When studying Mathematics, we see expressions with equality with one or more variables, such as $x+1=2$, I've always seen this as an expression which states $(number)+1 = 2$ where $number$ could be many numbers depending on potential values that can be subbed in yielding a $true$ or $false$ answer, a predicate that can depend on values of $x$.
Recently I have learnt of Existential quantification and I perceive something like the following:
$∃xP(x)$ as being 'there exists a value of the variable $x$ such that $P(x)$ (is true)'
I have seen it written as 'there exists a number $x$ such that (for example) $x+1=1$
This seems to contradict my view, as it seems $x$ is a particular number in their view of $x$ and can only represent this particular number, instead of being something than varies in a Domain $D$ and can be substituted into.
We also see terminology like where $x$ is a number, this seems ambiguous, is the correct interpretation 'there is one number $x$ or, $x$ represents a place where a number can be assigned or substituted for $x$. Is there a particular difference between 'there is a number $x$ such that…' and ' there exists a value of $x$ such that…'
Is there a distinction between a variable and what can be substituted into based on context as opposed to whether they are bound or free (as these could be considered free variables still).
Best Answer
The phrase "there exists" or the symbol $\exists$ does change the meaning of a sentence. Plain "$x+1=2$" by itself does not usually mean the same as "there exists a number $x$ such that $x+1=2$".
The meaning of just $x+1=2$ depends on context: if nothing is known about $x$, it's as you say a predicate which could be used with multiple values of $x$ to get a statement which is true or false. In predicate logic, we could say $P(x)$ is "$x+1=2$". Then $P(10)$ is false and $P(1)$ is true. But if $x$ was already defined to be some value, then $x+1=2$ is already a true or false statement.
Adding the existence quantifier, we also get a statement which is true or false. $\exists x P(x)$ (or $\exists x . P(x)$ or $\exists x : P(x)$) reads as "there exists a number $x$ such that $x+1=2$". This statement is true since the value $1$ satisfies the predicate.
Also note $\exists$ or "there exists..." actually means "there exists at least one ...". You may also see $\exists!$ to say "there exists exactly one..." or "there exists a unique..."
A few more examples:
$\exists x \in \mathbb{R}: x^2=1$ — "There exists a real number $x$ such that $x^2=1$ — true
$\exists x \in \mathbb{R} : x^2=-1$ — "There exists a real number $x$ such that $x^2=-1$ — false
$\exists! x \in \mathbb{R}: x^2=1$ — "There exists a unique real number $x$ such that $x^2=1$ — false