How to show disjunction can be expressed as a conditional

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

• I'm not quite sure what you are asking. Are you asking for a proof that A v B is logically equivalent to ¬A -> B where -> is the material conditional? Commented Oct 19, 2021 at 18:41
• not a homework forum... Commented Oct 20, 2021 at 5:29
• By saying "how to show", are you asking for a formal proof, or just a simple illustration? A truth table would do the trick for the latter.
– Cam
Commented Nov 30, 2021 at 19:50

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].

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.

• This is good! I was going to comment that the problem with the conventional definition is that a replacement is also needed for the Law of Excluded Middle, but this definition gets around that. It does, however, require a language with propositional quantifiers. Commented Dec 3, 2021 at 13:29
• This page also defines it as a ∨ b ≡ ∀ c ((a → c) → (b → c) → c). Commented Dec 3, 2021 at 21:05
• @user76284 - fine... (P → (Q → R)) is equiv to (P ∧ Q) → R) Commented Dec 4, 2021 at 15:29

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

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?

• Hello: This comment has been flagged because of its brevity. Could you expand a bit? This would help the Questioner and other users. Best - Geoffrey. Commented Dec 2, 2021 at 14:32
• @Geoffrey Thomas The answer posted by MathematicalPhysicist is shorter than mine and you didn't comment on it that it had been flagged for being too short. Commented Dec 2, 2021 at 18:06
• I was only responding to a user's comment. flagged to the invigilators. If you want to leave you answer as it is, then I won't intervene further, but don't you think a little expansion would help the Questioner? Commented Dec 3, 2021 at 9:05