1. ## Need logic help

Consider a first-order language with ternary predicate T and equality predicate =. We define the formulas F1, F2, F3 as follows.
F1: ∀x∀y∃zT(x,y,z).
F2: ∀x∀y∀z∀w ((T(x,y,z) ∧ T(y,x,w)) → =(z,w))
F3: ∀x∀y∀z (T(x,y,z) → T(y,x,z)))

a) Prove F1 ∧ F2 logically implies F3
b) Prove F1 ∧ F3 does not logically imply F2

How do I start this, I am very lost here...

2. ## Re: Need logic help

Assume that F1 and F2 are true in some interpretation. To prove that F3 holds in that interpretation, fix some x, y, z and assume T(x,y,z). Does there exist some w such that T(x,y,w)? What can you say about it in view of F2?

For b), let T(x,y,z) mean "point z is located between points x and y."

3. ## Re: Need logic help

For b)
Let domain be {0,1}
T(x,y,z): z is between x and y
Let x=0 y=3 z=1 w=2
Then F1 and F3 does not logically imply F2

Hows this? I am just trying to show a counter example, I hope this is enough to prove it though.

4. ## Re: Need logic help

Originally Posted by Sneaky
For b)
Let domain be {0,1}
T(x,y,z): z is between x and y
Let x=0 y=3

5. ## Re: Need logic help

For b)
Let domain be {0,1,2,3}
T(x,y,z): z is between x and y
Let x=0 y=3 z=1 w=2
Then F1 and F3 does not logically imply F2

Oops I missed that...

6. ## Re: Need logic help

OK. As I said in another thread, you should say that in the described interpretation, F1 and F3 are true and F2 is false, not that "then" F1 and F3 does not logically imply F2.

7. ## Re: Need logic help

For b)
Let domain be {0,1,2,3}
T(x,y,z): z is between x and y
Let x=0 y=3 z=1 w=2
Then F1 is true and F3 is true and F2 is false.
Therefore F1 and F3 does not logically imply F2.

edit: Do you mean I shouldn't have that last line "Therefore F1 and F3 does not logically imply F2." at all?