Logical inferences for quantifiers

Question 1

Universal insanitation

Answer 1

1. ∀x (P(x))
   (c) P(a) for any a

Question 2

Existential insanitation

Answer 2

1.∃x (P(x))
(c) P(a) for some a

Question 3

Pause remember for Universial we said for any a and for Existential we said for some a

Question 4

Universal Generalisation

Answer 4

1.P(a) for all a
(c) ∀x (P(x))

Question 5

Existential Generalisation

Answer 5

1. P(a) for some a
   (c) ∃x (P(x))

Question 6

We install from outside in, so how would you do ∀x ∃y (P(x,y))

Answer 6

1. ∀x ∃y (P(x,y))
2. ∃y (P(a,y)) for any a
3. (P(a,b)) for some b

Question 7

We generalise from inside out depending on the conclusion e.g if the conclusion was ∃y ∀x (P(x,y)) and we had step 3: (P(a,b)) for some b

Answer 7

3. (P(a,b)) for some b
4. ∀x (P(x,b))
5. ∃y ∀x (P(x,y))