My question is simple and I guess that it might be false even though I do not explain anything.
Suppose that the problem is to show that two compound propositions are "logically equivalent".
The generic approach for the problem is to simplify those propositions using well known relations that are logically equivalent.
Question: Can we use the rule of inferences to show that two propositions are logically equivelent?
My answer is No. This is because the rule of inferences to verify a validity of arguments only considers "True" premises. However, to show logically equivalent, we need to consider all possible cases.
I tried to find counter examples, but it failed.
Could you give some counter examples? or Is it possible to use the rule of inferences to show the logically equivalent?
Best Answer
$φ→ψ$ and $ψ→φ$ both being valid means precisely that $φ↔ψ$ is valid.
In other words, knowing that $$φ→ψ\tag1$$ and $$ψ→φ\tag2$$ are valid arguments is knowing that $φ$ and $ψ$ are logically-equivalent sentences.
To be clear, this requires neither $φ$ nor $ψ$ to be true in the current interpretation; the inference rules don't care about this. If $φ$ and $ψ$ are indeed also true, then we specifically call $(1)$ and $(2)$ sound arguments.