Tree method on propositional logic

Question

Why we cannot construct a tree to test logical equivalence for 2 formulas the same way we do with validity? Because if both formulas have the same logical truth negating one of them will close the tree right?

Answer 1

We can use the Truth Tree Method to test for logical equivalence.

2 wffs $\phi$ and $\psi$ are logically equivalent if, and only if, their valuations are equal, i.e., $v(\phi)=v(\psi)$. This means we can test $\vDash\phi\leftrightarrow\phi.$ in the usual way, i.e., see if $\lnot(\phi\leftrightarrow\psi)$ can be satisfied.

Proof

If $\phi$ is logically equivalent to $\psi$ then $\phi\vDash\psi$ and $\psi\vDash\phi$. We can apply the Semantic Deduction Theorem to both conjuncts, which gives us $\vDash\phi\to\psi$ and $\vDash\psi\to\phi$.

For reductio, assume $\vDash\phi\to\psi$ and $\vDash\psi\to\phi$, while $\nvDash\phi\leftrightarrow\psi$. If $\nvDash\phi\leftrightarrow\psi$ then $v(\phi)=1$ and $v(\psi)=0$, thus $v(\phi\to\psi)=0$, which contradicts $\vDash\phi\to\psi$, or $v(\phi)=0$ and $v(\psi)=1$, thus $v(\psi\to\phi)=0, which contradicts $\vDash\psi\to\phi$. Thus, $\vDash\phi\leftrightarrow\psi$.

That this is the case shouldn't be a surprise given that PL is sound and complete and a syntactic proof of logical equivalence demonstrates that $\phi\vdash\psi$ and $\psi\vdash\phi$.

As an aside, it can still be helpful to do 2 trees for particularly complex wffs that have a lot of branching, i.e., test $\phi\vDash\psi$ and $\psi\vDash\phi$. The reason for this is avoiding mistakes when reading off counterexamples/IPLIs.

Example$^1$:

1: https://www.umsu.de/trees/