A Two-Dimensional Logic for Two Paradoxes of Deontic Modality∗ Melissa Fusco Alexander W. Kocurek August 2020. Forthcoming in The Review of Symbolic Logic Abstract In this paper, we axiomatize the deontic logic in Fusco 2015, which uses a Stalnaker-inspired account of diagonal acceptance and a two-dimensional account of disjunction to treat Ross's Paradox and the Puzzle of Free Choice Permission. On this account, disjunction-involving validities are a priori rather than necessary. We show how to axiomatize two-dimensional disjunction so that the introduction/elimination rules for boolean disjunction can be viewed as one-dimensional projections of more general two-dimensional rules. These completeness results help make explicit the restrictions Fusco's account must place on free-choice inferences. They are also of independent interest, as they raise difficult questions about how to 'lift' a Kripke frame for a onedimensional modal logic into two dimensions. 1 Introduction The validity of or -introduction seems to be a basic rule of natural language disjunction: from φ, one may infer φ or ψ. Or-Intro. φ ( (φ or ψ) Yet the rule apparently fails in the scope of deontic modals. For example, the nonentailment known as Ross's Puzzle illustrates the failure of Or-Intro in the scope of 'ought' (henceforth O) (Ross, 1941): Ross's Puzzle. Oφ * O(φ or ψ) ∗Many thanks to Peter Fritz, Lloyd Humberstone, and an anonymous reviewer for their helpful feedback. Thanks also to the audience at the 2020 Philosophical Applications of Modal Logic conference. The authors contributed equally to this paper and are listed in alphabetical order. 1 As an illustration, observe that (1-a), famously, does not seem to entail (1-b): (1) a. You ought to post the letter. b. You ought to post the letter or burn it. Similarly, Or-Intro does not seem valid in the scope of 'may' (henceforth M). For instance, (2-a) does not seem to entail (2-b): (2) a. You may post the letter. b. You may post the letter or burn it. An attractive explanation for this is that (2-b) seems to entail that both disjuncts are permissible-an observation known as Free Choice Permission (Kamp, 1973): FCP. M(φ or ψ) ( Mφ ^Mψ For example, the inference from (3-a) to (3-b) and (3-c) seems valid: (3) a. You may have an apple or a pear. b. You may have an apple. c. You may have a pear. This would explain why (2-a) doesn't entail (2-b): if it did, then (2-a) would entail you may burn the letter, which is not the case!1 It is well-known that Ross's Puzzle and FCP cause trouble for standard deontic logic, which interprets O as a normal modal operator, with M as its dual. As a consequence, these operators are monotonic: if φ entails ψ, then Oφ entails Oψ, and Mφ entails Mψ. This means that standard deontic logic predicts that (i) Oφ entails O(φ or ψ) and Mφ entails M(φ or ψ), and (ii) if FCP holds, then Mφ entails Mψ for any φ and ψ. These predictions reveal a deep problem at the foundations of standard deontic logic. Ross's Puzzle is problematic for any deontic logic that validates two key principles: Necessitation. If ( φ, then ( Oφ K Axiom. ( O(φÑ ψ)Ñ (OφÑ Oψ) 1Note that while (3-a) seems to entail (3-b) and (3-c), it resoundingly fails to entail (i): (i) You may have an apple and a pear. This additional datum is sometimes called 'exclusivity'. EX. M(φ or ψ) * M(φ ^ ψ) See, inter alia, Simons 2005, Fox 2007, and Barker 2010. See also Fusco 2020a for empirical work on the nature of this exclusivity. 2 Fromtheseprinciples (togetherwith classical logic),we canderiveOφÑO(φ or ψ). Here is the proof: 1. φÑ (φ or ψ) Or-Intro 2. O(φÑ (φ or ψ)) Necessitation, 1 3. O(φÑ (φ or ψ))Ñ (OφÑ O(φ or ψ)) K Axiom 4. OφÑ O(φ or ψ) modus ponens, 2, 3 Thus, if one wishes to capture Ross's Puzzle in a deontic logic, one seems committed either to rejecting Necessitation or rejecting the K Axiom. But which is the culprit, and why? Fusco (2015) suggests that the culprit is Necessitation. To seewhy, let's consider why the inference from (1-a) to (1-b) seems bad. One reason it seems bad to say you ought to post the letter or burn it is that this seems to imply that both disjuncts are permissible. The following inference pattern-Free Choice Obligation-seems intuitively valid: FCO. O(φ or ψ) ( Mφ ^Mψ Thus, to illustrate, (4-a) seems to entail (4-b) and (4-c): (4) a. You ought to either do the dishes or take out the trash. b. You may do the dishes. c. You may take out the trash. Moreover, FCO follows from FCP together with a relatively plausible principle relating 'ought' and 'may': Ought Implies May. Oφ ( Mφ But if FCOholds, thenalreadyNecessitation is enough towreakhavoc: Necessitation plus FCO entail that everything is permissible:2 1. (φ or ¬φ) tautology 2. O(φ or ¬φ) Necessitation, 1 3. Mφ ^M¬φ FCO, 2 This provides one motivation for thinking that the way forward for deontic logic is to reject Necessitation. Two-dimensional semantics offers an elegant way of developing Necessitationfree modal logics: while (5-a) below is "necessary" in the sense of being knowable a priori, it is not metaphysically necessary, in that things could have been different from how they actually are (Crossley and Humberstone, 1977; Davies and Humberstone, 1980; Kaplan, 1989). 2Note it does not help here to restrict FCO to φ and ψ that are possible; that still would imply that for any contingent φ is permissible. 