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- W62329512 abstract "We show that for the adequate representation of some examples of normative reasoning a combination of different operators is needed, where each operator validates different inference rules. The combination of different modal operators imposes the restriction on the proof theory of the logic that a proof rule can be blocked in a derivation due to the fact that another proof rule has been used earlier in the derivation. In this paper we only use two operators and therefore we call the restriction the two-phase approach in the proof theory, which we formalize in two-phase labeled deontic logic (2ldl) and in two-phase dyadic deontic logic (2dl). The preference-based semantics of 2dl is based on an explicit deontic preference ordering between worlds, representing different degrees of ideality. The two different modal operators represent two different usages of the preference ordering, called minimizing and ordering. 1. Why deontic logic derivations must consist of two phases 1.1. Van Fraassen’s paradox Van Fraassen (1973) presents a logical analysis of dilemmas. In a logic that formalizes reasoning about dilemmas we cannot accept the conjunction rule, because it derives©(p∧¬p) from the dilemma©p∧©¬p, whereas ‘ought implies can’ ¬ © (p ∧ ¬p). On the other hand we do not want to reject the conjunction rule in all cases. For example, we want to derive ©(p ∧ q) from ©p ∧ ©q when p and q are distinct propositional atoms. That is, we have to add a restriction on the conjunction rule such that we only derive ©(α1 ∧α2) from ©α1 and ©α2 if α1 ∧ α2 is consistent. Van Fraassen calls the latter inference pattern Consistent Aggregation, which we write as the restricted conjunction rule (rand). He encounters a problem in the formalization of obliga2 L. VAN DER TORRE AND Y. TAN tions, and wonders if he needs a language in which he can talk directly about the imperatives as well. A variant of this problem is illustrated in the following example. Example 1 (Van Fraassen’s paradox). Assume a monadic deontic logic without nested modal operators1 in which dilemmas like ©p ∧ ©¬p are consistent, but which validates ¬©⊥, where ⊥ stands for any contradiction like p∧¬p. Moreover, assume that it satisfies replacement of logical equivalents and at least the following two inference patterns Restricted Conjunction rule (rand), also called consistent aggregation, and Weakening (w), where ↔ 3φ can loosely be read as φ is possible (or propositionally consistent). rand: ©α1,©α2, ↔ 3(α1 ∧ α2) ©(α1 ∧ α2) w: ©α1 ©(α1 ∨ α2) Moreover, assume the two premises ‘Honor thy father or thy mother!’ ©(f ∨ m) and ‘Honor not thy mother!’ ©¬m. The derivation of Figure 1 illustrates how the desired conclusion ‘thou shalt honor thy father’ ©(f ∨m) ©¬m ©(f ∧ ¬m) rand ©f w Figure 1. Van Fraassen’s paradox (1) ©f can be derived from the premises. Unfortunately, the derivation of Figure 2 illustrates that we cannot accept restricted conjunction and ©p ©(f ∨ p) w ©¬p ©(f ∧ ¬p) rand ©f w Figure 2. Van Fraassen’s paradox (2) weakening in a monadic deontic logic, because we can derive the counterintuitive obligation ©f from a deontic dilemma ©p ∧ ©¬p. The point of this paradox is that every ©(β), of which ©(f) is a special case, would be derivable. Van Fraassen asks himself whether restricted conjunction can be formalized, and he observes interesting technical questions. In this paper we pursue some of these technical questions. 2dl.tex; 15/06/2001; 17:08; no v.; p.2 TWO-PHASE DEONTIC LOGIC 3 ‘But can this happy circumstance be reflected in the logic of the ought-statements alone? Or can it be expressed only in a language in which we can talk directly about the imperatives as well? This is an important question, because it is the question whether the inferential structure of the ‘ought’ language game can be stated in so simple a manner that it can be grasped in and by itself. Intuitively, we want to say: there are simple cases, and in the simple cases the axiologist’s logic is substantially correct even if it is not in general – but can we state precisely when we find ourselves in such a simple case? These are essentially technical questions for deontic logic, and I shall not pursue them here.’ (van Fraassen, 1973) As far as we know, there is no discussion on Van Fraassen’s paradox in the deontic logic literature.2 We analyze Van Fraassen’s paradox in Example 1 by forbidding application of rand after w has been applied. This blocks the counterintuitive derivation in Figure 2 and it does not block the intuitive derivation in Figure 1, as we show below. Our formalization of two-phase reasoning works as follows. In the logic, the two phases are represented by two different types of obligations, written as phase-1 obligations ©1 and phase-2 obligations ©2 . The premises are phase-1 obligations, the conclusions are phase-2 obligations and the two phases are linked to each other with the following inference pattern rel." @default.
- W62329512 created "2016-06-24" @default.
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- W62329512 date "2000-01-01" @default.
- W62329512 modified "2023-09-26" @default.
- W62329512 title "Two-Phase Deontic Logic" @default.
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