"You can fool some of the people all of the time, and you can fool all of the people some of the time, but you can’t fool all of the people all of the time." (Abraham Lincoln)
Let
- $P$ be "fooling some of the people all of the time",
- $Q$ be "fooling all people some of the time",
- $R$ be "fooling all of the people all of the time".
$(P \lor Q) \rightarrow \neg R$
Is this a correct translation in propositional logic?
No, a correct formalization of Lincoln's sentence in propositional logic is the following:
$$(P \lor Q) \land \lnot R$$
Indeed, from a logical point of view, "but" as the same meaning as "and". Note that I translated the "and" between the first two propositions by an "or", because in this context the two propositions express an alternative.
By the way, propositional logic is not the best logic to formalize this kind of sentences. First-order logic and modal logic can express a more faithful formalization of Lincoln's sentence.