A4Q1b Solution?

[Roman]

if |= p -> n, then |= (for all x. p) -> (for all x. n)

My tutorial section did not give the solution to this question, can
someone explain how it is done?

[Curtis Bright]

Let (I, theta) be an arbitrary model with domain D. The general outline is:

I, theta |= ∀x.p
=> I, theta[x/v] |= p for all v in D
(use hypothesis |= p->n)
=> I, theta[x/v] |= n for all v in D
=> I, theta |= ∀x.n

Curtis

Post by Roman
if |= p -> n, then |= (for all x. p) -> (for all x. n)
My tutorial section did not give the solution to this question, can
someone explain how it is done?

[Roman]

Post by Curtis Bright
I, theta |= ∀x.p
=> I, theta[x/v] |= p for all v in D
(use hypothesis |= p->n)
=> I, theta[x/v] |= n for all v in D
=> I, theta |= ∀x.n
Curtis

Post by Roman
if |= p -> n, then |= (for all x. p) -> (for all x. n)
My tutorial section did not give the solution to this question, can
someone explain how it is done?

So you use the MP inference rule in the middle of the solution there?

[Curtis Bright]

Post by Roman
So you use the MP inference rule in the middle of the solution there?

Basically, though "inference rule" usually refers to syntactical rather
than semantical arguments.

Curtis

[Roman]

Post by Curtis Bright

Post by Roman
So you use the MP inference rule in the middle of the solution there?

Basically, though "inference rule" usually refers to syntactical rather
than semantical arguments.
Curtis

Thanks.