Is the question well defined?

Question

I have been set this question on the exam:

Using standard rules of inference and proper quantification, what conclusions can be drawn about Sam from the following set of premises?

1. All players are intelligent.
2. Sam is not intelligent.
3. All boxers are players.

I was puzzled by this question because nothing is defined about the conclusion, which conclusion is asked about to be true?

I drew the conclusion that 'Sam is not intelligent' which is the premise itself and my friend wrote 'Sam is not a player'.

It turned out that both of the answers were marked incorrect. The teacher told that the correct conclusion was 'Sam is neither a player nor a boxer nor intelligent'.

My question is, isn't this question ill defined because nothing is mentioned about the specific conclusion the teacher has in mind? And how do I convince my teacher if the question is ill defined?

Answer 1

Here is more detailed why there are infinitely many conclusions that one can draw. A complete answer would depend on the exact rules you were introduced to, so I'll be a little bit sketchy.

As one thing, you can introduce the double negation. From

Sam is not intelligent,

for example, you can conclude

It is not the case that it is not the case that Sam is not intelligent.

A little shorter may look like this:

Sam is not not not intelligent.

But then again, you can add a double negation, so that you get a sentence which has four times "It is not the case that" in front and "Sam is not intelligent" following. Thusly you can go on as long as you want, which already shows that you can draw infinitely many conclusions.

Something like this also works for disjunction. A common rule says that if you have a sentence $A$, then you can conclude $A \vee B$, whatever sentence $B$ is. So from

Sam is not intelligent

you can conclude

Sam is not intelligent OR Sam's teacher is lovely.

Again, from this you can conclude:

Sam is not intelligent OR Sam's teacher is lovely OR Sam's teacher is cool.

And so on. Such things can also be done with other operators like conjunction.

Answer 2

The question is not ill-defined, although it could be phrased better. The question asks "what conclusions can be drawn about Sam?", not "what is an example of something you can conclude about Sam?". The plain meaning of "what conclusions can be drawn" is to ask about all possible (relevant) conclusions. Still, it's easy to misunderstand the question and it would have been better to ask something like "What are all the conclusions that can be drawn..."

Answer 3

The question should better have been phrased "what is the strongest conclusion you can draw about Sam", meaning what is the maximum information you can collect about him.

The teacher wanted you to perform inferences 1 + 2 => Sam isn't a player, Sam isn't a player + 3 => Sam isn't a boxer.