How to show disjunction can be expressed as a conditional

Question

How to show that formula "A ∨ B" can be constructed from A and B using only the conditional connective (→).

Answer 1

We can use the following definition of "and": A ∧ B ≡ ∀X [ (A → (B → X)) → X].

With it, we have:

A ∨ B ≡ ∀X [ (A → X) ∧ (B → X) → X].

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We can also adopt another approach, taking into account that in quantified propositional logic we have that the Flasum constant can be defined as: ⊥ ≡ ∀X.X.

With it we can define negation: ¬P ≡ (P → ⊥) ≡ P → ∀X.X.

Thus, for classical logic, we can translate P ∨ Q ≡ ¬P → Q ≡ (P → ∀X.X) → Q.

Finally: P ∧ Q ≡ ¬(P → ¬Q) ≡ (P → (Q → ∀X.X)) → ∀X.X, that is basically the first formula above.

Answer 2

~AvB is equivalent to A->B. So (~A)->B will be equivalent to AvB.

Answer 3

How to show disjunction can be expressed as a conditional

This is not possible. A disjunction cannot be expressed as a conditional.

This idea comes from the false notions about logic which are fundamental to mathematical logic. In particular, it is not true that ¬A ∨ B is equivalent to "If A, then B". The consequence is that "If ¬A, then B" is not equivalent to A ∨ B.

No expression containing only conditionals is equivalent to a conjunction.

This is why we have conditionals to begin with. If "If A, then B" was equivalent to ¬A ∨ B, we wouldn't need conditionals. We would say "A is false or B is true" instead of "If A, then B".

I would be interested if the downvoter could articulate what he objects to in my answer that justifies downvoting it, because every I here here is not only true, but common knowledge. You don't like the style?