# elementary set theory – Propositional expansion of \$forall x P(x)\$

Wikiproof’s article on universal quantifier states that, considering the finite universe of discourse $$x_o, x_1 ldots x_n$$, the quantified formula $$forall x P(x)$$ is equivalent to $$P(x_0) land P(x_1) ldots P(x_n)$$. I’m having problems to understand this equivalence.

That $$forall x in {x_0, x_1 ldots x_n} P(x)$$ means $$P(x_0) land P(x_1) ldots P(x_n)$$ looks pretty straighfoward to me. However, how the formula $$forall x P(x)$$, without expliciting the quantification domain, can be expressed in propositional logic as a chain of logical conjunctions still remains a mistery to me. $$forall x P(x)$$ is equivalent to $$forall x in {x} P(x)$$ and to $$forall x in {x, x_0, x_1 ldots x_n} P(x)$$. The fact that at least one element in the domain of discourse must be equal to the bound variable $$x$$ makes impossible to eliminate the universal quantifier from the formula $$forall xP(x)$$ and, thus, to convert it to propositional logic.

There would be the possibility of defining the universal set $$U = {x : x=x}$$ and then stating that $$forall x P(x)$$ is the same as $$forall x in U P(x)$$. In this case, since the universal set contains all objects, it would be impossible to limit the universe of discourse to a defined set of elements $$x_0, x_1 ldots x_n$$ and state that formula $$forall x P(x)$$ is equivalent to $$P(x_0) land P(x_1) ldots P(x_n)$$; it could be equivalent, for example, to $$P(y_0) land P(y_1) ldots P(y_n)$$ or to a chain of predicates about any other variables.

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