3 (5) a. Everything is as it actually is. b. Necessarily, everything is as it actually is. Perhaps, then, the key to solving these puzzles is to move to a two-dimensional framework. This is the strategy pursued by Fusco (2015). The main idea behind Fusco's semantics is to introduce a two-dimensional entry for ' or ' on which it behaves classically in unembedded contexts but nonclassically under the scope of deontic modals. However, while this semantics captures Ross's Puzzle and FCP, there still remains a further question regarding what the complete logic of her semantics is. In this paper, we answer this question. We show how to axiomatize the logic of two-dimensional disjunction so that the introduction/elimination rules for boolean disjunction can be viewed as one-dimensional projections of more general rules. In addition, we prove several soundness and completeness results for two-dimensional deontic logics, which help make explicit the background assumptions and scope of Fusco's account. We take these completeness results to be of independent interest, as they raise interesting questions about how to "lift" a Kripke frame for a one-dimensional modal logic into two dimensions. These issues arise especially in the context of the picture of communication from Stalnaker 1978, where diagonalization plays a key role. Inparticular, two-dimensionalizingdeontic logic in a Stalnakerian framework seems to require an interesting and substantive metaphysical assumption about the nature of deontic accessibility, viz., thatwhichworlds are deontically accessible does not vary from world-to-world.3 Here is a brief outline. In § 2, we review the philosophical motivations, formal semantics, and logic of standard two-dimensionalism. In § 3, we present Fusco's two-dimensional semantics for disjunction and articulate different ways to axiomatize the logic of disjunction prior to adding deontic modals. In § 4, we extend two-dimensionalism with deontic modals and show how doing so can capture both Ross's Puzzle and FCP. The details of the completeness proofs are given in the technical appendices. 2 A Crash Course in Two-Dimensionalism 2.1 Two Notions of Necessity and Two Notions of Consequence In two-dimensional semantics, the truth of a well-formed formula φ in a model M is relativized to two parameters. The first, the world-as-actual, plays a role in 3This bears resemblance to a point made by Hawthorne and Magidor (2009) about epistemic accessibility. 4 determining the propositional content expressed by φ.4 The second, the world of evaluation, plays the role of the world in which that proposition's truth-value is interrogated; it is also the only parameter shifted by the standard modal operators ◻ and ◇. Two basic two-dimensional operators, @ (usually glossed as 'actually') and :, can be added alongside ◻ and ◇:5 (◻) y , x , ◻φ iff for all x1: y , x1 , φ (@) y , x , @φ iff y , y , φ (:) y , x , :φ iff x , x , φ One motivation for moving to a two-dimensional framework is the fact that we can regiment several philosophically important distinctions in a unified way (Davies and Humberstone, 1980). To illustrate, recall (5-a) and (5-b) from § 1: (5) a. Everything is as it actually is. b. Necessarily, everything is as it actually is. We observed that while (5-a) seems to be a logical truth, it does not seem to be necessarily true, i.e., (5-b) seems false. The problem is that we cannot accept (5-a) as a logical truth in a normal modal logic while denying (5-b). Let ' stand for pEverything is as it actually isq.6 In the two-dimensional framework, we can analyze this sentence as follows:7 (') y , x ,' iff y  x Then we have the following argument for (5-b): 1. ' is a logical truth 2. Necessitation holds, i.e., if φ is a logical truth, then so is ◻φ. 3. ◻ ' is a logical truth. 4We follow the literature here in defining a proposition to be a function from worlds to truthvalues. 5Technically, the satisfaction relation , should always be relativized to a model M. In our informal exposition, however, we will often drop mention of the model for readability. 6For similar sentential constants, compare (for "floating" actuality) 'I' fromHumberstone (2004, pg. 54), and (for "anchored" actuality) the constant 'n' in Meredith and Prior (1965). In a language with propositional quantifiers and @, we could express ' as ∀p(@p Ñ p). 7These truth conditions ignore complications that arise when x and y are qualitative duplicate worlds. We set this complication aside. 5 If we want to deny the conclusion, we need to reject one of the premises. But which? Two-dimensionalism provides us with an elegant answer: which premise you should give up depends on your notion of consequence. First, let us say an argument is strictly valid just in case it preserves truth over all points of evaluation, i.e., all pairs of worlds 〈y , x〉. Strict validity captures the thought that a valid argument is meant to preserve truth in every possible scenario. On this notion of logical consequence, Necessitation still holds: if φ is true at every point of evaluation 〈y , x〉, then so is ◻φ. However, the first premise of the argument for (5-b) is false, since there are non-diagonal points where the xand y-coordinates do not agree on what is the case-that is, whatever world is actual, it still could have been that things were different from how they actually are. Since ' only holds on diagonal points, it is not a strict logical truth. However, ' does hold on all diagonal points. An argument is diagonally valid just in case