We have that a formula $\alpha(x_1,x_2,\dots, x_k)$ is existential if it is of the form $$\exists t_1\exists t_2\cdots \exists t_l\beta(x_1,\dots,x_k, t_1,\dots,t_l)$$ where the formula $\beta(x_1,\dots,x_k, t_1,\dots,t_l)$ is quantifier-free, i.e., it doesn't contain the quantifiers $\exists$ and $\forall$.
If the formula $\beta(x_1,\dots,x_k, t_1,\dots,t_l)$ doesn't contain negation, it is called positive existential.
Which is the definition of an (positive) existential sentence?
Is the (positive) existential theory the set of all the (positive) existential formulas or the set of all (positive) existential sentences? Or is the definition of the (positive) existential theory entire different?
P.S. I have posted this question also here.
Disclaimer: Definitions may vary from person to person, book to book or article to article. Thus if you are reading a book or taking a course, you should refer to that litterature for the definition they mean. The bellow explanations are based on what I percieve as the most common definition.
A sentence is a formula without free variables. Thus a (positive) existential sentence is just a sentence on the form:
$ \exists t_1...\exists t_l\beta(t_1,...,t_l) $ where $\beta$ is quantifier free.
There is no The (positive) existential theory, but rather we say that a theory is existensial if it is a set of (positive) existensial sentences (or possibly possible to axiomatize using such sentences). If we would take the set of all existential sentences it would include things such as $\exists x (P(x) \wedge \neg P(x))$ and thus would be inconsistent.