I'm getting two different signals by reading about the validation of the logical argument. Others say that all premises should yield true and bi-conditionally agree with the conclusion. Others say that tautology should be inferred with the material implication (simple conditional).
Take this common Modus Ponens example:
$$ ¬P → Q\\ ¬P\\ -----\\ \therefore Q $$
Validity should be checked with the conjunction of the premises and possibly tautological material implication similar to this:
$$(((¬P → Q) ∧ ¬P) → Q) \space; True$$
Above would imply that the argument is valid, am I correct?
But if I use bi-conditional (if and only if), then the same case Modus Ponens argument validation "fails":
$$(((¬P → Q) ∧ ¬P) ↔ Q) \space; False$$
Could someone throw some clarity to this?