it preserves truth over all diagonal points, i.e., all pairs of the form 〈z , z〉. This captures the thought that a valid argument is meant to preserve truth no matter which world is actual. On this notion of logical consequence, the first premise of the argument for (5-b) is true, but the second premise is false-e.g., ' is diagonally valid, but ◻ ' is not. In general, whenever something is diagonally but not strictly valid, we'll have a counterexample to Necessitation. Quantification over diagonal points also allows one to represent the difference betweenmetaphysical necessity and a priority in a two-dimensional framework. This is just the difference between φ's being "horizontally" valid, i.e., holding everywhere along the x-axis, and φ's being diagonally valid.8 That is, fixing y as the actual world, φ is metaphysically necessary iff it holds at 〈y , x〉 for all x, whereas φ is a priori iff it holds at 〈z , z〉 for all z.9 The latter notion can be represented by introducing a new defined operator for a priori truth: Aφ B ◻:φ. It is easy to check that A has the following truth conditions: (A) y , x , Aφ iff for all z: z , z , φ. While ' ('everything is as it actually is') is not metaphysically necessary, it is a priori: that is, A ' is a strict logical truth. In fact, even though Necessitation does not hold for diagonal validity, the following variant of Necessitation does: A Priorization. If ( φ, then ( Aφ. That is: if φ is a diagonal logical truth, then so is Aφ. Strict validity is a formalization of necessary entailment, while diagonal validity is a formalization of a priori entailment. 8See Davies and Humberstone 1980; Kaplan 1989. For classic examples where metaphysical necessity comes apart from a priority, see Kripke 1980; Kaplan 1989. 9This idea can be found, inter alia, in van Fraassen (1977) and Lewis (1973, §2.8), who in turn cites Kamp (1971) and Vlach (1973). See also Chalmers (2004). 6 2.2 Stalnaker's Two Dimensionalism Another classic application of two-dimensionalism, which will provide us with a metalanguage (rather than object language) gloss onA, comes fromStalnaker 1978. There, Stalnaker uses two-dimensionalism tomodel conversations inwhich speakers informatively assert sentenceswhosepropositional content is under threat from indeterminacy. To clarify the problem, he introduces the notion of a context set: a set of possibileworlds left open bywhat is accepted in commonby the interlocutors of a conversation. Conversational update on an assertion of φ is a quasi-Bayesian procedure which aims to eliminate from the context set any worlds which are incompatible with φ's content. Suppose, however, that it is unknown-or perhaps, in the case of the open future, evenmetaphysically indeterminate-whichworld is actual. Then therewill be cases inwhich it is unknownor indeterminatewhat the content of an assertion of φ is, and thus unclearwhichworlds should be "thrown out" of the context set if the assertion is to be accepted. Stalnaker's flagship example involves the word 'you' in the asserted sentence pYou are a foolq (pg. 81). If it is unknown, or indeterminate, who the addressee of the context is, it will be unknown, or indeterminate, which person is being claimed to be a fool, and so it will be unknown or indeterminate which update is being proposed. Following Stalnaker, we can visualize the situation using two-dimensional matrices. In these matrices, the rows represent the world-as-actual (y-coordinate)-a role which contributes to determining the proposition expressed by a sentence (e.g., determining the referent of 'you')-while the columns represent the world of evaluation (x-coordinate), where the content, once determined, is assessed. If the content of a sentence depends on which world is actual, then the rows of the matrix will not be identical. So to use Stalnaker's example, suppose the speaker is addressing O'Leary in worlds i and j and to Daniels in world k, and moreover O'Leary is a fool in worlds i and k but not in world j, whereas Daniels is a fool only in j. In that case, we can represent the distribution of truth values of pYou are a foolq as in Figure 1. For instance, the cell on row i and column j contains the truth value of pYou are a foolq at the point 〈i , j〉, which is true iff the person that the speaker is addressing in world i is a fool in world j. Since at world i, the speaker is addressing O'Leary, and since O'Leary is not a fool at world j, pYou are a foolq is false at 〈i , j〉. Whichworlds need to be thrown out of the context set upon accepting an assertion of pYou are a foolq depends on which world is actual: if i or j is actual, we throw out j; if k is actual, we throw out i and k. But-to reiterate the problem Stalnaker is raising-which world is actual is exactly what is either unknown or indeterminate in context (op. cit., pg. 90). In such a predicament, Stalnaker proposes that a sentence φ can be rationally (re)interpreted as :φ. As Stalnaker puts it, this move, called diagonalization, 7 i j k i T F T j T F T k F T F pYou are a foolq Figure 1: A two-dimensional matrix for pYou are a foolq (Stalnaker, 1978, pg. 81). interprets φ as something like "what φ says is true", where "what φ says" is the proposition expressed by φ given a world as actual (pg. 82).10 This is visualized in Figure 2. Diagonalization restores a uniformity condition to matrices: the same proposition is expressed relative to each candidate for actuality.11 In a