In Full Predicate Calculus, some variable assignment q satisfies a disjunction under interpretation U if q satisfies one or both disjuncts under U, it satisfies a conjunction under interpretation U if q satisfies both conjuncts under U, and so on for conditionals, biconditionals, and negations.
Now, q satisfies "(EXISTS v)p" (where p is some sentence) if p is satisfied under U by q OR by some other variable assignment called a v-variant that is identical to q except for in the value that it assigns to v.
I don't think I intuitively understand why this is the case. It feels like we are making some sort of transgression by allowing variable assignments which are identical to the variable assignment in question, and then saying that the original variable assignment is satisfactory when in fact its v-variant was satisfactory.
Can someone clear this up for me?