# Translate an english sentence to first order logic

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").

• The amobiguous sentence is likely another one, about fooling some people all the time. At least I'm unsure what your italicazation is supposed to differentiate. Cf. the joke "This newspaper says that every twenty minutes a man gets run over by a car" - "What a poor fellow that guy is" – Hagen von Eitzen Nov 30 '14 at 10:47
• I meant to say both the sentences are the same, I'm unsure about the scope of "can't fool", as to whether he "can't fool all people" or that he "can't fool all the people all the time"? – nsane Nov 30 '14 at 10:51
• On the other hand, there is a difference between "I can fool A and I can fool B" vs. "I can fool A and B". Cf., if I you replace "fool" with "visit tomorrow" and A,B with Helsinki and Melbourne. I may be able to visit any of the two cities, but not both on the same day. – Hagen von Eitzen Nov 30 '14 at 10:52
• The context is a quote from A.Lincoln : "You can fool all the people some of the time, and some of the people all the time, but you cannot fool all the people all the time." Thus, I do not see any ambiguity... – Mauro ALLEGRANZA Nov 30 '14 at 13:22
• @MauroALLEGRANZA Doesn't that mean "Every politician CAN fool all people all of the time."? – nsane Nov 30 '14 at 13:47

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 :

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