# Is the validity of the argument determined by implication or bi-conditional rule?

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?

• I recommend you read "Logic, Sets and Recursion" by Robert Causey to get a complete understanding about these things. – Eric Jul 26 '17 at 14:49
• Btw, what does "bi-conditionally agree" mean? – Eric Jul 26 '17 at 14:51
• Biconditional agreement is the last example on my post. Except it is a disagreement. Maybe not a formal expression is this... – MarkokraM Jul 26 '17 at 15:33
• Looks like a suitable book, Eric. From the perspective of a programmer especially. Shame it is not sold as an ebook... – MarkokraM Jul 26 '17 at 15:59

Your very own example demonstrates that you should indeed not check whether the biconditional form is a tautology.

And checking the conditional form for being a tautology does work.

Here is why:

An argument with premises $\varphi_1, \varphi_2, ... \varphi_n$ and conclusion $\psi$ is valid if and only if (definition validity)

It is impossible for $\varphi_1, ... \varphi_n$ to all be true while $\psi$ is false iff (semantics $\land$)

It is impossible for $\varphi_1 \land \varphi_2 \land ... \land \varphi_n$ to be true and $\psi$ be false iff (semantics $\rightarrow$)

It is impossible for $(\varphi_1 \land ...\land \varphi_n) \rightarrow \psi$ to be false iff (definition tautology)

$(\varphi_1 \land ...\land \varphi_n) \rightarrow \psi$ is a tautology