December 1997 LSAT Section 2 Question 24

No mathematical proposition can be proven true by observation. It follows that it is impossible to know any mathemat...

Anna20 on August 13, 2020

Mathematical Proposition

How would you diagram this question / answers? Would no be a necessary indicator word? P: O --> not MPT P: missing C: KMPT I would then think that the missing premise here is: not MPT --> KMPT. But that doesn't make sense with any of the answer choices?! As a follow up, what about other indicator words - such as until (R will keep eating until X happens), always (W always eats lunch), cannot (Monica cannot run without her shoes), does not (the store does not open until 10pm). How can you tell whether these are sufficiency indicators or necessary indicators? Thank you so much!

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shunhe on August 13, 2020

Hi @Anna2020,

Thanks for the question! So there’s a couple of different ways to think about “no statements.” When we get a statement that says “No X are Y,” how can we diagram it? Well, what does it mean? Let’s look at the following example:

No avocados are vegetables.

How do we frame this in “if then” language? Well, it’s basically saying that if something’s an avocado, it’s not a vegetable! So this gets diagrammed

Avocado —> ~Vegetable

And in general, No X are Y gets diagrammed

X —> ~Y

Or the contrapositive, if you prefer, which is

Y —> ~X

So really, what matters is that you take one of the terms, put it in the sufficient condition, and take the other term, negate it, and put it in the necessary condition. So here, for example, we’re told that no math proposition can be proven true by observation. Well, that just means that if something’s a math proposition, it can’t be proven true by observation, or

MP —> ~PTO

And the contrapositive, which I think is what you diagrammed, is

PTO —> ~MP

The conclusion is then “it’s impossible to know any mathematical proposition to be true.” Well, this is just a fancy no statement. In other words, this is basically saying “no mathematical proposition can be known to be true.” And we diagram that the same way, so this is going to be

MP —> ~KT

So our argument is

Premise: MP —> ~PTO
Conclusion: MP —> ~KT

And it looks like we need something to connect ~KT and ~PTO. Specifically, if we can get ~PTO —> ~KT (or its contrapositive, KT —> PTO), we can make the conclusion follow. And that’s what (E) gives us.

As for other indicator words: If you must, “until” is basically “unless.” “Always” is kind of like “all the time,” so like “all,” which would introduce the sufficient, though it’s a bit weirder because you’re basically diagramming something like (using your example)

W —> Eat lunch

In the last example you use, “cannot” isn’t the indicator word, it’s “without.” And “without” is also like “unless.”

Hope this helps! Feel free to ask any other questions that you might have.

Anna20 on February 3, 2021

I have to say that the above is truly so helpful, I have referred back to this many times. Really appreciate it - thank you!!