

Postby TheMostDangerousLG » Thu Nov 22, 2012 3:35 pm

No, that wouldn't be valid. According to the LRB, neither "some" nor "most" statements have contrapositives. However, "some" statements are reversible: for "some X are Y", you can safely say "some Y are X" (but again, you can't take the contrapositive and infer "~Y then ~some X"). When you start with one side negated (e.g. "some X are not Y"), you can also reverse it and say "some not Y are X".

I'd thoroughly recommend checking out the LR Bible's chapter on formal logic for more on "some", "most", and "none" conditional reasoning.

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Zeta
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Re: conditional statement and 'some'

Postby Zeta » Fri Nov 23, 2012 2:28 am

haven't gotten that far yet but thanks!

bp shinners
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Re: conditional statement and 'some'

Postby bp shinners » Fri Nov 23, 2012 9:41 pm

Yep, to sum it up:

All statements: contrapositive is also valid
Most statements: just the original statement is valid
Some statements: converse of original statement is also valid
No statements: see All statements

If it's not on the list, then it's not valid (or not important for the LSAT).

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Zeta
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Re: conditional statement and 'some'

Postby Zeta » Fri Nov 23, 2012 10:57 pm

that's great. thanks for that.

TylerJonesMPLS
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Re: conditional statement and 'some'

Postby TylerJonesMPLS » Sat Nov 24, 2012 2:02 am

Venn diagrams make this sort of reasoning clear.
(I don't know how to draw the Venn diagram here, so I'll just describe it.)

Imagine one circle with the label X. And another circle with the label Y.
Because some X are (also) Y, the two circles intersect. But, just because they intersect, some Y are (also) X.
So, if you know that some X are Y, then you know that some Y are X.

TylerJonesMPLS
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Re: conditional statement and 'some'

Postby TylerJonesMPLS » Sat Nov 24, 2012 4:10 am

Sorry, I just realized that there may be some confusion between propositional logic and categorical logic (All, Most, Some, None logic).

Propositional logic deals with logical relationships between propositions (or statements). So, in propositional logic, X and Y stand for different statements. E.g. X = “Xander is rich”; Y = “Yosif is poor”.
If “X --> Y” is true, then its contrapositive, “~Y --> ~X” will also be true.

Categorical logic deals with categories (or groups or sets ) of things. In categorical logic, X stands for one category or group of things and Y stands for another category or group of things.

*Only* conditional statements in propositional logic have contrapositives.

Statements in categorical logic have converses, not contrapositives.
(The converse of a categorical statement is when the order of the names of the categories or groups are switched. E.g. the categorical statement “All X is Y”, has the converse “All Y is X”.)

So you can’t go from “Some X are Y” to “If ~Y then ~some X”. X and Y stand for entirely different things in propositional logic and in categorical statements.

(I don’t understand why bp shinners wrote above that “All” statements have a valid contrapositive. “All” statements don’t have contrapositives at all. Of course, there is a converse for “All” statements, e.g. “All roses are flowers” has the converse “All flowers are roses.” But this reasoning is obviously not valid. I think what bp shinners wrote must be a typo. Bp shinners: Please confirm! )

Here is an example in of valid reasoning in categorical logic:
“All X is Y”. It follows from this “All” statement that “Some Y is X”. If you think of categorical logic in terms of Venn diagrams, the deduction that “Some Y is X” is obvious. The circle labeled X contains the individual members of class X, the x’s. The circle labeled Y contains the individual members of the class Y, the y’s. If “All X is Y”, then all members of Circle X are also included in the Circle Y, so each individual x is also a y. It follows that some individual y’s are also x’s.

If “Most X is Y,” then the Circle X intersects with Circle Y so that most of Circle X is included in Circle Y. That means that most x’s are also y’s. And it follows that “Some Y is X”, because some y’s must be x’s as well, although not necessarily most of them.

And the same for “Some” statements: From “Some X is Y”, it follows that “Some Y is X”. And also the same for “No” statements: From “No X is Y”, it follows that “No Y is X”.

The moral is: In *propositional* logic there are *contrapositives*, which are always true if the original conditional is true.
In categorical logic there are no contrapositives. But there are converses, which are sometimes true but also sometimes false.

So:
The converse of “All” statements can be true or false. (From “All X is Y”- it does *not* follow that “All Y is X”)
The converse of “Most” statements can be true or false. (From “Most X is Y”- it does *not* follow that “Most Y is X”)
The converse of “Some” statements is always true. (From “Some X is Y”- it *does* follow that “Some Y is X”)
The converse of “No” statements is always true. (From “No X is Y”- it *does* follow that “No Y is X”)

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Zeta
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Re: conditional statement and 'some'

Postby Zeta » Sat Nov 24, 2012 12:52 pm

jesus, i have to take notes!
but really, thanks!

TylerJonesMPLS
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Re: conditional statement and 'some'

Postby TylerJonesMPLS » Sat Nov 24, 2012 2:10 pm

You are welcome! I am a philosophy professor who took early retirement, but I still enjoy explaining logic.




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