Question about v-variants of variable assignments

Question

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?

Answer 1

Think of variables as names, and a variable assignment as something that tells us what a variable names.

If you have an open formula, say Px, then Px will come out true when x names an entity which falls within the extension of P. If you consider the formula ∃xPx, x is bound by the existential quantifier. Since x is bound, what x happens to name under a given variable assignment is irrelevant to the truth of ∃xPx. What matters is that some assignment of x makes Px come out true. X-variants are defined to be alike in every respect to the original variable assignment, because often the formula to the right of the quantifier contains other variables, so we don't want to accidentally change their meaning. We fix the meaning (assignment) of every other variable in the formula, and only change up x, consider every possible thing x can name, and if x can name something that makes the formula to the right of the quantifier true, then the statement with the quantifier is true.

Consider a similar case, with the universal quantifier. If x names Socrates, and P is the predicate "is a man", then the open formula Px is true. But what about the statement ∀xPx (for all x, x is a man)? The particular assignment of x to Socrates doesn't help much in determining the truth of the universal statement. What matters is that no matter how we decide what x names (that is, for every X-variant), the statement to the right of the universal quantifier, Px, will come out true ("this is a man", "that is a man", etc).