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)?(x,y)?(?y)(?x)?(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)?(x)?(?x)¬?(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).