Problems and results for logics about imperatives

Deviating from standard possible-worlds semantics, authors belonging to what might be called the ‘imperative tradition’ of deontic logic have proposed a semantics that directly represents norms (or imperatives). The paper examines possible definitions of (monadic) deontic operators in such a semantics and some properties of the resulting logical systems. 1. Imperative Semantics and Basic Operators Deontic logic, i.e. the logical analysis of obligation, permission and prohibition, is usually modelled by a possible-worlds semantics, in which among the set of worlds a deontic alternative or preference relation selects some as ‘ideal’ or ‘better’ compared to others. However, when there is an explicit set of given imperatives, a code of norms, or a number of specified tasks to complete for the agent, the appeal to some prohairetic notion for the meaning of ought seems out of place. Instead, what is obligatory then depends on whether it serves to satisfy the imperatives, avoid norm violations, or complete all the tasks. In what might be termed the imperative tradition of deontic logic, a number of authors have deviated from the standard approach by giving semantics that relates the meaning of deontic operators to an explicitly given set of norms or imperatives.1 The general idea behind imperative semantics is that to each command or imperative there is a descriptive sentence that describes what must hold iff this imperative is satisfied. If a set of imperatives is under In [8] I mention semantics of Kanger [14], Stenius [20], van Fraassen [21], and Alchourrón/Bulygin [1]. A more comprehensive list should also include Smiley [19], and Niiniluoto [17]. Smiley, in a concept later endorsed by Ruth Barcan Marcus [15] and motivated by an Andersonian [2] definition of a constant representing the satisfaction of all normative demands, uses a ‘normative code’ consisting of propositions to define OA as meaning that there is a finite number of propositions p1, ..., pn in this code such that (p1 ∧ ... ∧ pn → A) is a logical truth. Niiniluoto represents commands by a tuple containing a proposition p, where the truth of OA then depends on whether A is logically implied by one such p. 2 J. Hansen / Problems and Results consideration, the set of corresponding descriptive sentences is then used of in the definition of deontic operators. The proper representation of this set is controversial: Directly representing imperatives by a set of descriptive sentences, as Kanger [14] and Alchourrón/Bulygin [1] have done, makes it appear as if norms can somehow be reduced to factual statements. Others like van Fraassen [21], Niiniluoto [17] and myself [8], have more cautiously represented imperatives by a set of objects that refer to states of affairs or propositions, thereby following the doctrine that norms bear no truth values. For reasons of readability I skip this intermediate step here, and use a set I ⊆ L [BL] of descriptive sentences that – in the language of a basic logic BL – mean the sentences associated with some imperatives in the above sense. Deontic operators are then considered as giving information about the properties of this set of imperativeassociated descriptive sentences. The following definitions of operators O1−5, differing in strength, will be employed throughout: I |= O1A iff A ∈ I I |= O2A iff ∃B ∈ I :|=BL B ↔ A I |= O3A iff ∃B ∈ I :|=BL B → A I |= O4A iff ∃B1, ..., Bn ∈ I :|=BL (B1 ∧ ... ∧Bn)→ A I |= O5A iff I |=BL A O1A is true iff A is one of the sentences in I, i.e. A (literally) corresponds to what an imperative demands. O2A is true iff the truth of A is BL-logically a necessary and sufficient condition for the fulfillment of what one imperative demands.2 Similar to Niiniluoto’s [17] operator, O3A is true iff A is BL-implied by what some imperative demands. Similar to Smiley’s [19] operator, O4A is true iff a finite subset of imperative-associated sentences BLimplies A. Finally and similar to Alchourrón/Bulygin’s [1] operator, O5A is true iff A is BL-implied by the set of all such sentences. Note that while all of O1 − O4 presuppose the existence of some sentence in I, O5A can hold for (BL-valid) sentences A even when I contains no object. Discussing the general framework, I showed in [8] how a number of well-known systems of monadic and dyadic deontic logic can be reconstructed with respect to imperative semantics. The present paper examines some details and problems connected with monadic deontic logics facilitated by the above semantics. Sec. 2 gives logical systems for each of the above operators. Sec. 3 then examines systems that include several of these operators. In sec. 4 we take a look at possible semantic restrictions on the content of imperatives, and see what changes must be applied to logical systems to correspond to them. Sec. 5 examines more closely the definitions van Fraassen [21] proposed for O-operators, and provides a monotonic system to resemble his reasoning, and sec. 6 addresses a ‘sceptical’ definition of ought proposed by Horty [12] to deal with normative conflicts. Finally, in sec. 7 I give systems and semantics for extended languages that permit iterated and nested deontic operators of the types presented above. In the terminology of Brown [4] O is a “type 2” operator, whereas operators O3−5 would be “type 1”, their scope being a necessary condition for norm satisfaction