Prove whether or not the following inferences hold using a suitable semantic method

Question

$\forall x,\;P(x)\to Q(x),\;\forall x,\;Q(x)\to R(x),\;\lnot R(a)\models\lnot P(a)$

I don't know how to use a truth table to prove it due to ∀ in formula。

Answer 1

In a semantic proof you must show that there can be no interpretation which evaluate all the premises as true but the conclusion as false.

In propositional calculus, a truth table does this by explicitly listing all of the interpretations. However, this is First Order Logic, with too many interpretations to list, so truth tables will not work.

You have to argue based on the meaning of the symbols:

• Suppose $\neg P(a)$ is valued as false, while the premises are all true. As $\forall x.(P(x)\to Q(x))$ entails that $P(a)\to Q(a)$, this will be held true too. … So on …