The use of 'A number' in Mathematical Language

Question

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).

Answer 1

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

Answer 2

In many mathematical texts, when a new variable is introduced, the type of quantification (existential / universal) is often implied and not explicitly stated. It would be wrong to assume existential quantification everywhere.

Existential quantification ($\exists x P(x)$) is used to talk about the existence of a value for $x$ such that $P(x)$ is true. But there is a way to talk about the whole range of values for which $P(x)$ is true, and then you can substitute any value for $x$ (and not just a particular number), as long as $P$ holds for that value.

The solution is to use universal quantification. For example, the statement $\forall x(P(x)\to Q(x))$ can be read as 'For all $x$ for which $P(x)$ holds, $Q(x)$ holds too'. Put differently, when the text starts with something like 'Let $x$ be such that $P(x)$' and then goes on to prove $Q(x)$, then the text uses universal quantification and formally it proves that $\forall x(P(x)\to Q(x))$.

Altogether, an equality that contains variables should not be read alone. One should look at the text to see how these variables are quantified. Phrases like 'Suppose some $x$ such that ...' or 'Assume any $x$ ...' or 'Let $x$ be such that ...' mean universal quantification, while 'There is an $x$ such that ...' means existential.

Answer 3

Consider the (satisfiable but logically non-valid) predicate $$(-1)^x=1.\tag#$$

The “truth status” of a predicate depends on its interpretation:

• In the context of $\mathbb R,$ predicate $(\#)$ is a conditional equation, i.e., true for some values in the domain of its variable $x,$ so

  1. $∃x\; (-1)^x=1$ is a true statement,
  2. $∀ x\; (-1)^x=1$ is a false statement.

• In the context of $2\mathbb Z,$ predicate $(\#)$ is an identity, i.e., true for all values of $x,$ so

  1. $∃x\; (-1)^x=1$ is a true statement,
  2. $∀ x\; (-1)^x=1$ is a true statement.

• In the context of $\mathbb Z{\setminus}2\mathbb Z$ (i.e., the set of odds), predicate $(\#)$ is inconsistent, i.e., true for no value of $x,$ so

  1. $∃x\; (-1)^x=1$ is a false statement,
  2. $∀ x\; (-1)^x=1$ is a false statement.