Question

Deontic logic often contains the axiom

O(p → q) → (Op → Oq)
where O is being used for "it is obligatory that".

This axiom strikes me as odd. It reads "If it is obligatory that p implies q, then if it is obligatory that p, then it is obligatory that q. For example, let

p = X has the ability to save Y's life
q = X saves Y's life
With these definitions, we haves for the premise of the axiom:

A. It is obligatory that if X has the ability to save Y's life, then x saves Y's life.

According to the axiom, A implies B:

B. If it is obligatory that X has the ability to save Y's life, then it is obligatory that X saves Y's life.

Isn't that an odd conclusion? In this case, □p doesn't even make sense, and in general the obligatoriness of a the premise of an implication doesn't seem related to the obligatoriness of the conclusion. From A, shouldn't we instead conclude C?
C. If X has the ability to save Y's life, then it is obligatory that X saves Y's life.

Answer 1

The axiom O(p → q) → (Op → Oq) in deontic logic consistently applies the normative principles of logical analysis by treating the premise as a sufficient condition for an obligation to lead to the obligation of the consequent. The student's alternative conclusion expresses a similar relation but assumes the antecedent's truth rather than its obligatoriness.

Step-by-step explanation:

The student's question revolves around an axiom in deontic logic, which deals with obligation. The given axiom, O(p → q) → (Op → Oq), might seem counterintuitive at first glance when considering the obligatoriness of an action based on another's ability. However, upon closer inspection through the lens of logical analysis and the understanding of necessary and sufficient conditions, the axiom holds up.

In your example, 'A' is an obligation: If it is obligatory that 'if X has the ability to save Y's life (p), then X saves Y's life (q)', it serves as a sufficient condition for the obligation. If it is indeed an obligation for 'X to have the ability to save Y' (Op), then it follows that 'X saves Y's life' (Oq) is also an obligation. This follows from the idea that in conditionals, the antecedent is a sufficient condition for the consequent; thus, making the consequent a necessary condition if the antecedent is obligatory.

Your alternative conclusion 'C' actually expresses the same relationship but in a different structure. It assumes that 'X has the ability to save Y's life' is true, and thus it directly imposes the obligation 'X saves Y's life'. While the axiom in question may seem strange when applied to moral scenarios, it is logically consistent given the structure and norms of deontic logic.