Question

Prove that each of the following conclusions (C) follows from the given premises.

P1: (Ǝx)(P(x) ∧ (∀y)(B(y) ⇒ R(x,y)))

P2: ~(Ǝx)(Ǝy)(P(x) ∧ F(y) ∧ R(x,y))

P3: (∀y) (F(y) ⇒ B(y))

C: ~(Ǝx)F(x)

Answer 1

The given premises of the argument are :

P1: (Ǝx)(P(x) ∧ (∀y)(B(y) ⇒ R(x,y)))

P2: ~(Ǝx)(Ǝy)(P(x) ∧ F(y) ∧ R(x,y))

P3: (∀y) (F(y) ⇒ B(y))

The conclusion to be proved is :

C: ~(Ǝx)F(x)

The argument is proved to be valid using the following table :

Statement no.	Statement	Justification
1	(Ǝx)(P(x) ∧ (∀y)(B(y) ⇒ R(x,y)))	given
2	~(Ǝx)(Ǝy)(P(x) ∧ F(y) ∧ R(x,y))	given
3	(∀y) (F(y) ⇒ B(y))	given
4	(Ǝx)(P(x) ∧ (B(c) ⇒ R(x,c)))	Universal instantiation on statement 1 with respect to variable y
5	P(b) ∧ ( B(c) ⇒ R(b,c) )	Existential instantiation on statement 4 with respect to variable x
6	~(Ǝx)(P(x) ∧ F(c) ∧ R(x,c))	Existential instantiation on statement 2 with respect to variable y
7	(∀x) ~(P(x) ∧ F(c) ∧ R(x,c))	Change of quantifier on statement 6
8	~(P(b) ∧ F(c) ∧ R(b,c))	Universal instantiation on statement 7 with respect to variable x
9	(F(c) ⇒ B(c))	Universal instantiation on statement 8 with respect to variable y
10	( B(c) ⇒ R(b,c) )	Simplification rule of inference on statement 5
11	(F(c) ⇒ R(b,c) )	Hypothetical Syllogism rule of inference on statements 9 and 10
12	~P(b) v ~F(c) v ~R(b,c)	De Morgan's Law on statement 8
13	P(b)	Simplification rule of inference on statement 5
14	~F(c) v ~R(b,c)	Disjunctive Syllogism on statements 12 and 13
15	~F(c) v R(b,c)	Material Implication rule of inference on statement 11
16	~F(c) v ~F(c)	Resolution on statements 14 and 15
17	~F(c)	Idempotent Law on statement 16
18	(∀x)~F(x)	Universal Generalization on statement 17
19	~(Ǝx)F(x)	Change of quantifier on statement 18

The conclusion ~(Ǝx)F(x) can be derived from the given premises.

Thus, the given argument is valid.