Formal Deduction - 
Prove by formal deduction that Paul is the son of John from the
     following premises:

Premises


*

*John is the father of Paul. 

*Paul is not the daughter of John.

*A child is either a son or a daughter.


Predicates && Constants && Domain:


*

*Domain: The set of all people

*$F(x,y)$: $x$ is the father of $y$. 

*$S(x,y)$: $x$ is the son of $y$. 

*$D(x,y)$: $x$ is the daughter of $y$.

*$J$: John

*$P$: Paul


Hint: the formal proof uses the


*

*disjunctive syllogism: ($A \lor B, \lnot A \vdash B$)

*($∀−$), universal instantiation.


What I am confused in this problem is,
for the 3rd premise, I set as
A child is either a son or a daughter: $\forall x \exists y(S(x,y) \lor D(x,y))$
Is this the correct interpretation of the English sentence?
Or $\forall x \forall y(S(x,y) \lor D(x,y))$?
If I have $\forall x\forall y(S(x,y) \lor D(x,y))$, I could easily prove.
Please, help. Thank you.

Using Adrian Keister's symbolization of the third premise, here is a formal proof:


Kevin Klement's JavaScript/PHP Fitch-style natural deduction proof editor and checker http://proofs.openlogicproject.org/
