سوال ۹۱ کنکور ارشد علوم کامپیوتر ۱۴۰۵

Validity is about form, not truth. An argument is valid iff whenever its premises are true its conclusion must be true — the actual truth (or even consistency) of the premises is irrelevant. Let $R,N,P,T$ denote real, negative, positive, natural.

Argument a. The form is $\forall x\,(R(x)\to N(x)),\qquad \exists x\,(R(x)\wedge P(x))\ \Longrightarrow\ \exists x\,(N(x)\wedge P(x)).$ Take a witness $x_0$ with $R(x_0)\wedge P(x_0)$. Since every $R$ is $N$, the universal premise gives $N(x_0)$, hence $N(x_0)\wedge P(x_0)$, so $\exists x\,(N(x)\wedge P(x))$. This is the classical valid mood Disamis ($\text{IAI}$, figure 3). Valid.

Argument b. The form is $\forall x\,\neg\bigl(N(x)\wedge P(x)\bigr),\qquad \exists x\,(P(x)\wedge T(x))\ \Longrightarrow\ \exists x\,(T(x)\wedge\neg N(x)).$ Take a witness $x_0$ with $P(x_0)\wedge T(x_0)$. Because no negative is positive, $P(x_0)\Rightarrow\neg N(x_0)$, so $\neg N(x_0)$, giving $T(x_0)\wedge\neg N(x_0)$ and thus $\exists x\,(T(x)\wedge\neg N(x))$. This is the valid mood EIO (Ferio-type). Valid.

Both arguments are valid, so the answer is Both — option 3.

Exam trick: Do not reject (a) just because its premises are false (even contradictory) in reality. Validity checks only whether the conclusion follows; carry the existential witness from the particular premise through the universal premise and the conclusion holds.