Is direction of logical consequence in FOL arbitrary?

Question

Wikipedia says about logical consequence:

A formula φ is a logical consequence of a formula ψ if every interpretation that makes ψ true also makes φ true. In this case one says that φ is logically implied by ψ.

But if φ and ψ are both true under some interpretations, then aren't they on equal footing? Why is one the logical consequence of the other?

In extension, if we have a set of expressions $S = \{X_{1}, X_{2}, X_{3}, X_{4}\}$ and this set is satisfied by an interpretation $I$, so that every expression $X$ in $S$ is satisfied by $I$, then couldn't we just choose the subset $S' = \{X_2, X_{3}\}$ and claim that $X_{1}$ and $X_{4}$ are logical consequences of $S'$? It seems to my (naive) eyes that all expressions in X stand in mutual entailment, which somehow seems wrong.

Answer 1

The point is that every interpretation making $\psi$ true, also has to make $\varphi$ true. This is not symmetric: maybe every interpretation making $\psi$ true also makes $\varphi$ true, but there are interpretations making $\varphi$ true which don't make $\psi$ true.

For instance, maybe $\varphi$ is "$\forall x\exists y R(x, y)$" and $\psi$ is "$\exists y\forall x R(x, y)$." Then $\varphi$ is a logical consequence of $\psi$; however, consider the structure with domain $\{1, 2\}$ and $R$ given by $R(a,b)\iff a\not=b$; then this structure satisfies $\varphi$ but not $\psi$.

Answer 2

Note that the quote says: if every interpretation that makes $\psi$ true, also makes $\phi$ true, then $\phi$ is a consequence of $\psi$. Thus it might be the case that there are interpretations that do not make $\psi$ true, but do make $\phi$ true. If such interpretations exist, saying that $\psi$ is also a consequence of $\phi$ does not make sense.