scenario illustrated by Figure 1, updating the context set with the diagonalized assertion amounts to eliminating worlds j and k from the context set. i j k i T F T j T F T k F T F pYou are a foolq i j k i T F F j T F F k T F F p:You are a foolq Figure 2: Visualizing diagonalization using matrices. Stalnaker's theory of assertion reveals a further distinction in the notion of logical consequence worth highlighting. In § 2.1, we introduced a classical notion of consequence, which defines validity in terms of preservation of truth: an argument is valid in this sense if the conclusion is true at any point of evaluation where the premises are true. But there is also an informational notion of consequence, which defines validity in terms of preservation of acceptance. An argument is valid in this sense if the conclusion is accepted at any context set where the premises are accepted.12 In a static one-dimensional framework, acceptance can be mod10For a suggestion that diagonalization can also occur in the scope of attitude verbs like 'believes', see Stalnaker (1981). 11In the literature, Hawthorne and Magidor (2009) object that diagonalization does not restore uniformity unless certain tendentious conditions on knowledge are met: namely, that the same worlds horizontally accessible on each row. We bracket this concern here, particularly because our discussion is applicable to nonepistemic as well as epistemic indeterminacy. 12For more on this notion, see Stalnaker 1975's notion of reasonable inference, as well as, inter 8 eled as global truth, i.e., truth relative to every world in a context set. Thus, Γ informationally entails φ whenever ◻Γ B {◻γ | γ P Γ} classically entails ◻φ.13 Γ classically entails φ B ∀x : (x , Γ ñ x , φ) Γ informationally entails φ B (∀x : x , Γ) ñ (∀x : x , φ) But what about diagonalization? In Stalnaker's two-dimensional framework, diagonalization may occur before acceptance into the common ground.14 Hence there is a strict-diagonal distinction to be had at the level of informational consequence as well as classical consequence. Informational strict consequence can be glossed like this: no matter which world is actual, if the content of the premises were accepted, so would the content of the conclusion. Stated in terms of classical strict entailment: Γ informationally strictly entails φ whenever ◻Γ classically strictly entails ◻φ. Informational diagonal consequence, by contrast, can be glossed like this: if the diagonalized content of the premises are accepted, then so is the diagonalized content of the conclusion. Stated in terms of classical strict entailment: Γ informationally diagonally entails φ whenever ◻:Γ B {◻:γ | γ P Γ} classically strictly entails ◻:φ. Because A B ◻: can be seen as a priority of the relevant, contextually-situated kind, we will also call informational diagonal consequence a priori consequence, which we will write as '▷'. This notion of consequence matches more closely with the preservation of acceptance in Stalnaker's framework. Moreover, it is a priori consequence which will be key to both validating FCP and blocking Ross's Puzzle on Fusco's semantics: it is a notion of consequence that is just weak enough to validate the former but not the latter.15 Thus, we have four notions of consequence-varying along two dimensions, alia, Veltman 1996; Yalcin 2007; Bledin 2015. 13The difference between classical and informational consequence mirrors the distinction between local and global consequence that one finds in the literature on modal logic (Blackburn et al., 2002). For unarticulated 'boxlike' modalities in natural language, it is worth noting Kratzer's influential proposal that bare indicative conditionals contain a covert necessity operator in the consequent Kratzer (1981, 1986). 14Indeed, Heim (2004, §9) suggests always diagonalizing, noting that diagonalization is superfluous if uniformity is already satisfied. 15One may object that a priori consequence is an odd notion of consequence to validate FCP in, since it validates the inference from p to Ap. It is important to bear in mind, however, that we are using the term "a priori" in a context-specific sense. In this sense, φ is a priori if, given the current conversational context, it is commonly accepted that φ, where common acceptance is cashed out recursively (everyone in the conversation accepts it, and everyone accepts that everyone excepts it, etc.). Thus, Aφ means something more like pit is a priori that φ given what is commonly accepted in the current conversationq. On that understanding, the inference from p to Ap is not surprising: if the speakers all commonly accept p, then they commonly accept that they commonly accept p. 9 strict vs. diagonal and classical vs. informational-illustrated in Figure 3.16 A priori consequence occupies the lower right-hand side of the relevant space of possibilities. classical informational strict Γ ( φ ◻Γ ( ◻φ diagonal :Γ ( :φ ◻:Γ ( ◻:φ Γ ▷ φ Figure 3: Different notions of consequence summarized. 