Where can we use quantifiers?

Question

I'm somewhat interested in how we can use quantifiers (mostly existential and universal to clarify the way mathematical systems work)

If given a constant '$a$' can we define $P(a)$ for this constant, and can we use the existential or universal quantifiers over this constant, or is the definition of a predicate only for a variable which takes values over a domain? Perhaps we can limit the domain to include only this constant value?

If I want to talk about a variable representing a quantity $x$ which changes, is it possible to use quantifiers to express this formula or system specifying it as a quantity that takes multiple values and not defined only for a particular or constant value?

Answer 1

If given a constant '$a$' can we define $P(a)$ for this constant, and can we use the existential or universal quantifiers over this constant, or is the definition of a predicate only for a variable which takes values over a domain?

When the open formula $P(x)$ is instantiated with the value $a$ from the domain of discourse, we obtain the closed formula $P(a).$

It is syntactically valid to quantify $P(a)$ even though it contains no free variable (i.e., is no longer a predicate); however, the quantification is redundant: $$∀x\;P(a)\equiv P(a)\equiv∃x\;P(a).$$

On the other hand, $∀a\;P(a)$ and $∃a\;P(a)$ are not well-formed formulae as $a$ is a constant rather than a variable.

Answer 2

Given a constant $a$ you can indeed talk about $P(a)$. For example, if we define $Even(x)$ as $x is a even$ over the natural numbers then $P(2)$, $P(4)$ and $\neg P(3)$.

I don't know what you mean about "using a quantifier over a constant". A quantifier bounds variables, an expression like $\forall x Even(2)$ is equivalent to $Even(2)$.

Your last question is a bit confusing, can you clarify what do you mean?

Answer 3

Well, it depends what you're working in. In propositional logic, quantification does not exist at all. In first-order logic, you can quantify, but only over sets etc, not over predicates, so things like $\forall \alpha (\varphi(\alpha))$ are allowed, but things like $\forall \varphi \exists \alpha (\varphi(\alpha))$ aren't, because $\varphi$ is a predicate, which we're not allowed to quantify over in first-order logics. In second-order logic, quantification over predicates is allowed. I don't think there has been a formal definition for third-order and higher logics, so I can't say about that.