only. J. Hansen / Problems and Results 3 2. Logics DL1−5 1 for Deontic Operators O 1−5 Let our basic logic be propositional logic PL. The alphabet of the language L [PL] has a set of proposition letters Prop = p1, p2, ..., truth-functional operators ‘¬’, ‘∧’, ‘∨’, ‘→’, ‘↔’ and brackets ‘(’, ‘)’. The set of sentences is defined as usual. ∧, ∨ in front of sets of sentences means their conjunction and disjunction, and e.g. ∧n i=1Ai further abbreviates {Ai, ..., An}. In the semantics, valuation functions v : Prop → {1, 0} define the truth of sentences A ∈ L [PL] as usual, ‖A‖ meaning the set of valuations v that make A true. > means an arbitrary tautology, and ⊥ an arbitrary contradiction. We suppose a sound and complete axiomatic system for PL. In this section, we examine deontic logics DL1 for operators O i, 1 ≤ i ≤ 5, defined above. The upper index in the system name (and the names of most axiom schemes) indicates the type of operator used, and the subscript indicates that the scope of Oi is L [PL], not a deontic language (in sec. 7 we give up this restriction). The alphabet of languages L [DL1], 1 ≤ i ≤ 5, is like L [PL] except for one additional operator Oi. L [DL1] is then the smallest set such that a) if A ∈ L [PL] then A ∈ L [DL1], b) if A ∈ L [PL] then OiA ∈ L [DL1], and c) if A, B ∈ L [DL1], so are ¬A, (A ∧B), (A ∨B), (A→ B), and (A↔ B). Note that we permit ‘mixed’ expressions like ‘p1 → Op2’. The axiomatic systems DL1 are then defined by the following clauses (we write `DLi1 A for A ∈ DL i 1): a) All L [DL1]-instances of PL-tautologies are in DL i 1. b) DL1 is closed under modus ponens and the following rule: (Ext1) If `PL A↔ B then `DLi1 O iA↔ OiB 2 ≤ i ≤ 5 c) For all A, B ∈ L [PL] (> being an arbitrary tautology): (M1) `DLi1 O i(A ∧B)→ (OiA ∧OiB) 3 ≤ i ≤ 5 (C1) `DLi1 (O iA ∧OiB)→ Oi(A ∧B) 4 ≤ i ≤ 5 (N1) `DLi1 O i> i = 5 As usual, a set of sentences Γ ⊆ L [DL1] is DL1-inconsistent iff there are A1,..., An in Γ, n ≥ 1, such that `DLi1 (A1 ∧ ... ∧ An) → ⊥, and Γ is DL i 1-consistent otherwise. A sentence A ∈ L [DL1] is DL1-derivable from a set Γ ⊆ L [DL1] (we write Γ `DLi1 A) iff Γ ∪ {¬A} is DL1-inconsistent. If ¬A is derivable from the empty set, A is called refutable. For the semantics, let v be as before, and I be a set of PL-sentences. The truth-definitions of DL1-sentences are relative to v and I, where the truth of a proposition letter is defined as before, and the truth of OiA is defined with respect to I as given in sec. 1, truth-definitions for Boolean operators being as usual. VerDLi1(I, v) denotes the subset of DL i 1-sentences defined true for I, v, and we write I, v |= A for A ∈ VerDLi1(I, v). A is DL i 1-valid (we write |=DLi1 A) iff A ∈ L [DL i 1] is true for arbitrary I, v, and Γ ⊆ L [DL1] DL1-entails A (we write Γ |=DLi1 A) iff I, v |= A for all I, v such that I, v |= B for all B ∈ Γ. If there is a pair I, 4 J. Hansen / Problems and Results v such that I, v |= B for all B ∈ Γ ⊆ L [DL1], we call Γ DL1-satisfiable. Note that Boolean valuations v are introduced just to give non-deontic formulae their usual interpretation, and the set of sentences I is not made relative to one such valuation. If imperatives are meant to arise in specific circumstances only, this is a matter of proper representation of (conditional) imperatives, for which here no means are provided. So for example p1 → Op2 states that either ¬p1 is true, or that p2 is what some imperative demands, regardless of the situation. Theorem 2.1. Each system DL1, 1 ≤ i ≤ 5, is sound and complete. Proof: Soundness is trivial with respect to the truth definitions employed. For completeness, we prove equivalently that if Γ is DL1-consistent then Γ is DL i 1-satisfiable: Let A1, A2,... be a fixed enumeration of L [DL1]. Let ∆ = ⋃ n ∆n, where ∆0 = Γ, and ∆n+1 = { ∆n ∪ {An+1} if this is consistent ∆n ∪ {¬An+1} otherwise . Clearly each ∆n must be DL1-consistent with either An+1 or ¬An+1, hence (i) ∆ is DL1consistent, (ii) for all A ∈ L [DL1] either A ∈ ∆ or ¬A ∈ ∆. Now we define: v = { 1 if p ∈ ∆ 0 otherwise I = { A | OiA ∈ ∆}. We prove I, v |= A iff A ∈ ∆ by induction on A. We only give the case for A = OiB, the others are trivial: Suppose OiB ∈ ∆. Then B ∈ I and for all operators Oi, I, v |= OiB. Suppose OiB / ∈ ∆, so ¬OiB ∈ ∆: i = 1: Assume I, v |= O1B, so B ∈ I, so O1B ∈ ∆, but then ∆ is DL1-inconsistent. i = 2: Assume I, v |= O2B, so there is a C ∈ I such that |=PL B ↔ C. So O2C ∈ ∆. But from (Ext1) and PL-completeness we have `DL21 O 2B ↔ O2C, so ∆ is DL1-inconsistent. i = 3: Assume I, v |= O3B, so there is a C ∈ I such that |=PL C → B, so O3C ∈ ∆. But from (Ext1), (M 3 1), and PL-completeness we have `DL31 O 3C → O3B, so ∆ is DL1-inconsistent. i = 4: Assume I, v |= O4B, so there is a non-empty finite set {C1, ..., Cn} ⊆ I such that |=PL (C1 ∧ ... ∧ Cn) → B. So { OC1, ..., O Cn } ⊂ ∆. But from (Ext1), (C1), (M1), and PL-completeness we obtain `DL41 ( OC1 ∧ ... ∧OCn ) → O4B, so ∆ is DL1inconsistent. i = 5: Assume I, v |= O5B, so I |=PL B. If I 6= ∅, then by strong completeness of PL there is a non-empty finite set {C1, ..., Cn} ⊆ I such that |=PL (C1 ∧ ... ∧ Cn) → B, so the r.a.a. is done as in case i = 4. If I = ∅ then `DL51 O 5B by (Ext1), (N 4 1), so ∆ is DL1-inconsistent. ut J. Hansen / Problems and Results 5 3. Combined Logics DL 1 for Operators O 1−5 The definitions of operators O1−5 all refer to a set I. Instead of having languages L [DL1] that make use of just one of