2.3 Axiomatization of Two-Dimensional Semantics One of our primary goals in this paper is to give an axiomatization of Fusco's twodimensional extension of deontic logic. So we should first review axiomatizations for ordinary two-dimensional logic before extending them with deontic modals. We give axiomatizations of classical strict consequence, as the other notions of consequence can be defined in terms of classical strict consequence using : and ◻ (as in Figure 3). Definition 1. Given a set Prop  {p1, p2, p3, . . . } of proposition letters, the basic two-dimensional language L2D is defined recursively in Backus-Naur form as follows: φ F Prop | ¬φ | (φ ^ φ) | ◻φ | @φ | :φ. ◇, _,Ñ andØ defined as usual; in particular, note that (φ _ ψ) B ¬(¬φ ^ ¬ψ). We also define Aφ (it is a priori that φ) as ◻:φ. Definition 2. A basic matrix frame is a Kripke frame 〈W W, R◻, R@, R:〉, where W ,  and: • 〈y , x〉R◻〈y1, x1〉 iff y  y1 16Note that in the two-dimensional setting, informational strict consequence does not align neatly with global strict consequence. Global consequence is usually defined in terms of points of evaluation: an argument is globally valid if its conclusion holds at every point of evaluation whenever its premises hold at every point of evaluation. Stated in terms of classical (i.e., local) strict entailment: Γ globally strictly entails φ whenever ◻:◻Γ B {◻:◻γ | γ P Γ} classically strictly entails ◻:◻φ. This is not the same as informational strict consequence, since the world-as-actual is held fixed from premises to conclusion. Thus, informational strict consequence is stronger than global strict consequence. On the other hand, informational diagonal consequence and global diagonal consequence do coincide. 10 • 〈y , x〉R@〈y1, x1〉 iff y1  x1  y. • 〈y , x〉R:〈y1, x1〉 iff y1  x1  x. Let M be the class of basic matrix frames. A basic matrix model is a pair of a basic matrix frame 〈W W, R◻, R@, R:〉 and a valuation function V : PropÑ ℘(W W). We define the satisfaction relation , between a pointed matrix model M , y , x and a L2D-formula φ in the usual manner for the Kripke semantics. Where Γ is a set of formulas, we'll write M , y , x , Γ to mean that M , y , x , γ for all γ P Γ. Finally, we define classical strict consequence: where Γ is a set of L2D-formulas and φ is a L2D-formula, Γ (2D φ iff for all pointed basic matrix modelsM , y , x, if M , y , x , Γ, then M , y , x , φ. Definition 3. Let 2D be the Hilbert-style proof system consisting of the following axioms and rules: (i) all propositional tautologies as axioms; (ii) the rules of Modus Ponens and Uniform Substitution; (iii) the K Axiom and Necessitation for each primitive modal operator; and (iv) the following axioms: T◻ ◻p Ñ p 5◻ ◇p Ñ ◻◇p @5◇ ◇@p Ñ@p G ◻p Ñ@p R@ @p Ø ¬@¬p R: :p Ø ¬:¬p XÑ :(p Ñ@p) YÑ @(p Ñ :p) 4A ◻:p Ñ ◻:◻:p 5A ◇:p Ñ ◻:◇:p We write Γ $2D φ if for some γ1, . . . , γn P Γ, the formula (γ1 ^       ^ γn)Ñ φ is derivable in 2D. The following is proven in Fusco 2020b. Theorem 1. 2D is sound and complete for classical strict consequence over M. A priori consequence can then just be defined in terms of classical strict consequence: Γ ▷2D φ iff for some γ1, . . . , γn P Γ: $2D (Aγ1 ^       ^ Aγn)Ñ Aφ. 2.4 Åqvist logic The matrix semantics given above is more general than the usual presentation of two-dimensionalism in at least one respect: we allowatomics to represent arbitrary matrices like the one provided for pYou are a foolq, whose content varies with the world-as-actual. By contrast, according to one common strand of two-dimensionalism, valuation functions interpret atomics as sets of worlds, not sets of pairs of worlds. This means 11 the truth of atomics never depends on the world-as-actual parameter. These "standard" truth conditions build in the requirement that the atomics are reserved for sentences that satisfy the Stalnakerian uniformity condition introduced in § 2.2, i.e., that 〈y , x〉 P V(p) iff 〈y1, x〉 P V(p) for any x , y , y1. For these sentences, necessity and a priority coincide.17 It is natural to ask how imposing this restriction on the valuations of atomics affects the logic given in § 2.3. We call such logics Åqvist logics (Åqvist, 1973; Segerberg, 1973). It turns out that their distinguishing axiom is Å: Å αØ :α for any atomic α (Segerberg, 1973, pg. 95) In terms ofmatrices,Å says that rows of thematrix of an atomicmust be duplicates, i.e., each column must either be a column of Ts or a column of Fs. Definition 4. Amatrix model 〈W W, R◻, R@, R:,V〉 is Åqvist iff for all x , y , y1 P W : 〈y , x〉 P V(p) ô 〈y1, x〉 P V(p). Let MÅ be the class of Åqvist matrix models. Note that we can define an "Åqvist operator" Å which applies to a formula φ iff φ's truth is insensitive to the world-as-actual parameter, i.e., iff it is interpreted the same way on every row: Åφ B ◻:◻(φØ :φ) Then we could reformulate Å as the axiom schema Åα for any atomic α. The Å operator turns out to be important for Fusco's explanation of FCP, as the inference only holds in her semantics for φ and ψ that have this "uniformity" feature. As Segerberg is at pains to emphasize, an Åqvist logic is not, in general, closed under uniform substitution.18 Rather, it is closed only under substitution within the 1D fragment of the language.19 Thus, in order to axiomatize Åqvist logics, we need to formulate proof systems that are not closed under Uniform Substitution, but still allow Uniform Substitution when no appeal to Å has been made. This is achieved as follows: 17One can compare here the PQTI sentences of Chalmers (2012). 18For discussion, see, inter alia, Smiley 1982; Holliday et al. 2013. 19This limitation on closure under substitution follows the spirit of, e.g., Williamson (2013), who remarks that "the obvious rationale for insisting on . . . closure. . . under uniform substitution in a propositional system is a reading of non-logical sentence letters as propositional variables" (Williamson, 2013, pg. 76, emphasis added; see also Burgess, 1999, pg. 176). Similarly, in a 2D framework Gregory (2001) remarks that "Goodmodal [arguments]-at least one variety-are ones which are informally sound...