Propositional Calculus - Validity

Question

I have the following question:

I have drawn a truth table below:

From the table I believe that the answer is not C. However I am not sure whether the premise is incorrectly defined as the premise and conclusion are entirely unrelated, or whether the premise simply does not entail the conclusion.

If anyone could help it would be much appreciated, thanks.

Answer 1

For simplicity sake, I will write $A$ for 'warm(mary)' and $B$ for 'warm(david)', ok? We need to determine whether $\neg A$ entails $B$, i.e. $\neg A \vdash B$. In order to do this, we consider the truth table of $\neg A \rightarrow B$. Is there a line it's false? We can look at your own truth table you drawed above, and we recall the following:

In a truth table, $ \phi \rightarrow \psi$ is false iff there is a line where the antecedent, $\phi$, is true and the consequent, $\psi$, false.

So it's easy to see it does, right? If this conditional is false in at least one line of your table your premise doesn't entail your conclusion and therefore your answer is (a).