The definition of equality and 'value' for variable expressions.

Question

The definition of equality that I've been given is that it 'asserts that two expressions have the same value'. This is adequate with well-defined expressions like $2+1=3$ or $2^2=4,$ but for expressions with variables it seems strange to assert that they have the same value when they don't necessarily have a value until we start exploring the valid assignments of the variable.

For example, if I have $x+1=2$ by itself, $x+1$ has no specific value until I say, "let $x=1.$" In this case, $x+1$ has a value of $2$ and the assertion is true.

Is there a better definition that explains equalities like $x^2=x+2$ where they are conditional?

Is the value of an expression something that can only be discussed under an assignment? Is there a definition of it where asserting that expressions have the same value makes sense when we aren't exploring a possible assignment on $x$?

Answer 1

It is a variable-free statement $2+1=3$ and equality and is a tautology, it is always true under the arithmetic you are using. On other hand, the statement $x+1=2$ is a equation and it is true for some $x$ in the referential from which $x$ is allowed to be taken (i.e. the truth value is now subject to the type of object that is $x$), e.g., if the referential is ${\bf R}$ then equality holds if, and only if, $x=1$. Strictly one should get the interpretation of the semantics one is using or the context of the statement.

Answer 2

You are looking for the notion of bound variables. In your example, $x +1=2$ the variable is not bound by any constraints, so the expression cannot be evaluated. If you had posed the same equality with some "constraints" on $x$, we would be able to evaluate it. Some "constraints" include:

$\bullet x\in$ "some set", where the expression would be evaluated as to whether it holds for said $x$ or not,

$\bullet$ The question of whether such an $x$ actually exists, where the expression would be evaluated to $x=1$,

$\bullet$ The question of how many or which $x$ satisfy the equation, where you would evaluate to either a number or a set.

The list goes on. I am using the word "constraints" here without defining what it actually is, hence the quotes. You can think of bound variables as being placeholders, where unbound variables create ambiguity.

Edit: To account for the edit to the question.

The words that you are using, "expression", "value", are shared as names for different notions. As you see from my examples, values can be different in nature, sets Vs numbers Vs yes or no answer Vs the list goes on. As a general rule, to talk about values is to assign meaning to otherwise free variables, but there are exceptions beyond the scope of this question.

Answer 3

yes, you have to add an existential quantifier.

$(\exists x)(x^2=x+2)$

In general a sentence can have a truth value if no variable in the expression is free.

Answer 4

Is the 'value' of an expression something that can only be discussed under an assignment?

No.

for expressions with variables it seems strange to assert that they have the same value when they don't necessarily have a value until we start exploring the valid assignments of the variable.

It is not strange to assert that both sides of the logical validity $$x=x$$ have the same value, even as neither has a fixed assignment.

The mathematical identity $$x^2-y^2\equiv(x+y)(x-y),$$ that is, $$\forall x\:\forall y\:\big(x^2-y^2=(x+y)(x-y)\big)$$ tells us that $x^2-y^2$ and $(x+y)(x-y)$ are mathematically always equal, regardless of value assignment.

Even without a value assignment, it is legitimate to assert that the conditional equation $$2x=6$$ is existentially true (is satisfied by some value of $x$), that is, $$\exists x\: 2x=6.$$