[meaning that] for any interpretation ofL, the propositions assigned to [the premises] entail the proposition assigned to [the conclusion] (op. cit., pg. 58, emphasis added). This restriction on closure will exclude arbitrary dipropositions, like ' and the (undiagonalized) pYou are a foolq, from the relevant standard of good argument. 12 Definition 5. Let 2DÅ be the Hilbert-style proof system consisting of Modus Ponens, Necessitation for each primitivemodal operator,Å, and the axiom schema: 2D φ, where φ is a theorem of 2D. In 2DÅ, we can appeal to Uniform substitution in deriving an ordinary theorem of 2D. But once we start reasoning with the consequences of Å, we are no longer allowed to appeal to Uniform Substitution andmust instead rely on the other rules and axioms. The following is proven in Fusco 2020b: Theorem 2. 2DÅ is sound and complete for strict classical consequence over MÅ. 3 Two-Dimensional Disjunction 3.1 An Actuality-sensitive Semantics for Disjunction We now return to Fusco's strategy for solving the deontic puzzles sketched in § 1. According to this two-dimensional strategy, what we need for Ross's Puzzle and FCP is a semantics for disjunction where Or-Intro is a priori, but not necessary. In this section, we review Fusco's proposal and explain how an axiomatization for the basic two-dimensional language can be extended to include her two-dimensional disjunction. This semantics is motivated by Groenendijk and Stokhof (1982) and Lewis (1982)'s two-dimensional approach to "whether p or q" attitude ascriptions, as in (6) Bob knows [whether [p or q]]. (7) Bob doesn't know [whether [p or q]]. The insight here is that (6) and (7) credit Bob with knowledge and ignorance, respectively, of the actually true answer to the question pp, or q?q. According to Groenendijk & Stokhof and Lewis, pwhether p or qq is a two-dimensional propositional concept, which expresses the proposition that p, if p rather than q is true in the world-as-actual, and q, if q rather than p is true in the world-as-actual. Thus a speaker of (6) or (7) can credit Bob with knowledge (or ignorance) of this proposition, even if he himself does not know it. Building off this, Fusco (2015) proposes we interpret bare disjunction along similar lines: if p but not q is actually true, then p or q expresses the proposition that p; and if q but not p is actually true, then p or q expresses the proposition that q. The key difference between the semantic entries for pwhether p or qq above and Fusco's semantics for p or q concerns the cases where either both disjuncts true or neither are: in that case, for Fusco, p or q just expresses the ordinary boolean 13 disjunction p _ q.20 We can state this semantics for or in terms of a simpified answerhood operator (Dayal, 1996, 2016). This operator takes and world w and disjuncts φ and ψ as arguments and outputs the true-in-w answers to the question of whether φ or ψ, if there are any. Otherwise, it outputs both answers: Ans(w , φ, ψ)   {φ} if w , w ,, φ and w , w ,. ψ {ψ} if w , w ,. φ and w , w ,, ψ {φ, ψ} otherwise. The truth conditions for disjunction can then be given as follows: ( or ) y , x ( (φ or ψ) iff ∃α P Ans(y , φ, ψ) : y , x , α. Figure 4 illustrates these truth conditions in matrix form. Notice that or is, as Humberstone (2020, §4.7) puts it, a kind of two-dimensional isotope of boolean_-that is, or and_ are equivalent along the diagonal. Thus, for diagonal consequence, all the standard rules governing boolean disjunction hold of or . But off the diagonal, or and _ come apart. For example, in Figure 4, w2, w3 . p or q even though w2, w3 , p. On this picture of disjunction, p or ¬p is like ', in that it is an a priori truth that cannot be necessitated. (Indeed, if there are only finitely many atomics p1, . . . , pn and all the rows of the truth table are represented by a unique world, then∧ i(pi or ¬pi) generates the same matrix as '.) The feature of disjunction will be responsible for nonclassical behavior in the scope of deontic modals (§ 4). Note also that (φ or ψ) will generally not be Åqvist, even if both φ and ψ are. Figure 5 shows how two-dimensional disjunction, like the @ operator itself, can build non-Åqvist matrices out of the matrices for Åqvist atomics. The diagonals of these matrices for disjunction reflect the classical profile of _: the more disjuncts a classical disjunction has, the more states in a classical (viz., one-dimensional) truth table it is true in. 20Groenendijk & Stokhof's entry presupposes that exactly one of {p , q} is true in the actual world (op. cit. pg. 184), leaving the proposition expressed by the disjunction undefined along the bottom row of the matrix in Figure 4. Lewis's entry (op. cit., pg. 52) would result in a matrix with F-F-F-F along the same row. But-to preview how the semantics will work under deontic operators like ought (O)-this will give the wrong intuitive results for the sentences in Ross's Puzzle and FCP. For example, Lewis's entry under O would entail that (4-a) is equivalent to pÔKq in cases where a lazy agent neither (actually) takes out the trash nor (actually) washes the dishes. It is also worth noting the great deal of post-Groenendijk and Stokhof 1982 work (not in a twodimensional tradition) on the general category of concealed questions into which (6) and (7) falls. For developments in treating this class in inquisitive semantics, see, e.g., Roelofsen 2019. 