How to show precisely that the conditional definition of validity is equivalent to the standard semantic definition.

Question

My favorite definition of validity is :

A reasoning from premises P1, P2, P3...Pn to conclusion C is valid iff the corresponding material conditional : (P1&P2&P3...&Pn) --> C is valid ( in other words, iff the conjunction of the premises logically imply the conclusion).

I'm looking for a rigorous proof of the fact that this definition is equivalent to the standard one :

a reasoning from a set of premises Gamma to a conclusion C is valid iff in all interpretation making all the members of Gamma true, the conclusion C is also true.

My question amounts, it seems to me, to : "how to show that the " logical consequence" relation is identical to the " logical implication" relation?" ( I said " identical" but maybe I should have said " is the converse of").

Answer 1

You have to consider that the "standard" definition allows that the set $\Gamma$ is an infinite set of sentences, while a formula is a finite string: this mean that we cannot have a conditional with an infinite number of antecedents.

In addition, you have to consider that the first definition holds only if the logical language has the conditional connective (this is not mandatory) and if the conditional is defined truth-functionally, like in classical logic.

Having said that, consider for simplicity a set $\Gamma = \{ \alpha \}$.

Assume that we have $\alpha \vDash \varphi$ and assume - for contradiciton - that $\alpha \to \varphi$ is not valid.

This means that there is an interpretation $\mathscr I$ such that $\mathscr \nvDash \alpha \to \varphi$, i.e. such that $\mathscr \vDash \alpha$ and $\mathscr \nvDash \varphi$.

But this contradicts the assumption that $\alpha \vDash \varphi$.

Similar for the other direction.