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

Geoffrey Thomas
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Hu Cares
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    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? – Bumble Oct 19 '21 at 18:41
  • not a homework forum... – Swami Vishwananda Oct 20 '21 at 05:29
  • 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 Nov 30 '21 at 19:50

3 Answers3

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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.

Mauro ALLEGRANZA
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  • 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. – Paul Ross Dec 03 '21 at 13:29
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    This page also defines it as a ∨ b ≡ ∀ c ((a → c) → (b → c) → c). – user76284 Dec 03 '21 at 21:05
  • @user76284 - fine... (P → (Q → R)) is equiv to (P ∧ Q) → R) – Mauro ALLEGRANZA Dec 04 '21 at 15:29
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~AvB is equivalent to A->B. So (~A)->B will be equivalent to AvB.

MathematicalPhysicist
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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?

Speakpigeon
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  • 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. – Geoffrey Thomas Dec 02 '21 at 14:32
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    @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. – Speakpigeon Dec 02 '21 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? – Geoffrey Thomas Dec 03 '21 at 09:05