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
Best Answer
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.