Are formulas without quantifiers universal formulas? A universal formula is defined to be a formula of the form $\forall x_1\forall x_2\ldots\forall x_n\alpha$ where $\alpha$ has no quantifiers. My question is, can $n=0$? That is, is a formula with no quantifiers considers a universal formula? I ask this because of this question which appears to actually be false if quantifier-free formulas are considered universal. This is because the question claims that for any wff $P$, the formula $\exists xP(x)$ is not equivalent to a universal formula, which is false as can be seen by taking a counterexample. Namely, if $P(x)$ is $x\neq x$.
The same question can be asked for existential formulas.
 A: There may be different definitions in different textbooks; in Shoenfield's classic "Mathematical Logic" the definition of a formula in prenex normal form (of which a universal formula is a special case in which only universal quantifiers and no existential quantifiers are permitted in the prefix) explicitly permits for an empty quantifier prefix.
Normally, unless specified otherwise, $n = 0$ is permitted in such definitions; and for the sake of the present notion it shouldn't hurt: The point of a universal formula is to have all quantifiers at the front and all quantifiers converted to universal ones; if there are no quantifiers altogether, it still matches that requirement.
An analogous argument can be made for existential formulas.  
In any case, I don't see how this would render the claim in the linked question false. A quantifier-free formula $\alpha$ is, in the general case, indeed not logically equivalent to an existentially quantified one: $$\alpha \not \equiv \exists x \alpha$$
Remember that two formulas are equivalent if they are true under all the same structures and variable assignments. But since the truth value of a formula with an free variable may vary between different variable assignments, while the truth value of a formula with the variable bound is constant within the same structure and hence if it is true under one assignment it is true under all of them, there may be variable assignments that satisfy $\exists x \alpha$ but not $\alpha$.
Of course, a universal formula without universal quantifiers is equivalent to an existential formula without existential quantifiers, because then you have syntactically identical formulas:  $$\alpha \equiv \alpha$$
