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**The dynamic logic of permission.**
*(English)*
Zbl 0855.03008

Traditional accounts of deontic logic have been unable to do justice to disjunctive sentences such as “You may have either icecream or fruit pie”. Such free choice permission expresses that the agent is allowed to perform either of two (or more actions). Intuitively, it should satisfy the condition \(P(p \vee q)\to Pp\) (where \(P\) stands for “permitted”). This condition is for several reasons difficult to accommodate in traditional deontic logic. One such reason, mentioned by the author, is that standard deontic logic satisfies the postulate \(Pq\to P(p \vee q)\), and from these two postulates we can derive the absurd conclusion that \(Pq\to Pp\). Another reason, which is applicable to a wider range of deontic systems, is that if obligation and permission are interderivable in the usual way, i.e., \(Pp\leftrightarrow \neg O\neg p\), then if \(P(p\vee q)\to Pp\) is a theorem so is \(Op\to O(p \& q)\), which is clearly undesirable.

The author follows McCarty and Meyer in attempting to give a more reasonable account of free choice permission in terms of dynamic logic. To each action (atomic action symbol) there is assigned a set of finite sequences of states, that represents the ways of executing the action in question. A subset of the set of pairs of states is defined to be the set of “green” transitions. A sequence of states is green just in case every pair of consecutive states in that sequence is green.

Two permission operators are defined. According to one of them, an action \(\alpha\) is non-forbidden (written: \(\diamondsuit \alpha\)) if and only if at least one execution of \(\alpha\) is green. According to the other definition, an action \(\alpha\) is free-choice permitted (written: \(\pi (\alpha)\)) if and only if all executions of \(\alpha\) are green. The \(\pi\) operator satisfies the postulate \(\pi (\alpha \cup \beta) \leftrightarrow \pi (\alpha) \& \pi (\beta)\), which corresponds to \(P(p \vee q)\to Pp\). The system is axiomatized, using Segerberg’s axioms for propositional dynamic logic, and proofs of both soundness and completeness are obtained.

This is a technically competent paper, but some questions of interpretation need to be further discussed.

1. Contrary to McCarty, the author assumes that a sequence of transitions is green if and only if each of the transitions is green. This is problematic, since arguably an action as a whole may be permitted although it contains parts that would not be permitted in isolation. (This simplification of the formal system seems to have been performed in order to obtain the completeness result.)

2. Following McCarty, the author construes an action as free-choice permitted if and only if each execution of the action is permitted. This does not seem plausible. Most desirable results can be obtained in undesirable ways, and consequently we should expect that permitted actions normally have forbidden variants. (I may be free-choice permitted to enter your house either through the front door or through the back door. This does not mean that I am allowed to enter through the front door using an axe to break open the lock.)

3. It does not seem to the present reviewer that a free-choice permission to perform either of the actions \(\alpha\) and \(\beta\) can be adequately represented as an extensional property of the sentence \(\alpha \vee \beta\). If it could, then it should be the case that if \(\alpha_1 \vee \beta_1\) and \(\alpha_2 \vee \beta_2\) are logically equivalent, then it is (free-choice) permitted to perform either \(\alpha_1\) or \(\beta_1\) if and only if it is (free-choice) permitted to perform either \(\alpha_2\) or \(\beta_2\). This, however, is clearly not the case. (Let \(\alpha_1\) denote that I take an afternoon walk and carry an umbrella, \(\beta_1\) that I take an afternoon walk and do not carry an umbrella, \(\alpha_2\) that I take an afternoon walk and shoot a policeman and \(\beta_2\) that I take an afternoon walk and do not shoot a policeman.) One way to avoid this problem would be to apply the free choice permission operator to sets of actions rather than to actions; we could then in this case have (with a variation of the author’s notation) \(\pi (\{\alpha_1, \beta_1 \})\) but not \(\pi (\{\alpha_2, \beta_2\})\).

A final point: The author says on p. 469 that the new operators have been defined to avoid some of the deontic paradoxes, but then goes on to say that these operators are “not intended to have a direct correspondence to natural language constructions”. There seems to be a need for clarification here. The deontic paradoxes refer to problematic relationships between natural language and logical representations of natural language concepts. Therefore, a solution to these paradoxes will have to be connected in a reasonably precise way to natural language constructions.

The author follows McCarty and Meyer in attempting to give a more reasonable account of free choice permission in terms of dynamic logic. To each action (atomic action symbol) there is assigned a set of finite sequences of states, that represents the ways of executing the action in question. A subset of the set of pairs of states is defined to be the set of “green” transitions. A sequence of states is green just in case every pair of consecutive states in that sequence is green.

Two permission operators are defined. According to one of them, an action \(\alpha\) is non-forbidden (written: \(\diamondsuit \alpha\)) if and only if at least one execution of \(\alpha\) is green. According to the other definition, an action \(\alpha\) is free-choice permitted (written: \(\pi (\alpha)\)) if and only if all executions of \(\alpha\) are green. The \(\pi\) operator satisfies the postulate \(\pi (\alpha \cup \beta) \leftrightarrow \pi (\alpha) \& \pi (\beta)\), which corresponds to \(P(p \vee q)\to Pp\). The system is axiomatized, using Segerberg’s axioms for propositional dynamic logic, and proofs of both soundness and completeness are obtained.

This is a technically competent paper, but some questions of interpretation need to be further discussed.

1. Contrary to McCarty, the author assumes that a sequence of transitions is green if and only if each of the transitions is green. This is problematic, since arguably an action as a whole may be permitted although it contains parts that would not be permitted in isolation. (This simplification of the formal system seems to have been performed in order to obtain the completeness result.)

2. Following McCarty, the author construes an action as free-choice permitted if and only if each execution of the action is permitted. This does not seem plausible. Most desirable results can be obtained in undesirable ways, and consequently we should expect that permitted actions normally have forbidden variants. (I may be free-choice permitted to enter your house either through the front door or through the back door. This does not mean that I am allowed to enter through the front door using an axe to break open the lock.)

3. It does not seem to the present reviewer that a free-choice permission to perform either of the actions \(\alpha\) and \(\beta\) can be adequately represented as an extensional property of the sentence \(\alpha \vee \beta\). If it could, then it should be the case that if \(\alpha_1 \vee \beta_1\) and \(\alpha_2 \vee \beta_2\) are logically equivalent, then it is (free-choice) permitted to perform either \(\alpha_1\) or \(\beta_1\) if and only if it is (free-choice) permitted to perform either \(\alpha_2\) or \(\beta_2\). This, however, is clearly not the case. (Let \(\alpha_1\) denote that I take an afternoon walk and carry an umbrella, \(\beta_1\) that I take an afternoon walk and do not carry an umbrella, \(\alpha_2\) that I take an afternoon walk and shoot a policeman and \(\beta_2\) that I take an afternoon walk and do not shoot a policeman.) One way to avoid this problem would be to apply the free choice permission operator to sets of actions rather than to actions; we could then in this case have (with a variation of the author’s notation) \(\pi (\{\alpha_1, \beta_1 \})\) but not \(\pi (\{\alpha_2, \beta_2\})\).

A final point: The author says on p. 469 that the new operators have been defined to avoid some of the deontic paradoxes, but then goes on to say that these operators are “not intended to have a direct correspondence to natural language constructions”. There seems to be a need for clarification here. The deontic paradoxes refer to problematic relationships between natural language and logical representations of natural language concepts. Therefore, a solution to these paradoxes will have to be connected in a reasonably precise way to natural language constructions.

Reviewer: S.O.Hansson (Uppsala)