these operators, it makes sense to permit mixed statements such as ‘Op1 ∧¬Op1’ which expresses that though p1 is not implied by what a single imperative demands, it is implied by a finite subset of of all imperative demands. So let Op be a subset of operator indices {1, 2, 3, 4, 5}, card(Op) > 1, and let L [DL 1 ] be like L [DL1] except that the alphabet contains all of Oi, i ∈ Op, and clause b) in the definition of sentences now reads b) if A ∈ L [PL] then OiA ∈ L [DL 1 ], for all i ∈ Op Let the description of systems DL 1 follow that of DL i 1, where the clause a) is now relative to the language L [DL 1 ], and clauses b), c) hold for all i ∈ Op satisfying the additional requirements. We have the following additional axiom schemes: (OiOj) ` DL 1 OiA→ OjA {i, j} ⊆ Op, i < j (O3O2) ` DL 1 O3⊥ → O2⊥ {2, 3} ⊆ Op (O4O3) ` DL 1 O4> → O3> {3, 4} ⊆ Op (O5O3) if 0PL A then `DLOp 1 O 5A→ O3> {3, 5} ⊆ Op (O5O4) if 0PL A then `DLOp 1 O 5A→ O4A {4, 5} ⊆ Op Theorem 3.1. All systems DL 1 are sound and weakly complete. Proof: Soundness is again trivial with respect to Oi-truth definitions. For weak completeness, we have to prove that if DL 1 A then ` DL 1 A. We assume 0DLF1 A so ¬A is not refutable in DL 1 . One disjunct δ in a disjunctive normal form of ¬A is then not refutable in DL 1 . From the non-deontic conjuncts of δ we obtain a valuation v that satisfies them. As for the deontic conjuncts, for all i ∈ Op let Oi = {B ∈ L [PL] | OiB is a conjunct of δ}, and Qi = {B ∈ L [PL] | ¬OiB is a conjunct of δ}. Let π1, π2 be proposition letters that do not occur in δ. We define I = O1 1 ∈ Op (a) ∪ {(B ∧ (π1 ∨ ¬π1)) | B ∈ O2} 2 ∈ Op (b) ∪ {(B ∧ π1) | B ∈ O3} 3 ∈ Op (c) ∪ {π2, (π2 → (B ∧ π1)) | B ∈ O4} 4 ∈ Op (d) ∪ {π2, (π2 → (B ∧ π1)) | B ∈ O5 and 0PL B} 5 ∈ Op (e) Obviously, for all B ∈ Oi, I, v OiB. We prove that for each B ∈ Qi, I, v ¬OiB: B ∈ Q1: If B ∈ I then B / ∈ O1 by PL-consistency of δ. But all other B ∈ I contain proposition letter π1 not occurring in any B ∈ Q1. B ∈ Q2: For all C ∈ I by clause (a) or (b), 2PL (B ↔ C) by PL-consistency of δ and axiom scheme (O1O2). All other C ∈ I derive some contingent formula π2 or (π2 → π1). Since B contains no occurrence of π1, π2, if PL B → C then PL ¬B. The C ∈ I generated by clauses (d), (e) all are contingently true or false, so if there is a C ∈ I such that PL ¬C it must be due to clause (c). But then O3C DL 1 -derives O2C by (O3O2) and O2C DL 1 -derives O 2B by (Ext1), so δ is DL Op 1 -refutable. 6 J. Hansen / Problems and Results B ∈ Q3: For all C ∈ I by clause (a),(b), or (c), 2PL (C → B) by PL-consistency of δ and axiom schemes (O1O3), (O2O3), (M1), (Ext 3 1). So only by virtue of clauses (d), (e) can there be a C ∈ I such that PL (C → B). But no single C generated by these clauses PL-implies B unless PL B. But then there is some conjunct O4C in δ, or some conjunct O5D with non-tautological D, from which DL 1 -derives O 3C by use of (O5O3) or (O4O3), (M1), (Ext 4 1). So δ is DL Op 1 -refutable. B ∈ Q4: If O4B is true, then I 6= ∅, Oi 6= ∅ for some i ∈ Op. By the construction PL ∧ I → B iff PL ∧ ( ⋃4 i=1O ∪ { C ∈ O5 |0PL C } ) → B. For any C in 4i=1Oi ∪ { C ∈ O5 |0PL C } we derive O4C by (OiO4), i ∈ Op, and O5B using (M1), (Ext1), so δ is DL 1 -refutable. B ∈ Q5: If PL B then δ is DL 1 -refutable by (N1), so again I 6= ∅, Oi 6= ∅ for some i ∈ Op. The proof relies on axiom schemes (OiO5), i ∈ Op, and is done as in case Q4. ut Theorem 3.2. No DL 1 , Op 6= {4, 5}, is strongly complete. Proof: The semantics of DL 1 for Op 6= {4, 5} is not compact, i.e. there are finitely satisfiable sets Γ ⊆ L [DL 1 ] such that Γ is not satisfiable: Suppose {i, j} ⊆ Op. We give non-satisfiable sets Γ and an If for each finite Γf ⊆ Γ such that If , v satisfies Γf , v being arbitrary. i = 1, j = 2: Let Γ = {¬O1A ||=PL A↔ p1 } ∪ O2p1 } . For any finite subset Γf of Γ there is a proposition letter π that does not occur in any A ∈ Γf . Each set Γf is satisfied by If = {((π ∨ ¬π) ∧ p1)} . i = 1, 2, j = 3: Let Γ = {¬OiA ||=PL A→ p1 ∪O3p1 } . Let Γf be any finite subset, and π be as before. Each set Γf is satisfied by If = {(π ∧ p1} . i = 1, 2, 3, j = 4, 5: Let Γ = {¬OiA |2PL A } ∪ Ojp1 } . Let Γf be any finite subset, and π be as before. Each set Γf is satisfied by If = {(π → p1), π} . ut Theorem 3.3. DL{4,5} 1 is (strongly) complete. Proof: We adapt the completeness proof of DL1 in theorem 2.1: ∆ is now defined with respect to an enumeration of DL{4,5} 1 -sentences and DL {4,5} 1 -consistency. I, v is constructed as for DL 4 1, in particular I = { A | O4A ∈ ∆} . The proof that I, v |= O4A iff O4A ∈ ∆ is as before. To prove that I, v |= O5A iff O5A ∈ ∆, assume first O5A ∈ ∆. Either O4A ∈ ∆, but then A ∈ I, and I |=PL A is trivial. Or O4A / ∈ ∆, then ¬O4A ∈ ∆, but then A must be tautological, for otherwise by use of (O5O4) the finite subset {¬O4A,O5A} ⊆ ∆ is DL{4,5} 1 -inconsistent. So |=PL A and I, v |= O5A is true. For the other direction assume O5A / ∈ ∆, so ¬O5A ∈ ∆. For r.a.a. assume I, v |= O5A, so I |=PL A. If I 6= ∅ then I, v |= O4A, so by completeness of PL O4A ∈ ∆, but then due to (O4O5) the set {¬O5A,O4A} is DL{4,5} 1 -inconsistent. If I = ∅ then A is tautological, so by using (N1), (Ext1) {¬O5A} is DL{4,5} 1 -inconsistent. ut J. Hansen / Problems and Results 7 4. DL1-Logics and Semantic Restrictions So far, a set of sentences has been used to model what a set of imperatives demands, there being neither restrictions on the size