Obligation as Weakest Permission: a strongly Complete Axiomatization

In (Anglberger et al., 2015, Section 4.1), a deontic logic is proposed which explicates the idea that a formula φ is obligatory if and only if it is (semantically speaking) the weakest permission. We give a sound and strongly complete, Hilbert style axiomatization for this logic. As a corollary, it is compact, contradicting earlier claims from Anglberger et al. (2015). In addition, we prove that our axiomatization is equivalent to Anglberger et al.’s infinitary proof system, and show that our results are robust w.r.t. certain changes in the underlying semantics. §1. Intro. In Roy et al. (2014, 2012) and Anglberger et al. (2015), a logic is developed for “obligation as weakest permission”.1 The semantics proposed in Anglberger et al. (2015) is meant to capture the deontic aspects of reasoning in strategic games, where we speak about properties of the best actions available to a given agent. Whereas usually in formal models of such games, actions and/or agents are modeled explicitly at the object level, the present logic only speaks about action tokens (which correspond to states in a Kripke-model) and action types (sets of action tokens). Let us explain this briefly—we refer to the cited works for a more elaborate discussion. Consider a situation in which an agent can choose from a number of distinct action tokens, where at least some of these are optimal. Whereas the agent is permitted to perform one of those optimal action tokens, his sole obligation (if there is one at all – mind this important caveat) is to perform one of the optimal action tokens. This means that the deontic operators O and P can be read as follows, where φ refers to an arbitrary action type: Oφ: “φ is the (only) action type that is obligatory”, or more elaborately: “an action token is optimal if and only if it is of type φ” Pφ: “if an action is of type φ, then it is optimal” Anglberger et al. moreover introduce an alethic modality , which they interpret as a universal modality. φ thus means that all available action tokens are of type φ. They then propose what they call a “minimal logic” 5HD for these three operators. However, as they argue, 5HD only captures one half of the notion of “obligation as weakest permission”. That is, if φ is obligatory, then the logic stipulates that φ is the weakest permitted action type. The converse does not hold: something can be the weakest permitted action type without being obligatory. Received: September 30, 2015. 1 In more recent work Dong and Roy (2015); Van De Putte (2015), the logic is compared to other constructions in deontic logic. c © Association for Symbolic Logic, 2016 370 doi:10.1017/S1755020316000034 OBLIGATION AS WEAKEST PERMISSION 371 In the fourth section of Anglberger et al. (2015), a brief discussion of this converse direction is given, and it is shown how this translates to the semantics of 5HD. Let us call the resulting logic 5HD∗; it will be defined in Section 2. It is argued in Anglberger et al. (2015) that 5HD∗ is not compact, and a proof system with an infinitary rule (R-Conv) is shown to be (weakly) sound and complete w.r.t. 5HD∗. The main aim of the present paper is to give a sound and strongly complete, Hilbert-style axiomatization for 5HD∗ (Section 3). As a corollary, this consequence relation is compact, contradicting the claims mentioned in the previous paragraph. We prove in addition that the proof system proposed by Anglberger et al. is equivalent to 5HD∗ (Section 4). Finally, we show that these results can be generalized to other, similar logics for “obligation as weakest permission” (Section 5). §2. Definitions. This section is meant to fix notation; it contains no new material. See Anglberger et al. (2015) for the original definitions and notation. We work with a modal propositional language, obtained by closing the set of propositional letters S = {p1, p2, . . .} and ⊥, under boolean connectives ¬,∨,∧,⊃,≡ and the unary operators , O, P . Call the resulting set of formulas W . We treat only ¬,∨,⊥, O, P, as primitive; ∧,⊃,≡ are defined in the usual way. In the remainder, let the metavariables φ,ψ, . . . range over arbitrary members of W and , , . . . over arbitrary subsets of W . DEFINITION 2.1. A strict deontic frame F is a quadruple 〈W, R , n P , nO〉, where W is a non-empty set (the domain of F), R = W × W , and n P : W → ℘(℘(W )) and nO : W → ℘(℘(W )) satisfy the following conditions (OR) If X ∪ Y ∈ n P (w), then X ∈ n P (w) and Y ∈ n P (w) (WP) If X ∈ nO(w) and Y ∈ n P (w), then Y ⊆ X (OP) If X ∈ nO(w) then X ∈ n P (w) (OC) If X ∈ nO(w), then X = ∅ (Conv) If X ∈ n P (w) and for all Y ∈ n P (w), Y ⊆ X, then X ∈ nO(w) If a frame obeys all the above conditions except (possibly) (Conv), it is just a deontic frame. A (strict) deontic model is a (strict) deontic frame F together with a valuation v that maps every propositional atom to a subset of the domain of F. DEFINITION 2.2. Let M = 〈W, R , nO , n P , v〉 be a (strict) deontic model and w ∈ W . M, w | ⊥ M, w | p iff w ∈ v(p) M, w | ¬φ iff M, w | φ M, w | φ ∨ ψ iff M, w | φ or M, w | ψ M, w | φ iff M, w′ | φ for all w′ ∈ R (w) M, w | Oφ iff ‖φ‖M ∈ nO(w) M, w | Pφ iff ‖φ‖M ∈ n P (w), where ‖φ‖M = {u ∈ W | M, u | φ}. DEFINITION 2.3. 5HD∗ φ iff for all strict deontic models M: if M, w | ψ for all ψ ∈ , then M, w | φ. 372 FREDERIK VAN DE PUTTE §3. Axiomatization of 5HD∗. DEFINITION 3.1. The set of 5HD∗-theorems is the closure of the set of all instances of the following axiom schemas (CL) All tautologies of classical propositional logic (S5 ) S5 for (EQO) (φ ≡ ψ) ⊃ (Oφ ≡ Oψ) (EQP ) (φ ≡ ψ) ⊃ (Pφ ≡ Pψ) (FCP) P(ψ ∨ φ) ⊃ (Pψ ∧ Pφ) (Ought-Perm) Oφ ⊃ Pφ (Ought-Can) Oφ ⊃ φ (Weakest-Perm) Oφ ⊃ (Pψ ⊃ (ψ ⊃ φ)) (Taut-Perm) P ⊃ O under the following rules: