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The following question has been asked in the Indian entrance examination "GATE CS 2021", and it has stirred up a lot of confusion regarding how one should interpret "either-or" when coming to logical questions where we are given certain premises and have to determine if something is true via deduction.

Q. Given below are two statements 1 and 2, and two conclusions I and II

Statement 1: All bacteria are microorganisms. Statement 2: All pathogens are microorganisms.

Conclusion I: Some pathogens are bacteria. Conclusion II: All pathogens are not bacteria.

Based on the above statements and conclusions, which one of the following options is logically CORRECT?

  1. Only conclusion I is correct
  2. Only conclusion II is correct '
  3. Either conclusion I or II is correct
  4. Neither conclusion I nor II is correct

refer this link for the debate - https://gateoverflow.in/357468/gate-cse-2021-set-1-ga-question-9 refer this link for my reasoning - https://gateoverflow.in/357468/gate-cse-2021-set-1-ga-question-9?show=358167#a358167

In short - I feel option D is the correct choice, because neither conclusions follow from the premises, but on the other side a lot of people feel that option C is right because according to them option c is a tautology.

Also there is another confusion regarding the statement of conclusion 2 "All pathogens are not bacteria". Does this mean that there is no pathogen which is also a bacteria, or does it mean there exists some pathogens who are definitely not bacteria? Because when we say a similar statement like "All students are not geniuses", we generally mean that there are only few students who are geniuses, and most of the students are not geniuses. We don't mean to say that there isn't even a single student who is not a genius.

Can anybody please provide an unambiguous, mathematically backed argument for choosing the correct answer, or is the given question and options inherently ambiguous?

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    $\begingroup$ Option C is not a tautology. $\endgroup$
    – MJD
    Feb 21 at 17:52
  • $\begingroup$ Hey MJD, care to elaborate? "Either some pathogens are bacteria, or none of the pathogens are bacteria" - This statement is clearly trivially true, and this is the argument of people who are saying that option 3 represents a tautology, and hence the correct choice. $\endgroup$ Feb 21 at 18:13
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    $\begingroup$ Although "I or II" is always true, neither of these two options is a correct conclusion from the given information. As far as the given information goes, it might be that I is true (and then II is not) but it might also be that II is true (and then I is not). So we can't infer I from the given information, and we can't infer II either, even though we know that "I or II" is true. $\endgroup$ Feb 21 at 18:30
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Intro This type of questions tends to be problematic because you can't be really sure at what level of precision is the author of the question working. So sometimes what looks like a purely logical exercise (which should have a clear answer) ends up being rather vague or weird.

I think that there is one more ambiguity that is missing in the discussion of the question and that is the difference between correct and valid conclusion. Where correctness is more about the truth-value of the stamen whereas the validity of a conclusion means that it logically follows from premises (and a valid conclusion can be incorrect in the case of false premises).

So I think that the problem isn't about interpreting either-or which is commonly understood as xor. But the problem lies in understanding what author of the question means by logically correct. I would interpret it as valid...

Argument

Statements 1 and 2 both that both pathogens and bacterias are subsets of the bigger set of microorganism. Nothing more. Namely, you can't tell whether these two (sub)sets intersect (some pathogens are bacteria) or not (All pathogens are not bacteria).

So neither conclusion 1, not conclusion 2is valid. They don't follow from the premises.

So this gives us option 4.

However, when we speak only of the truth value of statements, then indeed, we get options 3 tautologically (with no need to look at the premises, i.e. staments 1 and 2) as mentioned in comments.

About big quantifier The formulation used by the author of the question is rather weird. I think that reason behind it is, that it tries to mimic the way logical formula looks -with quantifier (all pathogens) at the beginning and the statement about them at the end ( x is not a bacteria).

In normal English, I guess, one would prefer no pathogen is bacteria. Which is less ambiguous.

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This is a nice question. Let $\Gamma$ be the two premises (statement 1 and 2) and let us write $\Gamma\models A$ for "$A$ is a valid conclusion from $\Gamma$".

Then for example the formalization of "Only conclusion $1$ is correct" is the (false) statement $$(\Gamma\models\text{conclusion 1})\text{ and }(\Gamma\nvDash\text{conclusion 2}).$$

The ambiguity arises because there are two different readings of the option "3. Either conclusion I or II is correct." This could either be the false statement $$ (a)\quad\text{either }(\Gamma\models\text{conclusion 1})\text{ or }(\Gamma\models\text{conclusion 2})$$ or it could be the true statement $$ (b)\quad\Gamma\models\text{either conclusion 1 or conclusion 2}$$

I think $(a)$ is more natural, so that option 3 is false (but I am not a native speaker).

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Conclusions I and II can be restated as "some pathogens are bacteria" and "no pathogens are bacteria".

Whether option C is a tautology or not depends on the meaning of "some", and whether it is to be interpreted as "at least some" (i.e. some and possibly all) or "only some" (I.e. some but not all).

It is only a tautology if "some" includes the possibility of "all", otherwise it is not a tautology since there is a third possibility that "all (and not merely some) pathogens are bacteria", and both the alternatives expressed in option C could then be false.

And since the question asks us only to resolve the matter by reference to the statements and conclusions, and we cannot infer the exact meaning of "some" from them, then I would say option D is correct, that neither of the conclusions are logically correct.

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