Giving a formula a truth value?

Question

Let's take a formula with 'free' variables $x+1=x+1-1$ in some texts I see that if we can write that the formula is true, for all $x$ in a domain of discourse $D$, we give the formula a constant truth value, so for the formula above over $R$, we can universally quantify it over the reals and in some examples the formula is given a value of true. This seems a little bit odd, because a formula like '$x+1=2$ being true is just a statement and doesn't really have much of a meaning until I define what $x$ actually is. Is this just a shorthand for not needing universal quantification. I.E. for any $x$ we know that we will have, $[P(x)]= 1$

Answer 1

It is a frequent practice to drop wide-scope quantifiers for a less crowded and more natural reading when the context is clear enough and allows the inferences to be carried out without taking trouble with explicit quantifier operations. So,

$\forall x\forall y\forall z\big((Rxy\wedge Ryz) \rightarrow Rxz\big)$ can be expressed as $(Rxy\wedge Ryz) \rightarrow Rxz$

and

$\exists x(x^{2}-4 = 0)$ as $x^{2}-4 = 0$.

However, if the reader is left grappling with ambiguity, then it is an expository fault.

These are open formulas by syntactic form, while semantically meant to be quantified. They should not be confused with "genuine" open formulas in which free variables occur on purpose. For example, consider the following sentence

$\text{The students }\underbrace{\text{who volunteered to participate in the experiment}}_{relative\: clause}\text{ are going to discuss the results}$

where an open formula represents the relative clause in some linguistic and philosophical formalisms. As this example suggests, some notions of logic whose applications may not be mathematically interesting can find significance in other fields.

A possible structural representation of the formulas in the questions are $f(x) = g(f(x))$ and $f(x) = c$, where $f$ and $g$ are term-forming functions, $c$ is an individual constant (hence, a referring term) and $=$ is the identity predicate. As such, they are well-formed formulas (in standard predicate logic) and their closures should be either clear from the context or supplied explicitly as mentioned in the examples above.