Universal Quantification definition question

Question

I have a question on how the universal quantifier is defined. Church Intro to Metamathmatics defines universal quantification as "$\forall x$ __ is True if the value of __ is True for all values of x" and that "$\forall x$ __ is False if the value of __ is False for any value of x. I'm assuming "all values of x" to mean the domain of discourse, which is the universe for the current system. Frege Begriffsschrift states that $\forall x P(x)$ "whatever we may take for its argument, the function is a fact" and this is similar to Church's definition.

wiki defines quantification as "a quantifier specifies the quantity of specimens in the domain of discourse that satisfy an open formula." This definition is different from the above definitions as it suggest elements from the DOD are chosen to satisfy an open formula. I would suspect this means $\forall x$ must always be true as only elements from the DOD that satisfy the formula are selected to be substituted for $x$. Is this wiki definition correct?

Given the Church and Frege definitions for quantification would this formula be false because all $n$ from the DOD may not be members of $\mathbb{N}$?

$\forall n\in \mathbb{N} P(n)$

which can be written: $\forall n (n \in \mathbb{N} \land P(n))$

or is the $\forall n \in \mathbb{N}$ specifying the the DOD? What specifies the DOD?

Answer 1

The correct way to switch from restricted quantifiers to unrestricted quantifiers is \begin{align}\forall x\in A, P(x)&\equiv \forall x, (x\in A\to P(x))\equiv \forall x,(P(x)\lor \neg x\in A)\\ \exists x\in A, P(x)&\equiv \exists x,(x\in A\land P(x))\end{align}

This covers natural language, ordinary usage in matematics and basically everything (possibly even some material truth).

Answer 2

In ordinary mathematics the domain of discourse is expressed in a statement like this: "for all $x \in A$ the property $P(x)$ holds". You can translate this to say "for all $x$, if $x \in A$ then the property $P(x)$ holds", which can then be written as $$\forall x, \, x \in A \implies P(x) $$ Note that this is not equivalent to what you suggested in your post, namely $x \in A \land P(x)$. The correct logical connector is the implication $\implies$, not the conjunction $\land$.

Keep in mind that it is not necessary to express a domain of discourse, nonetheless it is very commonly done in ordinary mathematics.