Reference no: EM132546469

Question 1: Prove the following theorems (a(x), β(x) etc. represent FOL wffs.):

(d) |- (∀x)(α(X) => β(x)) => ((∀x) ~ β(x) => (∀x) ~ α(x)).

Question 2: Prove the following theorem. This another part of the above question.

? (∀x)(∀y)β(x, y) <=> (∀x)(∀y)(∼ α(x, y) => β(x, y)) ∧ (∀x)(∀y)(α(x, y) => β(x, y)).

Question 3: Prove 'that each of the following conclusions (C) follows from the given premises

P1: ( ∃x)(P(x) ^ (∀y)(B(y) => R(x, y)))
P2: ~(∃x)(∃y)(P(x) ^ F(y) ^ R(x,y))
P3: (∀y)(F(y) = B(y))
C : ~ (∃x)F(x)

i. P1: (∀x)(P(x) = ((∃y)(T(y) A B(x, y)) (∃z)(G(z) ^ R(x, z))))
P2: (∀x) ~ G(x)
C : (∀x)(∀y)((P(x) ^ T(y)) => ~B(x,y))

Question 4: Prove that each of the following arguments is valid.

(a) People who comsume neither meat nor fish food are vegetarians.
Vegans consume no animal food nor milk products.
Meat is animal food
Fish food is animal food
Hence, vegans are vegetarians.