14 w1 w2 w3 w4 w1 T T T F w2 T T F F w3 T F T F w4 T T T F p or q w1 w2 w3 w4 w1 T T T F w2 T T T F w3 T T T F w4 T T T F p _ q Figure 4: Matrix for p or q, compared with that of p _ q, where w1 is an arbitrary (p^ q)-world, w2 is a (p^¬q)-world, w3 is a (¬p^ q)-world, and w4 is a (¬p^¬q)world. p q r p or q q or r p or (q or r) Figure 5: Matrices with 8 worlds per axis. Shading indicates 'true'; white 'false'. 15 3.2 Axiomatics What is the logic of two-dimensional or like? While some classical inference patterns governing _ (e.g., Or-Intro) are not strictly valid for or , some are. For instance, the following principles are still classically strictly valid evenwith Fusco's two-dimensional or : Idempotence. φ ( (φ or φ) Commutativity. φ or ψ ( ψ or φ Associativity. φ or (ψ or χ) ( (φ or ψ) or χ21 Or-Elim. φ or ψ,¬φ ( ψ Thus, the logic for or is not entirely divorced from the logic of_ off the diagonal. But it does raise thequestionofwhichprinciples, exactly, are the result of projecting from the two-dimensional or into a one-dimensional framework. So we turn to the question of how to axiomatize our basic two-dimensional language L2D when extended with or . In a sense, this task is trivial; as Humberstone (2020) notes, φ or ψ on Fusco's semantics can be expressed using @:22 (@(φ ^ ¬ψ)Ñ φ)^ (@(¬φ ^ ψ)Ñ ψ)^ (@(φØ ψ)Ñ (φ _ ψ)) Thus, we could simply view or as a defined connective. However, this definition is long-winded; it would be nice to know whether there are more illuminating axioms governing or directly. Indeed, there are. LetL2D( or ) be the result of extendingL2D with a primitive or . Then, as noted above, we could axiomatize 2D extended with or using the following defining axiom: or df (p or q)Ø [(@(p ^ ¬q)Ñ p)^ (@(¬p ^ q)Ñ q)^ (@(p Ø q)Ñ (p _ q))] Alternatively, we could replace or df with the following six axioms, which more closely mirror the standard introduction and elimination rules for disjunction: or I1 (@p ^ p)Ñ (p or q) or I2 (@q ^ q)Ñ (p or q) or I3 (@¬(p _ q)^ (p _ q))Ñ (p or q) 21The validity of Associativity can be verified by cases based on which of φ, ψ, and χ is satisfied at 〈y , y〉. We omit the proof, as it is tedious and unilluminating. Note also that the converse of Associativity follows from Associativity and Commutativity. 22In fact, the opposite is true, too: @φ can be expressed as (φ or ¬φ) Ø φ. 16 or E1 (p or q)Ñ (p _ q) or E2 ((p or q)^@¬p)Ñ (@¬q _ q) or E3 ((p or q)^@¬q)Ñ (@¬p _ p) It is straightforward to show that 2D + or df is equivalent to 2D + or I1– or E3. The latter, however, more clearly manifest the way in which the rules governing _ can be seen as one-dimensional projections from more general two-dimensional rules governing or . Since @φ is diagonally equivalent to φ, or I1 and or I2 reduce to the standard disjunction introduction rule; or I3 diagonally follows from explosion; or E1 is already an articulation of the disjunction elimination rule in terms of disjunctive syllogism; and or E2 and or E3 diagonally follow from the fact that everything implies a tautology. Using these axioms, we can directly derive the collapse of or and _ along the diagonal. To facilitate the reasoning, observe that the following rule is admissible in 2D for any primitive modal operator△: RK. If φ1, . . . , φn ( ψ, then△φ1, . . . ,△φn (△φ. Here, then, is an axiomatic proof of◻:((φ or ψ)Ø(φ_ψ))-the apriori equivalence of or and _ (with instances of Uniform Substitution suppressed): 1. (φ or ψ)Ñ (φ _ ψ) or E1 2. :((φ or ψ)Ñ (φ _ ψ)) Necessitation, 1 3. :(φÑ@φ) XÑ 4. :(ψÑ@ψ) XÑ 5. (@φ ^ φ)Ñ (φ or ψ) or I1 6. (@ψ ^ ψ)Ñ (φ or ψ) or I2 7. :((@φ ^ φ)Ñ (φ or ψ)) Necessitation, 5 8. :((@ψ ^ ψ)Ñ (φ or ψ)) Necessitation, 6 9. :(φÑ (φ or ψ)) RK, 3, 7 10. :(ψÑ (φ or ψ)) RK, 4, 8 11. :((φ or ψ)Ø (φ _ ψ)) RK, 2, 9, 10 12. ◻:((φ or ψ)Ø (φ _ ψ)) Necessitation, 11 This offers an axiomatic illustration of how the two-dimensional entry for or is an isotope of the standard boolean _. 17 4 Deontic Modality 4.1 Two-Dimensional Deontic Logic Here is an interim summary of where we are. In §§ 2.1–2.2, we presented several motivations for the two-dimensional framework as well as four notions of consequence that can be distinguished within it. The axiomatization of this framework was given in § 2.3. In § 2.4, we motivated exploring Åqvist logics with their Åcompliant atomics. Finally, in § 3, we sketched a two-dimensional semantics of disjunction and presented an axiomatization for it. With this setup, we are ready to enrich the language with the deontic operators O and M. Wedo this by enriching themodal domain of ourmatriceswith a deontic accessibility relationRO . Thegoal of this section is to articulate the logic that results from lifting standard one-dimensional deontic logic into two-dimensions. We then show how doing so can solve the deontic puzzles with which we began. First, however, we face a choice point regarding the deontic accessibility relation, similar to the choice point faced by the fully general vs. Åqvist approach to atomics sketched in § 2.4, viz., should the deontic accessibility relation be uniform across each row of the matrix? A 'yes' answer suggests that at its foundations, deontic accessibility cannot depend on facts which are (metaphysically or