Here is the question explained with another example which will hopefully help people think freshly without thinking about the actual question.
Statement 1 – all rats who lost one of their limbs are 3 limb mammals
Statement 2 – all rodents who lost one of their limbs are 3 limb mammals
Conclusion 1 – some 3 limb rats are 3 limb rodents
Conclusion 2 – all 3 limb rats are not 3 limb rodents
Cases possible from given statements 1 and 2:
Now,
1. can we conclude that SOME 3 limb rats are 3 limb rodents? No. Counter example is figure 1
2. can we conclude that ALL 3 limb rats are not 3 limb rodents? No. Counter example is all other figures.
So neither conclusion 1, nor 2 is correct. There is no ambiguity here right? because neither of the conclusions were derivable from the premises.
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Now why is – either conclusion 1 or 2 is correct – wrong? The answer lies in interpretation of the question (semantics), as well as the definition of a conclusion.
Saying that either conclusion 1 or conclusion 2 is correct implies that one of the above conclusions can possibly be deduced from the statements, meaning either we were able to deduce that some 3 limb rats are 3 limb rodents, or we were able to deduce that all 3 limb rats aren’t 3 limb rodents.
Definition of the conditional statement from Rosen which mentions what is called a “conclusion” :
Let p and q be propositions. The conditional statement p → q is the proposition “if p, then
q.” The conditional statement p → q is false when p is true and q is false, and true otherwise.
In the conditional statement p → q, p is called the hypothesis (or antecedent or premise)
and q is called the conclusion (or consequence).
Another definition from https://www.britannica.com/topic/logic#ref535920
“An inference is a rule-governed step from one or more propositions, called premises, to a new proposition, usually called the conclusion. An inference rule is said to be valid, or deductively valid, if it is necessarily truth-preserving. That is, in any conceivable case in which the premises are true, the conclusion yielded by the inference rule will also be true.”
Stmt 1 AND Stmt 2 → Conclusion 1 stmt ? – False
Stmt 1 AND Stmt 2 → Conclusion 2 stmt ? – False
But do the individual statements of conclusion 1 and conclusion 2 together form a tautology? Definitely.
Although here is where we need to differentiate between making a whole new proposition by interpreting option c as “stmt C1 OR stmt C2”, and what the logical english interpretation of the options should be, i.e. either conclusion one can be deduced or conlusion two can be deduced. (a conclusion being correct implies that the conclusion CAN be deduced. Refer the definitions above).
But we clearly can’t deduce either of the conclusions from the given premises. There is a clear difference between saying “one of the conclusion holds and is correctly deduced” vs “one of the statements of both the conclusions has to be true”, the 2nd one being an assertion, which doesn’t take into account what a conclusion is! and simply considers both conclusions as statements OR’ed together, which we shouldn’t implicitly assume.
I feel like my argument is sound, and I’m not dismissing the fact that the other way of interpretation of the question may have some merit to it. (don’t come after me @Nikhil_dhama, blame the english language), but either option D should be correct, or both option C and D should get marks. The case where only option C is correct is clearly wrong.
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EDIT 1: As pointed out by @debmalya, statement of conclusion 2 “All pathogens are not bacteria” is probably being wrongly interpreted by everyone.
For example when we say that “All students are not toppers”, we don’t mean to say that there are no students who are toppers. Instead we mean that most of the students are not toppers.
Therefore the venn diagram of conclusion 1 and conclusion 2 both are actually identical, and they are not mutually exclusive, i.e one is not the negation of the other, and therefore both of them together in fact dont form a tautology, making option D the only correct choice.
