1. (∀x)(∀y)[(Px • Py) ⊃ Pf(x,y)]
2. (∃x)[Px • (∃y)(Py • x≠y • Rxy)]
-Which of the following propositions is derivable from the given premises in FF?
A) (∃x)(∃y)[Pf(x,y) • Rxy • x≠y]
B) (∃x)(∃y)[Pf(x) • Rxy • x≠y]
C) (∃x)(∃y)[Pf(f(x)) • Rxy • x≠y]
D) (∀x)(∀y)Pf(x,y)
E) (∀x)Pf(x,x)

Question

Quizplus · Accepted Answer

From the first premise, we can derive (∀x)(∀y)[Px ⊃ (Py ⊃ Pf(x,y))] using conditional proof.
Then, from the second premise, we can derive (∃x)(Px • (∃y)(Py • x≠y • Rxy)) using existential instantiation.
Next, using (∀x)(∀y)[Px ⊃ (Py ⊃ Pf(x,y))] and (∃x)(Px • (∃y)(Py • x≠y • Rxy)), we can derive (∃y)[Py • x≠y • Rxy • Pf(x,y)] using universal instantiation and modus ponens.
Finally, using (∃y)[Py • x≠y • Rxy • Pf(x,y)], we can use existential instantiation twice to get (∃x)(∃y)[Pf(x,y) • Rxy • x≠y].

1. (∀X)(∀y)[(Px • Py) ⊃ Pf(x,y)]