why is AxEy(X[xy]vX[yx]) Universal and ExAy(X[xy]vX[yx]) existential?
The first formula translates to:
"For all x, there exists a y such that..."
The second formula translates to:
"There exists an x for which all y..."
In other words, the outermost denominator determines whether the formula is universal or existential (if it has a global scope). As you can see above, interchanging the place of the quantifiers changes the meaning of the formula.
I mean for the second formula all y are within the scope of Ay, so it should be universal as well??
The formula states that there exists an x for which all y (within that subset) share a property, not that all y in the universe share that property.
And if you have a formula like this:
(Ex X[xy]) V (X[yx]),
where V denotes a binary connective, would this be a binsary connective formula >or an existential formula?
Essentially, this formula isn't necessarily existential or universal since there is no shared global scope.
What it translates to (if for example we take V to be OR) is:
"There exists an x which has an edge between x and y OR there is an edge between y and x".
To answer the question; this is a combination of two formulas, the first of which is existential (denoted by the E) and the second of which is atomic with a truth-value that is determined by the meaning of x and y (as they are free variables since they do not fall under the scope of a quantifier).