of I, nor restrictions on the logical type of sentences in I. But concerning the size of I, it may be argued that a deontic logic should be applied only if there is at least one demand to be considered, since reasoning about what one ought to do when nothing is explicitly obligatory seems a borderline case. Concerning the elements of I, it may be argued that imperatives that demand the logically impossible or logically necessary are not ‘proper’, and one may also want to use a ‘rationality restraint’ to the effect that what the imperatives demand should be (jointly) satisfiable. Consider the following restrictions: [R-0] I 6= ∅ (Non-Triviality) [R-1] ∀A ∈ I : 2BL ¬A (Excluded Impossibility) [R-2] I 2BL ⊥ (Collective Satisfiability) [R-3] ∀A ∈ I :2BL A (Excluded Necessity) Consider the logics DL1 defined in sec. 2 and the following additional axiom schemes: (XI1) if `PL ¬A then `DLi1 ¬O iA (D1) if `PL ¬(A1 ∧ ... ∧An) then `DLi1 ¬(O A1 ∧ ... ∧OAn) (XN1) if `PL A then `DLi1 ¬O iA Warranted by the following theorems, the table below lists, for languages L [DL1], 1 ≤ i ≤ 5, axiom systems sound and complete with respect to DL1-semantics conforming to any of the above restrictions individually (square brackets indicate weak completeness only): L [DL1] i = 1 i = 2 i = 3 i = 4 i = 5 [R-0] [DL1] DL i 1+(N i 1) DL 5 1 [R-1] DL1+(XI i 1) DL i 1 [R-2] DL1+(D i 1) [R-3] DL1+(XN i 1) [DL i 1] DL 5 1 Theorem 4.1. The following systems are sound and (strongly) complete: 1. DL1 + (N i 1) for DL i 1-semantics restricted by [R-0], i = 3, 4 2. DL1 for DL 5 1-semantics restricted by [R-0] 3. DL1 + (XI i 1) for DL i 1-semantics restricted by [R-1], i = 1, 2, 3 4. DL1 for DL i 1-semantics restricted by [R-1], i = 4, 5 5. DL1 + (D i 1) for DL i 1-semantics restricted by [R-2], i = 1, 2, 3, 4, 5 6. DL1 + (XN i 1) for DL i 1-semantics restricted by [R-3], i = 1, 2 7. DL1 for DL 5 1-semantics restricted by [R-3] Proof: Soundness is again obvious. The completeness proofs are forthcoming adaptations of that for the DL1-systems. When there is a specific axiom, it obviously ensures that the constructed I conforms to the given restriction. Consider cases 2,4,7 where there are no specific axioms: Case 2: O5> ∈ ∆ by (N1), so by the construction > ∈ I 6= ∅. 8 J. Hansen / Problems and Results Case 4: We construct ∆ as before, but now define I = {A | OiA ∈ ∆ and 2PL ¬A}. Suppose OiA ∈ ∆ and A / ∈ I, so |=PL ¬A. Then |=PL A→ p1, |=PL A→ ¬p1, so by (Ext1) we have Op,O¬p1 ∈ ∆, so {p1,¬p1} ⊆ I, so I |=PL A. The remainder is as before. Case 7: Let ∆ be as before, but now I = {A | O5A ∈ ∆ and 2PL A}. Suppose O5A ∈ ∆ and A / ∈ I, so |=PL A. But then trivially I |=PL A, and the remainder of the proof is as before. ut Theorem 4.2. DL1-semantics, i = 1, 2, is not compact if restricted by [R-0]. DL i 1-semantics, i = 3, 4 is not compact if restricted by [R-3]. Proof: To show that DL1-semantics, i = 1, 2, is not compact if restricted by [R-0], let Γ = {¬OiA | A ∈ L [DL1]}. To satisfy all of Γ, I must be empty, which is excluded by [R-0], so Γ is not satisfiable by [R-0]-restricted semantics. But any finite subset Γf ⊆ Γ is satisfied by If , v, where If = {π}, and π is a proposition letter that does not occur in any A ∈ Γf , v being arbitrary. For the proof that DL1-semantics, i = 3, 4 is not compact if restricted by [R-3], let Γ = {¬OiA |2PL A} ∪ { Oi>}. To satisfy all of Γ, I must contain tautologies only, which is excluded by [R-3], yet it also cannot be empty due to the expected truth of Oi>, so Γ is not satisfiable by [R-3]-restricted semantics. But any Γf is again satisfied by If , v, as described above. ut Theorem 4.3. The following systems are sound and weakly complete: 1. DL1 for DL i 1-semantics restricted by [R-0], i = 1, 2 2. DL1 for DL i 1-semantics restricted by [R-3], i = 3, 4 Proof: Soundness is again obvious. For weak completeness, suppose {A} is DL1-consistent. To demonstrate that there are I, v such that I, v A, we do a construction as in the proof of Theorem 3.1, i.e. form a disjunctive normal form of A, from the non-deontic conjuncts of a non-refutable disjunct obtain a valuation v that satisfies these, and from the deontic conjuncts construct sets Oi and Qi as before. Again, let π be a proposition letter not occurring in A. Case 1: Let I = { Oi, if Oi 6= ∅, {π}, otherwise. The construction ensures that I 6= ∅ (R-0), and for all B ∈ Oi, I, v OiB. If i = 1 and B ∈ Q1, then B / ∈ O1 by PL-consistency of {A}, and B 6= π since π does not occur in A, so B / ∈ I and I, v ¬O1B. If i = 2 and B ∈ Q2, then no C ∈ O2 is PL-equivalent to B due to (Ext1) and PL-consistency of {A}, and no formula not containing π is PL-equivalent to π. So there is no C ∈ I s.t. PL (B ↔ C), and I, v ¬O2B. Case 2: Let I =    Oi − {A ∈ L [PL] | PL A}, if this is non-empty, {π}, otherwise and if Qi ∩ {A ∈ L [PL] | PL A} = ∅, ∅, otherwise. J. Hansen / Problems and Results 9 Obviously I contains no tautological element (R-3). Suppose B ∈ Oi. If there is a nontautological C ∈ Oi, for all B ∈ Oi, tautological or non-tautological, I, v OiB. If there are tautological B ∈ Oi (only), there cannot be a tautological C ∈ Qi due to (Ext1) and PL-consistency of {A}, so π ∈ I and again I, v OiB. For each B ∈ Qi, I, v ¬OiB: B ∈ Q3: If there is a tautological C ∈ Q3, O3 = ∅ due to (Ext1), (M1) and PL-consistency of {A}, and I = ∅ by the above construction, so I, v ¬O3B. Otherwise, no C ∈ O3 PL-implies B again due to (Ext1), (M 