Let Dx = "x is a detective", U12 = "1 is the uncle of 2", and appropriate lowercase letters for names.
- If there are any detectives, John is the only detective.
- No detective is John's uncle.
∃x(Dx ∧ (Dj ∧ ∀y (Dy → y=j)) ∧ x=y). I know this isn't correct, because it doesn't take the if/then form. But my thought process is the following, which seems to capture what the sentence is saying? There is at least one x such that: (i) x is a detective, (ii) John is a detective, and for all y, if y is a detective, y is John, and (iii) x is y (and thus is John). Am I at least on the right track here?
¬∃x (Dx ∧ Uxj). I'm thinking: There is no x such that x is a detective and x is the uncle of John.