Formal Logic Q.

[quiz555]

My question involves the language indicators for necessary and sufficient conditions.

What I have noticed is that, the words "the only" coupled together, always represent a sufficient condition.

I have not seen anything that illustrates the contrary. Can this be deemed a universal rule?

Thanks!

[quiz555]

Wow,

Now I have some reason to doubt that it can't be applied universally. I just encountered the following question that includes this sentence:

"But if the Association wants a mayor who will attract more businesses to the town, Cooper is the only candidate it could support." (PowerScore Virtual Course Book, pg. 10-98.)

It seems here that the words "the only" are not representing the sufficient condition in this sentence, since the sufficient condition is clearly stated before it with the words "But if the...)

Hmm,

Help please.

Thanks!

[Nikki Siclunov]

The word "only" (or the phrase "the only") both modify a necessary condition, not a sufficient one. Let me give you a few examples:

The only way to cure headache is to take Aspirin.

Cure Headache --> Aspirin

(note that "the only" refers to the "way" to cure headache, not to curing headache per se. So "the only" modifies the idea of taking Aspirin).

The only children playing in the park are boys.

Play in the park --> Boys

(again, the phrase "the only" refers not to the idea of playing in the park, but to who is playing, i.e. to the boys).

Do not automatically assume that the indicators must immediately precede the sentence element they modify. You need to think about the conditional structure that underlies the argument based on the sentence construction you are given.

[quiz555]

Got it!

Thank you!

[quiz555]

I think it's safe to say that "the only" always introduces/precedes a sufficient condition, but, at the same time, modifies and refers to the necessary element.

There's no example out there, that I can think of where this is not true...

Even in the aforementioned examples stated above, "the only" is introducing/preceding the sufficient condition, but at the same time referring to the necessary element.

[Jon Denning]

Thanks for the question. As Nikki pointed out, "the only" expresses necessity, so whatever it modifies/describes is the necessary condition. What you're seeing is simply a product of the English language: sentences and statements can be constructed in a wide variety of ways, and yet still have equivalent meanings.

Take a few variations of the same idea:

At the conference, the only people in attendance are lawyers.
Lawyers are the only people attending the conference.
The only people attending the conference are lawyers.
For anyone wishing to attend the conference, the only requirement is that he/she be a lawyer.

Each of those is essentially Attend Conference --> Lawyer.

The placement of "the only" can vary, but what it modifies in each instance--attendees/lawyers--is always necessary.

Be careful when trying to apply too much of a formulaic/mathematical type of analysis to something as subtle and malleable as language; most of the time you'll be alright, but the test makers are masters at twisting the phrasing of things around to try to confuse and trick you. Your best bet is to couple your knowledge of modifying phrases with a genuine understanding of what an author is actually attempting to convey from a meaning standpoint. That'll get you the proper setup every time.

Thanks!

[quiz555]

Great, Thank you Jon!