Backward entailment

Question

An valid argument (p⊩c) is one where the premises (p) necessarily lead to the conclusion (c) , with truth table one check its validity by showing that p⟹c is a tautology ( ⊩p⟹c ) .in such manner we can deduce the conclusion B from the set of premises { A , A⟹B } by showing that ⊩ (A ∧ (A⟹B)) ⟹ B is tautology. My question is whether it is possible to proceed backward from a tautology to a valid argument

Answer 1

Let me rephrase:

One of the rules in logic is from a deduction $A \vdash B$ you get an inference $\vdash A \to B$. The question is can you get the deduction back from the inference?

The answer is yes. Here are the steps:

1. From $\vdash A \to B$ you get $A \vdash A \to B$ by weakening the empty premise.
2. $A \vdash A$.
3. Using Modus Ponens (MP) you get $A \vdash B$.

This is the Deduction Theorem. Sorry I didn't recognize this in my previous answer.

Answer 2

For a single tautology? Yes. Using one-half of the deduction meta-theorem we can move from a tautological conditional to a valid argument. Prefix notation comes as preferable for this discussion, since the type of tautology according to its main connective comes as relevant. We can move from some tautology $\vdash$ ($\rightarrow$ A B) to A $\vdash$ B (note A and B are not propositional variables).

For an arbitrary tautology? No. At least not in any straightforward way. There is no valid argument that any disjunction, conjunction, or equivalence translates into. For example, the tautology, in Polish notation,

E EEpqr EpEqr (equivalence associates)

doesn't have any straightforward translation into a valid argument. At least not if the equivalence connective 'E' comes as a primitive connective.