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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?

3 Answers3

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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.

ryang
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  • is $∃aP(a)$ possible? –  Feb 27 '22 at 16:38
  • @user1007028 Edited. – ryang Feb 27 '22 at 16:39
  • so for example if I had defined a=2 to be a true statement then I would be an abuse to then write either $∃aP(a)$ or ∀aP(a)? (explicitly in terms of a, e.g not something like $∃2P(2)$ which would be completely illogical (hence the variable binding of such an expression not allowing us to specify an explicit number) –  Feb 27 '22 at 17:12
  • @user1007028 If $a:=2$ ($a$ has been assigned a fixed value), then the symbol $a$ essentially is a constant, not a variable. – ryang Feb 27 '22 at 17:25
  • @user1007028 Well, both constants and variables can be represented by letter symbols. -) – ryang Feb 27 '22 at 17:44
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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?

Zanzag
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  • essentially can you have a constant $a=2$ and have $∃aP(a)$ or $VaP(a)$ and it have a meaning or would we reduce the domain D to contain only the value of $a$? –  Feb 27 '22 at 15:17
  • the last part is can we specify an equality to be true such as y=f(x) using quantifiers and predicates. –  Feb 27 '22 at 15:17
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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.

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