3 1) and PL-consistency of {A}, and if π does then B is tautological, but this was excluded. B ∈ Q4: As before, if C ∈ Q4 is tautological then O4 = ∅ and I = ∅, so I, v ¬O4B. Otherwise, B is neither implied by any {C1, ..., Cn} ⊆ O4 due to (Ext1), (M1), (C1) and PL-consistency of {A}, nor by I = {π} since a tautological B was excluded. ut If the semantics employs more than one of the above restrictions, then generally there is a sound and (strongly) complete axiomatic system iff each of the semantics with just one of these restrictions has such a system, and the axiomatic system is obtained by including all axioms and rules of the systems for the singularly restricted semantics. However, this rule fails in one case: Theorem 4.4. DL1-semantics is not compact if restricted by [R-0], [R-3]. Proof: To show incompactness, let Γ = {¬O5A |2PL A}. To satisfy all of Γ, I cannot contain anything but tautologies, which is excluded by [R-3], yet it also cannot be empty due to [R-0], so Γ is not satisfiable by [R-0][R-3]-restricted semantics. But any finite subset Γf is again satisfied by If , v, where If = {π}, π being as before, and v being arbitrary. ut Theorem 4.5. DL1 is sound and weakly complete for [R-0][R-3]-restricted DL 5 1-semantics. Proof: Soundness is again obvious. For completeness we assume that {A} is DL1-consistent and construct an I, v such that I, v A as in the proof of theorem 4.3, now letting I = { Oi − {A ∈ L [PL] | PL A}, if this is non-empty, {π}, otherwise Obviously I satisfies [R-0] and [R-3]. For all B ∈ O5, if B is non-tautological then B ∈ I and I, v O5B, and trivially so if B is tautological. Suppose B ∈ Q5: B cannot be tautological due to (N1) and PL-consistency of {A}, so also B cannot be PL-implied by {π}, so if I PL B by completeness of PL there must be some {C1, ..., Cn} ⊆ O5 such that `PL ((C1 ∧ ...∧Cn)→ B), but this is excluded by (Ext1), (M1), (C1) and PL-consistency of {A}. ut 10 J. Hansen / Problems and Results 5. Van Fraassen’s Imperative Logic Van Fraassen [21] discusses three truth definitions of O-operators with respect to given imperatives. For all these operators, axiom scheme (C) does not hold, for van Fraassen is concerned with possibly conflicting imperatives, and argues that when there is a demand for A, and a demand for ¬A, we should admit the truth of OA ∧ O¬A, but the derivation of O(A ∧ ¬A) should be blocked. In van Fraassen’s semantics, to each imperative i there is a class of possible outcomes i+ in which i is fulfilled. His first definition of an O-operator reads: [Df-F1] OA is true iff, for some imperative i that is in force, i+ is part of the set of possible outcomes in which A is true. In the terms used here, van Fraassen’s definition may be reformulated as [Df-F1*] I |= OA iff ∃B ∈ I :|=PL B → A Since van Fraassen also presupposes that no single imperative may be impossible to fulfill, van Fraassen’s first logic coincides with DL1+(XI 3 1) above. However, van Fraassen thinks definition [Df-F1] too ‘simple minded’, for two reasons: For one, imperatives are conditional, and according to van Fraassen a conditional imperative can be fulfilled or violated only if its condition is the case. His second logic therefore introduces conditional imperatives and a corresponding dyadic O-operator. I leave this definition aside, since we are not concerned with conditional imperatives here. The other argument against [Df-F1] is that it should be allowed to draw conclusions from imperatives that do not conflict, e.g. from two imperatives that demand p1 ∨ p2 and ¬p2 respectively conclude that p1 is obligatory. But since there is no single imperative that demands p1, [Df-F1] does not allow the derivation of Op1. Van Fraassen therefore introduces the notion of a score of an outcome v, i.e. the set of imperatives that v fulfills, and defines: [Df-F3] OA is true iff there is a possible state of affairs v in ‖A‖ whose score is not included in the score of any v′ in ‖¬A‖. Again, this definition may be reformulated in the terms used here: [Df-F3*] I |= OFA iff ∃v ∈ ‖A‖: ∀v′ ∈ ‖¬A‖: {B ∈ I | v |= B} * {B ∈ I | v′ |= B} Let the set of maximally PL-consistent subsets I −⊥ of I be the set of subsets I ′ ⊆ I such that (i) I ′ 0 ⊥, and (ii) there is no I ′′ such that I ′ ⊂ I ′′ ⊆ I and I ′′ 0 ⊥. It can then be proven that [Df-F3*] is equivalent to [Df-F3**] (cf. Horty [12] p.30 Th. 2): [Df-F3**] I |= OFA iff ∃I ′ ∈ I −⊥: I ′ `PL A Let the language L [DL1 ] be like any of the L [DL i 1], except that O F replaces Oi in all DL1-sentences. Let the axiomatic system DL F 1 be like DL 3 1+(XI 3 1)+(N 3 1), except that O F replaces O3. Let DL1 -semantics be defined like DL 3 1-semantics, except that truth definition [Df-F3**] replaces that for O3. We then have the following results: Theorem 5.1. DL1 is sound and weakly complete. J. Hansen / Problems and Results 11 Proof: Concerning soundness, the validity of (Ext11 ), (M F 1 ), (XI F 1 ) and (N F 1 ) is immediate. Concerning (weak) completeness, we have to prove that if DL1 A then `DLF1 A. We assume 0DLF1 A so ¬A is not refutable in DL F 1 . As before (cf. Theorem 3.1), let δ be a non-refutable disjunct in a disjunctive normal form of ¬A. From its non-deontic conjuncts we again obtain a valuation v that satisfies these. Concerning the deontic conjuncts of δ, let OF be the set of PL-sentences B such that OFB is a conjunct of δ, and QF be the set of PL-sentences B such that ¬OFB is a conjunct of δ. For any set Γ ⊆ L [PL] let minPLΓ = {A ∈ Γ | ∀B ∈ Γ : if `PL B → A then `PL B ↔ A} Let n = card(minPLO ). Let {π1, ..., πn} ⊂ Prop be an arbitrary set of n proposition letters not occurring in δ, and σ be a function that maps minPLO onto the set {π1,¬π1 ∧ π2, ...,¬π1 ∧ ...