Philosophy
The Logic of Deontic Modalities and Permission
Quick fact
In standard deontic logic, permission is not simply the absence of prohibition—it is logically equivalent to 'not obligatory that not' (Pφ ≡ ¬O¬φ), meaning that what is permitted is exactly what is not forbidden, revealing a subtle logical symmetry.
Why this is interesting
We say 'You may go' and 'You must stay' with ease, but how do we reason with such statements? What does it really mean for something to be permitted?
Read the full explanation
Understanding The Logic of Deontic Modalities and Permission
Deontic logic extends classical logic to handle normative concepts. The basic operators are O for obligation, P for permission, and F for prohibition. You can think of these as 'must', 'may', and 'must not'. The key is that they are interdefinable: something is permitted if and only if it is not the case that it is obligatory to do the opposite. For example, if it is not required that you stay silent, then you are permitted to speak. To give these a rigorous semantics, we use possible worlds. Imagine a set of possible worlds that are 'ideal' with respect to the norms under consideration. A proposition is obligatory if it holds in all of those ideal worlds; it is permitted if it holds in at least one of them. This gives a precise way to evaluate statements like 'You may take a cookie'—it means there is at least one ideal world where you take the cookie.
A deeper explanation
The mechanism of deontic modalities rests on an accessibility relation between worlds: a world w 'sees' a set of worlds that are deontically ideal relative to w. Permission is then defined as truth in at least one of these accessible worlds: Pφ is true at w iff there exists a world v accessible from w where φ holds. Obligation is truth in all accessible worlds: Oφ is true iff φ holds in every accessible world. This mirrors the diamond and box operators of modal logic. Consequently, P relates to O by Pφ ≡ ¬O¬φ, and similarly Fφ ≡ O¬φ. This elegant formalization, however, generates paradoxes. For instance, the Good Samaritan paradox: if a man is helped, he is obliged to be helped (O h). If he was robbed, then being robbed (r) entails being helped (r → h). From O r, one can derive O h, but the obligation to be helped seems to imply the obligation that a robbery occur, which is absurd. This shows that standard deontic logic can misrepresent the subtleties of permission and obligation, prompting alternative systems such as dyadic deontic logic or defeasible deontic reasoning.