C_ = _ is a Celt , R_ = _ is a Roman , c = Claudius , e = Elona , L__ = _likes _
If some Roman likes every Celt, then every Celt is liked by some Roman.
∀x∃y( (Ry ∧ Cx ∧ Lyx) → (Cx ∧ Ry ∧ Lyx) ) which can be simply ∀x∃y(Ry ∧ Cx ∧ Lyx)
If Claudius likes Elona and Elona is a Celt, then if Claudius is a Roman, at least one Roman likes at least one Celt.
(Lce ∧ Ce) → ( Rc → ∃x,y(Rx ∧ Cy ∧ Lxy) )
Are these symbolizations correct? :) Thanks in advance.
Second one is correct, but for the first one you need separate quantifiers on either side of the conditional.
Example:
If all elephants are big, then all ducks like water
If you use just one quantifier, you get:
$$\forall x ((Ex \rightarrow Bx) \rightarrow (Dx \rightarrow Wx))$$
But this doesn't work: the statement is an 'if ... then ...' statement where both sides are quantificational statements. So, you should use a quantifer for both parts:
$$\forall x (Ex \rightarrow Bx) \rightarrow \forall x (Dx \rightarrow Wx)$$
And by the way, 'some Roman likes every Celt' is not the same as 'every Celt is liked by some Roman'. The first statement says that there is a very special Roman who likes every Celt .. so every Celt is liked by this very one Roman. In the second statement every Celt is liked by some Roman .. but it doesn't have to be the same one very special Roman that likes every Celt. Compare: 'Everyone is liked by someone' (plausibly true) vs 'someone likes everyone' (definitely false!)