¬πn−1 ∧ πn} of n mutually exclusive PL-sentences. We then define: I = {B ∧ σ(B) | B ∈ minPLO } Note that I − ⊥ = {{i} | i ∈ I}, since σ(B) is PL-inconsistent with any σ(C), B,C ∈ minPLO and B 6= C. No B ∈ OF is a contradiction due to (XI1 ), and for each consistent B ∈ OF the definition of minPLO ensures that there is an I ′ ∈ I − ⊥ such that I `PL B, so OFB is true. Suppose B ∈ QF but I, v OFB, so there is an I ′ ∈ I − ⊥ : I ′ `PL B. Then there is a C ∈ minPLO such that I ′ = {C ∧ σ(C)} and `PL C ∧ σ(C) → B. Since σ(C) contains no sentence letters occurring in either B or C, `PL C → B. But then ¬OFC DL1 -derives from ¬OFB by (M1 ), (Ext1 ), so δ is DL1 -inconsistent. ut Theorem 5.2. There is no strongly complete axiomatic system DL1 . Proof: Let Γ = {OF p1, OF p2,¬O (p1 ∧ p2),¬O¬p1,¬O¬(p1 ∧ p2),¬O¬(p1 ∧ p2 ∧ p3), ...}. Γ is finitely satisfiable: Let π be a proposition letter that does not occur in some sentence of a finite Γf ⊂ Γ. Then I = {p1 ∧π, p2 ∧¬π} satisfies Γf . However, Γ is not satisfiable: Suppose there is a set I that satisfies Γ. OF p1, OF p2 ∈ Γ, so there are PL-consistent finite sets Γ1,Γ2 ⊆ I that derive p1 and p2 respectively. Γ1 ∪ Γ2 is inconsistent, since otherwise there is some I ′ ∈ I −⊥ such that Γ1 ∪Γ2 ⊆ I ′ and I ′ ` p1 ∧ p2, so I would not satisfy ¬OF (p1 ∧ p2). Now 0PL ( ∧ Γ1 ∧ p1 ∧ ... ∧ pn) → ⊥ for any n ≥ 1, for otherwise Γ1 `PL ¬(p1 ∧ ... ∧ pn) would contradict ¬O¬(p1∧...∧pn). The same holds for Γ2. Let n be the highest index of proposition letters occurring in Γ2. Since Γ2 ∪ {p1, p2, ..., pn} is PL-consistent, i.e. (p1 ∧ ... ∧ pn) is the first disjunct in the disjunctive normal form of ∧ Γ2, we have `PL ( ∧ Γ2 ∧ p1 ∧ ... ∧ pn) ↔ (p1 ∧ ... ∧ pn). So we obtain 0PL ( ∧ Γ1 ∧ ∧ Γ2 ∧ p1 ∧ ... ∧ pn) → ⊥. But Γ1 ∪ Γ2 was PLinconsistent. ut After explaining that semantics based on [Df-F3] satisfies the ‘basic criteria’ (the axiom schemes of DL1+(XI 3 1), van Fraassen notes that in the semantics we have the following additional truth: If there are two imperatives i1, i2 in force, and if there is a v that belongs to i1 and also to i + 2 , then if [all such] v ∈ ‖A‖, OFA is true. He then asks: 12 J. Hansen / Problems and Results “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.” If what van Fraassen means by ‘the logic of the ought-statements alone’ is a monotonic axiomatic system, then van Fraassen’s first question deserves a negative answer: There is a logic of ought-statements DL1 that is weakly complete, but the only additional truth if compared to [Df-F1] is OF>.3 In particular, DL1 does not provide the inferences van Fraassen desires. And Theorem 5.2 shows that there is no strongly complete axiomatic system in terms of the O-operator used. But, regarding his additional question, it seems to me that the “ought language game” extends to operators like O1 or O2 that more “directly talk about imperatives”. So perhaps a more powerful language can provide the missing inferences. It is immediate that the counterexample used in Theorem 3.2 to disprove compactness of DL{2,4} 1 also disproves compactness of any semantics that employs operators of type O1 or O2 in addition to van Fraassen’s operator OF . So for an improved characterization of van Fraassen’s third semantics it must suffice to give a weakly complete axiomatic system: Let L [DL{2,F} 1 ] be the language that is like L [DL {2,3} 1 ] except that O F replaces O3 in all DL{2,3} 1 -sentences. For the semantics, the truth of O 2, OF is defined with respect to a set I as above, and for proposition letters and Boolean connectives it is defined as usual. Let the axiomatic system DL{2,F} 1 contain all instances of L [DL {2,F} 1 ] in PL-theorems, be closed under modus ponens and (Ext1), (Ext F 1 ), contain all L [PL]-instances in the axiom schemes (M1 ), (XI F 1 ), (N F 1 ) of DL F 1 , and additionally contain all L [PL]-instances in the following schemes: (F-1) If 0PL ¬ ∧n i=1Ai then `DL{2,F} 1 ∧n i=1O Ai → OF ∧n i=1Ai (F-2) `DL{2,F} ( ∧n i=1O Ai ∧OFB ∧ ¬OF (B ∧ ∧n i=1Ai))→ OF (B ∧ ¬ ∧n i=1Ai) According to (F-1), agglomeration of contents of O2-obligations is admissible if these are jointly PL-consistent.4 (F-2) then states that if O2-contents may not be added to the content of an OF -obligation, then there is an obligation to the contrary that goes with that content. So (F-1) and (F-2) now properly define the ‘simple cases’, where agglomeration of contents is admissible. Theorem 5.3. DL{2,F} 1 is sound and weakly complete. Van Fraassen seems to have overlooked the additional truth of O>. To avoid its truth, the O-type definition [Df-F3**] may be changed into a more O-type definition: I |= OA iff ∃I ′ ∈ I −⊥: ∃B1, ..., Bn ∈ I ′ |=BL B1 ∧ ... ∧Bn → A It can then be shown that DL1+(XI 3 1) is sound and weakly complete with respect to the changed definition, and the counterexample in Theorem 5.2 again disproves compactness. A difficulty in axiomatizing van Fraassen’s semantics was pointed out by Horty [11] p. 50: If background imperatives are coded into ought-statements, as they are here using operator O, then the set {OA | A ∈ L [BL]} must derive all BL-consistent sentences, though for some basic logics like first order logic these are not recursively enumerable. But then due to (F-1) the set of DL {2,F} 1 -axioms is not decidable and DL {2,F} 1 not recursively enumerable, so for such basic logics no axiomatization in the usual sense is provided here. J. Hansen / Problems and Results 13 Proof: For soundness of the new axiom schemes, validity of (F-1) is immediate from the fact that if there are B1, ..., Bn ∈ I equivalent to A1, ..., An respectively, and A1, ..., An are PL-consistent, then some I ′ ∈ I −⊥ contains {B1, ..., Bn}, so I ′ `PL A1 ∧ ...