# Justification for axiom OB-RE in deontic logic

Let OB p denote "p is obligatory". Axiom OB-RE is (p ↔ q) → (OB p ↔ OB q). This axiom seems false to me (under the interpretation of obligation). For example, let p denote "don't lie to me" and q denote "hurt me". Suppose p Λ q, that is, you're not lying to me but you are hurting me. Then it is true that p ↔ q, since ⊤ ↔ ⊤. Suppose furthermore that OB p, that is, you're obligated not to lie to me. Then OB-RE would imply OB q, that is, you're obligated to hurt me. This clearly doesn't seem right. Am I interpreting this axiom incorrectly?

• No. The entry states "If p ↔ q is a theorem". You left that crucial part out. p ↔ q is not a theorem in your example. – Eliran Dec 8 '18 at 0:54
• @Eliran How is p ↔ q not a theorem? Since p is true, q → p (any material conditional with a true consequent is true). Similarly since q is true, p → q. Therefore, p ↔ q. – user76284 Dec 8 '18 at 1:04
• A theorem in logic means that it can be derived without assumptions. In other words (and given completeness), p ↔ q is a theorem when p ↔ q is a tautology. That doesn't hold in your case. – Eliran Dec 8 '18 at 1:07
• – Eliran Dec 8 '18 at 1:08