سوال ۹۱ کنکور ارشد علوم کامپیوتر ۱۴۰۲

We test each argument for logical validity — does the conclusion follow from the premises regardless of the actual meaning of the terms? Let $R,N,I,P,\text{Pos},\text{Neg}$ denote rational, natural, integer, prime, positive, negative.

Argument (a)

• P1: $\exists x\,(R(x)\wedge\neg N(x))$ — "some rationals are not natural."
• P2: $\forall x\,(N(x)\to I(x))$ — "all naturals are integers."
• C: $\exists x\,(R(x)\wedge\neg I(x))$ — "some rationals are not integers."

The middle term $N$ appears in P1 only as $\neg N$ and in P2 only as $N$. Knowing a rational is not natural tells us nothing about whether it is an integer.

Counterexample (abstract): let the "rationals that are not natural" be exactly the negative integers. They are all integers, so the conclusion "some rational is not an integer" fails while both premises hold. The form is the invalid mood (it commits the fallacy of the illicit use of the middle/major term). $\Rightarrow$ (a) is invalid.

Argument (b)

• P1: $\forall x\,(P(x)\to \text{Pos}(x))$ — "all primes are positive."
• P2: $\exists x\,(\text{Neg}(x)\wedge\neg \text{Pos}(x))$ — "some negatives are not positive."
• C: $\forall x\,(\text{Neg}(x)\to\neg P(x))$ — "no negative is prime."

The conclusion is a universal claim about every negative number, but P2 is only an existential claim about some negatives. A universal conclusion cannot be drawn from a particular premise here: the premises do not rule out a negative number that is positive (impossible in reality, but logic must use only what the premises state) being prime.

Formally, P1 and P2 are jointly satisfiable while C is false — e.g., a model in which some negative object is both positive and prime, and the negative object guaranteed by P2 is a different one. Both premises hold, the conclusion fails. $\Rightarrow$ (b) is invalid.

Conclusion

Neither argument is valid, so the answer is option 4.

Exam trick: Validity is purely about form, not about whether the statements are true in the real world. A red flag for invalidity: a universal conclusion ("no…", "all…") drawn from a particular ("some…") premise, as in (b); and a middle term that is negated in one premise but affirmed in the other, as in (a).