∧An. For the validity of (F-2), suppose OA1, ..., OAn. If ¬OF (B ∧ ∧n i=1Ai), then no I ′ ∈ I −⊥ may derive B and all Ai, so if OFB is true and there is an I ′ that derives B, it must be inconsistent with ∧n i=1Ai and consequently derive B ∧ ¬ni=1Ai. For (weak) completeness, we have to prove that if A is DL{2,F} 1 -valid, then A ∈ DL{2,F} 1 . We assume A / ∈ DL{2,F} 1 , so ¬A is not refutable in DL{2,F} 1 . As now usual, let δ be a nonrefutable disjunct in a disjunctive normal form of ¬A. From its non-deontic conjuncts we again obtain a valuation v that satisfies them. As for the deontic conjuncts of δ, first let O2 = {B ∈ L [PL] | O2B is a conjunct of δ} Q2 = {B ∈ L [PL] | ¬O2B is a conjunct of δ} Let L [PL]δ be the set of PL-sentences that contain only proposition letters occurring in some deontic conjunct of δ. Let r(L [PL]δ) be a set of 22 n mutually non-equivalent representatives of L [PL]δ, n being the number of proposition letters occurring in the deontic conjuncts of δ. By writing PL-sentences we now mean their unique representatives in L [PL]∆. We construct a finite set ∆ with the following properties: a) If OFB,¬OFC are conjuncts of δ, then OFB,¬OFC are in ∆. b) For all B ∈ r(L [PL]δ), either OFB ∈ ∆ or ¬OFB ∈ ∆. c) {δ} ∪∆ is DL{2,F} 1 -consistent. In a), we may have to replace B,C by r(L [PL]δ)-equivalents using (Ext1 ). We now define: OF = {B ∈ L [PL] | OFB ∈ ∆} QF = {B ∈ L [PL] | ¬OFB ∈ ∆} Lemma 5.1. For all B ∈ O2: (a) if 0PL ¬B then there is a C ∈ minPLO such that `PL C → B, and (b) for all C ∈ minPLO , if {B,C} is PL-consistent, then `PL C → B. Proof: (a) is immediate from (F-1) and the definition of minPLO (cf. the proof of Theorem 5.1). Concerning (b), either OF (B ∧ C) ∈ ∆ or ¬OF (B ∧ C) ∈ ∆ (clause b in the construction of ∆). If OF (B ∧C) ∈ ∆ then `PL C ↔ (B ∧C) by definition of minPLO , so `PL C → B. If ¬OF (B∧C) ∈ ∆ then O2B,OFC,¬OF (B∧C) DL{2,F} 1 -derives OF (¬B∧C) by use of (F-2), so OF (¬B ∧C) ∈ ∆ by clause (c) in the construction of ∆. But then `PL C ↔ (¬B ∧C) by definition of minPLO , so {B,C} is not PL-consistent. ut Let σ : minPLO → L [PL] be the function defined in the proof of Theorem 5.1. Let I = O ∪ {B ∧ σ(B) | B ∈ minPLO } Lemma 5.2. For all I ′ ∈ I − ⊥, B ∈ L [PL]δ: if I ′ 6= ∅ and I ′ `PL B then there is a C ∈ minPLO such that `PL C → B. 14 J. Hansen / Problems and Results Proof: Immediate from Lemma 5.1, the construction of I, and the fact that σ(C) contains no proposition letters occurring in B or C. ut It remains to prove that I satisfies all deontic formulas in δ and ∆: O2: For all B ∈ O2 we have B ∈ I, so O2B is true. Q2: For all B ∈ Q2, if C ∈ I and PL B ↔ C, then C / ∈ O2, for otherwise δ is refutable by (Ext1). All other C ∈ I derive some πi not occurring in δ, but no B ∈ Q2 does. OF : Unless B ∈ OF is a contradiction, which is excluded by (XI1 ) and DL{2,F} 1 -consistency of ∆, by definition of minPLO and the construction of I there is a I ′ ∈ I − ⊥ such that I ′ `PL B, so OFB is true. QF : For all B ∈ QF , there is no I ′ ∈ I − ⊥ such that I ′ `PL B: For otherwise there is a C ∈ minPLO such that `PL C → B (Lemma 5.2). Then OFC ∈ ∆, and ¬OFC ∈ ∆ due to (Ext1 ), (M F 1 ), so ∆ is DL {2,F} 1 -inconsistent. ut 6. Saving a Twin Without Guilt The Sceptical Ought O In the long-standing philosophical dispute on the existence of moral dilemma and conflicting obligations, Barcan Marcus [16] provided the following Buridan’s ass type example: Identical twins are in danger of being crushed to death by a rock. They are pinned down such that only one can be pulled free at a time. If nothing is done, the rock will soon kill them both, but if either twin is removed this will cause the rock to slide and kill the other. A mountain guide is liable for the lives of both twins. What are her obligations? As the example is set up, it suggests the co-existence of two conflicting obligations, i.e. the guide has to save one twin (T1), and also save the other (T2), T1 contradicting T2. In logics DL1 and DL 5 1 we have O T1 and OT2, and also O4/5 (T1 ∧ T2) . But then a contradiction is obligatory, so we cannot also accept the principle ‘ought implies can’ expressed by (XI 1 ). In van Fraassen’s logic we maintain OT1 and OT2, and derivation of OF (T1 ∧ T2) is blocked by the absence of (C1 ) and inconsistency of T1 ∧ T2 (axiom F-1). However, in addition we have O¬T1 and O¬T2: by saving one twin the mountain guide will violate her duties towards the other. So why not simply walk away? Conee [5], Donagan [6], and Brink [3] have argued that morally there is no conflict: according to Conee, either act is permitted and neither absolutely obligatory, while for Donagan and Brink there only exists an obligation to save one twin or the other, but not two conflicting obligations to save either. Deontically, the dilemma was examined by Jacquette [13] and Horty [12]. Jaquette points out that van