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Let's take an expression like $x+x$ and $2x$ and form a statement which is always true in my domain D.

For example, $x+x=2x$ this statement is true for all $x$, so for any value of changing $x$, why can I not then define this expression to be true just as given with free variables, instead of only giving it a truth value under a particular assignment.

user37577
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  • The above formula is true for every value of x, but this is not always true. Consider è.g. the formula 2x=3. – Mauro ALLEGRANZA Aug 18 '22 at 17:14
  • "why do I need to actually assign it a value before the expression can be defined as 'true' or 'false'?" What do you mean by "assigning a value"? – Natalie Clarius Aug 18 '22 at 19:14
  • So I need to specify I'm considering a particular assignment e.g $x=1$ before considering a truth value. – user37577 Aug 18 '22 at 20:04
  • "true for all $x$" means true under any particular assignment. Evaluating an expression with free variables as true by verifying that it holds "for any value changing $x$" is exactly what we're doing. We iterate over the possible values of $x$, find that the statement holds true for each of them, and thus conclude that the formula with free variables is true in the structure. – Natalie Clarius Aug 19 '22 at 15:14
  • A formlula (possibly with free variables) is true for a domain if and only if it is true under all variable assignments - that's what we're doing. – Natalie Clarius Aug 19 '22 at 15:20
  • @lemontree but importantly making such a conclusion is equivalent to a universal quantifier on D? So P(x) itself has no truth value only when we quantify? – user37577 Aug 19 '22 at 19:14
  • $P(x)$ has a truth value by itself, and it's equivalent to the truth value of $\forall x P(x)$. – Natalie Clarius Aug 19 '22 at 22:58

2 Answers2

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We don't assign a value to $x$ to ensure that $x+x = 2x$ is true because like you said, the statement is true for all $x$ (assuming we are working with basic algebra we learn in grade school).

It's like asking if this is true:

$$\text{"If an integer } x \text{ is a multiple of $6$, then } x \text{ is even."}$$

That is a true statement regardless of what integer $x$ is. After all, all multiples of $6$ are even, so no matter which multiple of $6$ the integer $x$ happens to be, it is even. But do we assign $x$ to be some integer value before the statement can be defined as true or false? No.

Does this answer your question? The question is somewhat hard to follow, so hopefully, my response helps.

Accelerator
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  • Formulas with free variables are regularly assigned truth values if they are true for all variable assignments. For example, Definition 1.7.9 in Leary’s mathematical logic text. – Joe Aug 18 '22 at 19:27
  • I feel like 'if $x$ is an integer' is a form of saying 'for any integer $x$' and hence universally quantified, a more vague statement is equivalent to an equality. – user37577 Aug 18 '22 at 20:09
  • @Joe I've not seen that before, that seems likely, would you have another example? – user37577 Aug 18 '22 at 20:44
  • @user37577 It is a common convention in the field of model theory. I can try to find other specific books, but I recall the convention is used without explanation in some proofs in David Marker’s model theory book. – Joe Aug 18 '22 at 23:00
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Suppose we have a formula $\phi(x)$ with some “unassigned variable” $x$. Formally, this $x$ is called a “free variable”. For example, your formula $\phi(x) : \equiv x + x= 2x$. Note I am using “$: \equiv$” to distinguish “$\phi$ equals …” from the ordinary equals sign $=$.

We can take that formula and turn it into a sentence with no free variables as follows. Consider the sentence “for all $x$, $\phi(x)$ is true”. This is denoted by “$\forall x \phi(x)$”. The upside-down A is called the “universal quantifier” because it “quantifies” over all possible values for $x$ in our “universe”. Our universe, in this sense, is all the values in our domain of discourse $D$. Maybe $D$ is the naturals or the reals.

This new sentence $\forall x \phi(x)$ has no free variables, so there’s nothing left to assign! We can talk about this sentence as being true or false without any concern. Saying the formula $\phi(x)$ is true is usually just a shorthand for saying the sentence $\forall x \phi(x)$ is true.

Joe
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  • So it is generally not common practice to give a statement with free variables a truth value? It's more of a shorthand? – user37577 Aug 19 '22 at 07:52