Answer to Question #178543 in Discrete Mathematics for Ayesha

Question

Consider Premises: If Claghorn has wide support, then he’ll be asked to run for the senate. If Claghorn yells “Eureka” in Iowa, he will not be asked to run for the senate. Claghorn yells “Eureka” in Iowa. Conclusion: Claghorn does not have wide support. Determine whether the conclusion follows logically from the premises. Explain by representing the statements symbolically and using rules of inference

Answer 1

Let $p$ = "Claghorn has wide support", $q$ = "Claghorn will be asked to run for the senate", $r$ = "Claghorn yells “Eureka” in Iowa". Then the premises are $p\to\ q,\ r\to \overline{q},\ r$ and conclusion $\overline{p}$.

Using to the premises $r\to \overline{q}$ and $r$ the rule of inference Modus Ponens, we conclude $\overline{q}$. Then using to $p\to\ q$ and $\overline{q}$ the rule of inference Modus Tollens, we have the conclusion $\overline{p}$. Therefore, the conclusion follows logically from the premises.