Reasoning/logic question: Premises, conclusion?

I would lean towards C. There is not enough information to arrive at a conclusion.

It’s an inductive argument, so it’s only going to give you a probabilistic conclusion—and that is even stated with “probably”.

The conclusion appears to falsify the argument, as the result is the opposite of what was expected.

But since the conclusion was only ever “probable”—it’s not necessarily falsified the argument.

I think the answer is D.

As for the “why,” if this helps:

Try reading the premise and then predicting what will happen. Then compare that to the conclusion that’s offered in the problem.

Setup: If a non-profit organization receives a grant increase, [then] volunteers who are already on staff at the time of the increase will receive pay for their work. This will probably attract more volunteers, who will hope that they will eventually be paid.

You’re then told: The non-profit organization’s grant was increased. Based on the setup, what do you expect will happen next? (Don’t worry about anything beyond this point yet. Just look back through the setup. What things are known or predicted will happen as a result of the grant increase?)

Now look at the conclusion. How does that compare to what you expect will happen?

…

Or another way to look at it, if this helps more:

If grant increase (X), then current volunteers get paid (Y), and probably more people volunteer (Z).

>> If X, then Y and probably Z.

The grant is increased.

>> X is true (aka, Given X).

Conclusion: No more people volunteer.

>> Not Z.

Whole problem condensed:

If X, then Y and probably Z.
Given X, not Z. True or false?

(And then see @Kropotkin‘s post about the significance of the “probably”).

I go along with @Kropotkin and @Soubresaut .

The statement has the form: if X then Y and probably Z.
We are told X is true.
Therefore Y and probably Z
Y is true.
Z is probably true.

You are asked to evaluate not Z.
Z being probably true is the same as not Z being probably false.
The correct answer is D.

Note that the wording of the problem allows for the possibility that there might be more volunteers apart from any cash incentive. This does not change things. It would only make Z more probable.

^^ That’s a better way to explain it. Cleaner/clearer than mine.

Since the question is multiple choice, I’m going to assume that it comes from either a logic textbook or an assignment (though it could also come from a test of some kind). This means that one of the most relevant questions is “what concepts have I been learning/working on recently?” It would therefore be helpful if you could give us the context of the problem. Is this from a class/book on introductory logic? Modal logic? Inductive logic? The last of these seems most likely, so I’m going to answer with that in mind.

To start, I wouldn’t formalize the argument in the same way others have above. Let’s look at the premises separated from one another for why:

1. If a non-profit organization receives a grant increase, volunteers who are already on staff at the time of the increase will receive pay for their work.

Let G = “the grant is increased,” and let P = “the current volunteers get paid.” Translating the sentence into logical form then gives us: G -> P (“if G, then P”).

2. This will probably attract more volunteers, who will hope that they will eventually be paid.

Let M = “the organization attracts more volunteers,” and let ¶ be an operator representing “probably” or “it is probable that.” Importantly, “this” refers to the current volunteers getting paid, so the sentence is a disguised conditional: P -> ¶M (“if P, then probably M”).

3. The non-profit organization’s grant was increased.

This one is pretty simple to translate: G (“the grant is increased”). So that just leaves the conclusion:

C: There were no volunteers to work in the program in the subsequent year.

The most elegant way to translate this is ¬M (“not M”). We could give it a unique letter, but that would just invalidate the argument from the start by introducing a term in the conclusion that doesn’t appear in the premises.

This makes the argument:

1. G -> P
2. P -> ¶M
3. G
C: ¬M

So what is the status of the conclusion? We are given five options, which—using [] to represent the necessity operator—can be translated as:

A. []¬M
B. ¶¬M
C. indeterminate
D. ¶¬¬M (which can be simplified to ¶M)
E. []¬¬M (which can be simplified to []M)

And this brings us why I prefer my formalization of the argument (though I should stress that they are logically equivalent): my formalization makes it a bit more obvious that the premises jointly entail ¶M (thanks to the rule of hypothetical syllogism). Since we know that the argument entails ¶M, and since that is one of the answers, we know that the answer is D.

We can also get there indirectly, through elimination. If we know that ¶M, then we also know that the conclusion (that is, ¬M) is neither necessarily true (since []¬M entails ¬¶M, which is inconsistent with the premises) nor necessarily false (since ¬M is neither contradictory nor contrary to ¶M). We can also know that the answer is not ¶¬M because this is logically equivalent to ¬¶M, which contradicts a known entailment of the premises (namely, ¶M). And we know the answer is not indeterminate because there is an answer that matches an entailment of the premises (again, ¶M). Therefore, the answer is again D.