[Math] Translate an english sentence to first order logic

Question

Here's an English statement –

Politicians can't fool all of the people all of the time.

(𝈗x for all things, P(x) x is a person, Q(x) x is a politician, T(x) x is a time and F(x, y, z) x can fool y at t).

I believe this statement seems ambiguous as it is not clear whether

Politicians can't fool all the people all the time

or

Politicians can't fool all of the people all of the time.

Am I correct? BTW, the original question had two more statements similar to the above one and a note at the end reading "One of the above may be ambiguous").

Answer 1

It seems to me that we have to translate :

Politicians can't fool all of the people all of the time

as follows :

$\forall x[Q(x) \rightarrow \lnot \forall y \forall z ((P(y) \land T(z)) \rightarrow F(x,y,z))]$

i.e.

$\forall x[Q(x) \rightarrow \exists y \exists z \lnot ((P(y) \land T(z)) \rightarrow F(x,y,z))]$.

Using the equivalence between : $\lnot (p \rightarrow q)$ and $(p \land \lnot q)$, we can rewrite it as :

$\forall x[Q(x) \rightarrow \exists y \exists z (P(y) \land T(z) \land \lnot F(x,y,z))]$

or in prenex form :

$\forall x \exists y \exists z[Q(x) \rightarrow (P(y) \land T(z) \land \lnot F(x,y,z))]$.