is quantifier domain tied to predicate argument

Question

I have trouble understanding the domains of predicate quantifiers.

P(x) =“x knows calculus”.

Domain for x is all rabbits.

$ \forall x\neg P(x) $

To me it seems like for all rabbits no all rabbits knows calculus. Which makes no sense. So if I were to provide an actual value for the predicate P(pygmy rabbit) the statement would not make sense. So the actual x is merely the domain I suppose.

If we were to make something that could actually provide some different outcomes based on the inputs

P(x) =“x is a mammal”.

Domain for x a set of animals

$ \forall x P(x) $

let x be $\{ Rabbit, Human, Whale \}$

$ \forall x P(x) \equiv true$

The above observations leads me to believe that the predicate variable has to be presented as an argument inside of the predicate, I would like for someone to clarify this for me

Answer 1

A One variable predicate of the form $ P(x) $ is a sentence, not a proposition, which depends on $ x $. The parameter $ x $ lies in a certain set $ E $. To become a proposition, we must either specify the value of $ x $, or use quantifiers.

Example :

$$E=\Bbb R\;\; P(x)\;:\; x>3$$

$x>3$ is not a proposition.

If $ x=2 $, $ P(x) $ is a false proposition.

$(\forall x\in \Bbb R) \; P(x)$ is a false proposition.

$(\exists x\in \Bbb R)\;:\; P(x)$ is a true proposition.

Answer 2

I don’t understand what you mean by your last paragraph, but you do seem to have some misunderstandings about domains and the translation of $\forall x\big(\neg P(x)\big)$ into English.

The domain in your first example is the set of all rabbits. The expression $\forall x\big(\neg P(x)\big)$ means for any rabbit it is not the case that that rabbit knows calculus; in ordinary English this is simply no rabbit knows calculus, there is no rabbit that knows calculus, or the like. Note that pygmy rabbit is not in the domain: it is not a single specific rabbit.

Similarly, in your second example Rabbit, Human, and Whale are not in the domain: they are not single, specific animals. Jonas Grønbek and Brian M. Scott are in the domain, as is Shaman, my partner’s cat, and substituting any of them for $x$ in $P(x)$ results in a true statement.