I have a habit of trying out correctness about some logical statements with worlfram alpha by generating truth table for them. For example, I can try if this:
((¬x→y)∧(¬x→¬y))→x
is correct or not by geerating truth table for ((¬x→y)∧(¬x→¬y)) which turns out to be the same as x column in the same truth table. Hence the above is correct. However is there any way I can check same for biconditionals involving nested existential and universal quantifiers and predicates? For example can I somehow verify rules of this kind?:
(∀x)(∀y)ϕ(x,y)⇔(∀y)(∀x)ϕ(x,y)
Update
I am able to do following check ∀x,y(x∨y) as follows:
Resolve[ForAll[{x,y}, x or y]]
which correctly returns False as (x∨y) does not hold for all x and y.
So now I thought I can do something similar to obtain True for following (which is a general fact): ¬(∀x)ϕ(x)⇔(∃x)¬ϕ(x). I tried this:
Resolve[ForAll[x,(not ForAll[x, x]) xnor (exists[x,not x])]]
But it did not work. Note that ⇔ is nothing but xnor. So how do I do this especially something like following also correctly returns True:
which stands for ¬∀x(x).
