How to read an equivalence of open sentences

Question

It is well-known that $(x+1)(x-1)=0 \iff x=-1~\text{or}~x=1$. Does this mean that the open sentences $(x+1)(x-1)=0$ and $x=-1~\text{or}~x=1$ are equivalent because they have the same truth set $\{-1,1\}$?

Answer 1

Given two formulas $\phi,\psi$ with free variables in $X$ with $x\in X$ taking a value in the set $D_x$, they are equivalent if and only if for any valuation $\sigma : X \to \bigcup_{x\in X}D_x$ (that is $\sigma x \in D_x$), $\phi_\sigma \iff \psi_\sigma$, where $\xi_\sigma$ is $\xi$ in which each variable $x$ has been substituted by its value $\sigma x$.

Answer 2

• The statements $$\forall x\;\;(x+1)(x-1)=0$$ and $$\forall x\;\;\big(x=-1\:\:\text{or}\:\: x=1\big)$$ have the same truth value (false), so they are equivalent to each other; i.e., $$\forall x\;\;(x+1)(x-1)=0 \iff \forall x\;\;\big(x=-1\:\:\text{or}\:\: x=1\big).\tag A$$
• The open formulae $$(x+1)(x-1)=0\tag1$$ and $$x=-1\:\:\text{or}\:\: x=1\tag2$$ have the same truth set, so, for each $x,$ they are equivalent to each other; i.e., $$\forall x\;\Big(x+1)(x-1)=0\iff \big(x=-1\:\:\text{or}\:\:x=1\big)\Big).\tag B$$

When $(1)$ and $(2)$ are said to be equivalent to each other (typically in the context of solving equations), the intended meaning is $(\mathrm{B})$—because the universal quantifier has been informally dropped—rather than the weaker $(\mathrm{A}).$

On the other hand, $$x=x\iff y=y$$ doesn't require a universal quantification because both open formulae $x=x$ and $y=y$ have a fixed truth value.