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/