Fraassen’s approach is unsatisfactory, since it does not hold that an obligatory act is also permitted: OFA→ PFA is not a truth in DL1 , with PFA defined as ¬OF¬A. So according to van Fraassen’s notion of ought, either twin must be saved, but can only be saved at the price of guilt (for doing something forbidden). As an alternative to van Fraassen’s logic, Horty, in accord with Donagan’s and Brink’s disjunctive proposal, considers a ‘sceptical theory’ of obligations, where something can be obligatoryS only if there is no consistent set of norms that demand the contrary. With respect to our semantics, Horty’s definition reads: J. Hansen / Problems and Results 15 [Df-S] I |= OSA iff ∀I ′ ∈ I −⊥: I ′ `PL A So A is obligatory iff all maximally consistent subsets of imperative-associated sentences derive A.5 Let L [DL1 ] be like any of the L [DL i 1], except that operator O S replaces Oi. Let the truth of DL1 -sentences be defined with respect to [Df-S] and the usual definitions for proposition letters and Boolean connectives. Let DL1 be like DL 5 1 + (XI 5 1). We then obtain: Theorem 6.1. DL1 is sound and (strongly) complete. Proof: Immediate. For completeness, ∆ is constructed just as in the proof of Theorem 2.1. ut The result is welcome at first, for it seems that standard deontic logic, including agglomeration (C1 ) and ‘ought implies can’ (XI S 1 ), is the proper tool to represent the sceptic notion of ought. However, the problem is that now all the problems have disappeared: Conflicting imperatives and their impact on what is “sceptically obligatory” have become imperceivable in DL1 . To see what goes on behind the curtain, and illustrate the relation of sceptically defined oughts with van Fraassen’s logic, we must again admit the use of more than one O-operator: Let the language L [DL{2,F,S} 1 ] be like L [DL {2,F} 1 ] except that we have the additional operator OS . The truth of DL{2,F,S} 1 -sentences is defined as for DL {2,F} 1 -sentences with the additional truth definition [Df-S]. Let the system DL{2,F,S} 1 contain all axiom-schemes and rules of DL{2,F} 1 and DL S 1 , and additionally the following: (FS-1) (OFA ∧OSB)→ OF (A ∧B) (FS-2) ( ∧n i=1O Ai ∧ ¬OF¬( ∧ Γ1 → ∨k j=2 ∧ Γj))→ OS( ∧ Γ1 → ∨k j=2 ∧ Γj) (FS-3) ( ∧n i=1O Ai ∧ ¬OF¬ ∨k j=1 ∧ Γj)→ OS ∨k j=1 ∧ Γj (FS-4) ( ∧n i=1O Ai ∧ ¬OF ∧n i=1Ai)→ OS¬ ∧n i=1Ai In FS-2 and FS-3, Γj is a nonempty subset of {A1, ..., An}. Note that (FS-3), (FS-4) are special cases of (FS-2). (FS-3) derives (O2A∧O2B∧¬OF¬(A∨B))→ OS(A∨B), which is a syntactic version of the disjunctive solution to Marcus’s dilemma: the disjunction (A ∨B) of two possibly conflicting imperative demands A, B is obligatory, unless there is a consistent set of imperatives that demands even ¬A∧¬B. Hence one twin or the other must be saved. From (FS-1) and (XI1 ) we derive O FA → ¬OS¬A, so what is obligatory according to van Fraassen’s ought is at least ‘permitted’ in the sceptical sense. If guilt is a notion that attaches to a violation of a ‘sceptical ought’ only, then one may save a twin without guilt, but walking away remains forbidden. From the results of sec. 5 it is immediate that DL{2,F,S} 1 cannot be strongly complete. However we obtain the following result: It might seem awkward that if I contains contradictory imperative demands, e.g. if I = {p1,¬p1, p2} then with [Df-S] the set of truths is the same as for I ′ = {p2}, i.e. the same as for an I ′ without these demands. If the fulfillment of one of the demands is to remain obligatory, an alternative would be to have a “disjunction of oughts” instead of the “ought of a disjunction”, and define: [Df-DS] VerDS(I, v) := {A ∈ L [DL1] | ∀I ′ ∈ I −⊥: A ∈ DL51 ′, v)} In the above example, Op1 ∨O¬p1 is now true under the new definition, O (p1 ∨ ¬p1) remains valid, and O⊥ remains false. Ignoring operator indices, we have V DLS1 (I, v) ⊆ VerDS(I, v) ⊆ V DL51 v). 16 J. Hansen / Problems and Results Theorem 6.2. DL{2,F,S} 1 is sound and weakly complete. Proof: Concerning soundness, for the validity of (FS-1) assume OFA, so some I ′ ∈ I−⊥ derives A. If OSB is true, then all I ′ ∈ I−⊥ derive B, so I ′ derives A∧B and OF (A∧B) is true. For the validity of (FS-3) and (FS-4) suppose OAi is true for all i, 1 ≤ i ≤ n. Concerning (FS-3), assume ¬OF¬kj=1 ∧ Γj , so all I ′ ∈ I − ⊥ are PL-consistent with at least one disjunct, so all I ′ contain some Γj , 1 ≤ j ≤ k, so all I ′ derive ∨k j=1 ∧ Γj . Concerning (FS-4), assume ¬OS¬ni=1Ai, so some I ′ ∈ I −⊥ is PL-consistent with ∧n i=1Ai, so there is some I ′ ∈ I −⊥ that contains each Ai. For the validity of (FS-2), assume again OAi true for all i, 1 ≤ i ≤ n, and assume ¬OF¬(Γ1 → ∨k j=2 ∧ Γj). Those I ′ ∈ I − ⊥ that contain a set Γj also derive ∧ Γ1 → ∨k j=2 ∧ Γj . Consider an I ′ that does not contain any Γj : Since I ′ is not consistent with all Ai ∈ Γj , it must derive ¬ ∧ Γj for each j, so it derives ¬ ∨k j=2 ∧ Γj . If I ′ is consistent with Γ1, it derives ∧ Γ1 ∧ ¬ ∨k j=2 ∧ Γj , but then ¬OF¬( ∧ Γ1 → ∨k j=2 ∧ Γj) is false. So I ′ must be inconsistent with Γ1, so it derives ¬ ∧ Γ1 and hence also ∧ Γ1 → ∨k j=2 ∧ Γj . For (weak) completeness we use the construction employed in the proof of Theorem 5.3 with the following modifications: ∆ is now defined with respect to DL{2,F,S} 1 -consistency (clause c in the construction of ∆), and additionally we have sets OS and QS : OS = {B ∈ r(L [PL]δ) | {δ} ∪∆ ` DL {2,F,S} 1 OSB} QS = {B ∈ r(L [PL]δ) | {δ} ∪∆ ` DL {2,F,S}