“If all politicians are showmen and no showman is sincere then some politicians are insincere.” Ans: F:= $(\forall x\,(P(x) \to ShMan(x)) \land \not \exists y\,( ShMan(y) \to Sinc(y))) \to \exists z\,(P(z) \to \neg Sinc(z)) $
Is the resulting formula valid in classical logic?
my answer is yes. I have checked like : $\neg(F)$ which shows it's unSat $ \to $ F is valid
Prove your answer in terms of semantical arguments. can someone explain it for me. I mean how do it? answer is it's not valid but How to show it?
You actually need modifications in your translation: The connectives in each existential (or negation of existence) need to be $\land$, not $\rightarrow$.
I'll use $S(x)$ to mean "is a showman", and $H(x)$ to mean "x is sincere (honest)". Then what follows is the correct translation of the statement “If all politicians are showmen and no showman is sincere then some politicians are insincere.”
$$(\forall x\,(P(x) \to S(x)) \land \not \exists y\,( S(y) \land H(y))) \to \exists z\,(P(z) \land \lnot H(z))$$
The correctly translated implication is invalid, because we do not have enough information to prove the existence of a politician, let alone the existence of a politician that is also insincere. *This is a fine point.
I'll repeat @Andreas Blass's nice semantic argument (see comment below) here: "A quick way to formulate the semantic argument is to imagine an ideal world, where everyone is sincere and there are no politicians and no showmen."