epistemically) indeterminate relative to the context set. A 'no' answer, on the other hand, suggests that it can. Here, we present completeness results for both paths from this choice point. However, a number of interpretative complications arise on the latter, non-uniform approach. (What does it mean, exactly, for deontic ideality to depend both on the world-as-actual and theworld of evaluation?) And, at any rate, Fusco (2015) adopts a uniform deontic accessibility relation. Since our main goal is to axiomatize this system, we will allow uniformity to play a role in explaining Ross's Puzzle and (more critically) FCP. We leave it to future investigation to settle whether the uniformity approach is warranted. Second, there is another choice point concerning the properties of the deontic accessibility relation. In standard one-dimensional deontic logic, it is assumed that the deontic accessibility relation is at the very least a serial and shift-reflexive subrelation of R◻ (generalized to the two-dimensional setting; for readability, we leave off the world-as-actual parameter, which is held constant throughout). May Implies Can. ∀w , v : wRO v ñ wR◻v Seriality. ∀w∃v : wRO v Shift-Reflexivity. ∀w , v : wRO v ñ vRO v 18 These constraints correspond respectively to the following axioms (where Ô is defined to be the dual of O; we separate Ô from M conceptually for reasons that will be made clear in § 4.3): MC Ôp Ñ ◇p DO Op Ñ Ôp OTO O(Op Ñ p) Adding DO and OTO to theminimal normal modal logic results in themodal logic KDU, i.e., "standard deontic logic+" in the terminology of McNamara 2010. Stronger constraints could be imposed on RO , though. In particular, it is natural to assume that RO is rigid, meaning which worlds are "deontically ideal" does not vary from world to world: Deontic Rigidity. ∀w , v , u : wR◻v & wRO u ñ vRO u This corresponds to the following axiom: DR Ôp Ñ ◻Ôp On this analysis, RO effectively partitions W into the classes of deontically ideal and nonideal worlds.23 This allows us to reduce deontic logic to alethic logic via the standard Anderson-Kanger reduction using ◻ and a privileged atomic d standing for pThe deontic ideals are metq (Anderson, 1958; Kanger, 1971): Oφ B ◻(d Ñ φ) Mφ B ◇(d ^ φ) Again, for the sake of neutrality, we present completeness results for both options. It turns out that this option affects what we say about FCP. Because Fusco only adopts the weaker framework, without Deontic Rigidity, she needs to revise the relationship between O and M in order to explain FCP. Assuming Deontic Rigidity, however, this revision is not required. Let us now turn to the question of how to axiomatize each combination of choices. Definition 6. The two-dimensional deontic language LD2D is defined recursively as follows: φ F Prop | ¬φ | (φ ^ φ) | ◻φ | @φ | :φ | Oφ. In addition to the other abbreviations, we define Ôφ B ¬O¬φ. 23This is similar to the way the deontic selection function of MacFarlane and Kolodny (2010) works. See also Kratzer 1977, 1981's ordering-source approach. 19 Definition 7. A deontic matrix frame is a tuple F  〈W  W, R◻, R@, R:, RO〉, where 〈W, R◻, R@, R:〉 is a basic matrix frame and RO is a serial, shift-reflexive subrelation of R◻. A deontic matrix model is a pair of a deontic matrix frame with a valuation function V : PropÑ ℘(W W). A deontic matrix frame (model) is rigid if in addition RO satisfies Deontic Rigidity. Let DM be the class of deontic matrix models and DMR the class of rigid deontic matrix models. Definition 8. Let D2D be the logic that results from adding MC–OTO as axioms to 2D, and let D2DR be the result of adding DR as well. The following is proved in § A. Theorem 3. (a) D2D is sound and complete for DM. (b) D2DR is sound and complete for DMR. Now for adding the other options. Definition 9. LetM  〈W W, R◻, R@, R:, RO ,V〉 be a deontic matrix model. Say RO is uniform if the following condition is met for all x , y , w , v P W : 〈〈x , w〉, 〈x , v〉〉 P RO ô 〈〈y , w〉, 〈y , v〉〉 P RO . We'll say M is uniform if RO is. The definition of Åqvist matrix models from Definition 4 carries over to deontic matrix models. Let DMÅU be the class of Åqvist uniform deontic matrix models and DMÅUR the class of Åqvist, uniform, and rigid deontic matrix models. To state the proof system, we first need a definition. Definition 10. A LD2D-formula is explicitly 1D if it is @-free and :-free. Definition 11. Let F B D2DÅU be the proof system axiomatized by Modus Ponens, Necessitation for each primitive modal operator, Å, and the following axiom schemas: D2D φ, where φ is a theorem of D2D. U OφØ :Oφ where φ is an explicitly 1D-formula. Let FR B D2DÅUR be defined similarly except we replace D2D with: D2DR φ, where φ is a theorem of D2DR. 20 F is the proof system that axiomatizes Fusco (2015)'s semantics, while FR is the same proof system extended with DR. The following is proved in § B.24 Theorem 4. (a) F is sound and complete for DMÅU. (b) FR is sound and complete for DMÅUR. F and FR give us a pair of one-dimensional deontic logics "lifted" into two dimensions-that is, into the space of matrices-in a way that preserves the original, one-dimensional interpretation of atoms.25 4.2 Ross in the context of Deontic Matrix Models Now that we've seen various ways of lifting deontic logic into two dimensions, what happens when we adopt Fusco's two-dimensional semantics for or ? As it turns out, both Ross's Puzzle and FCP can be accounted for (though, as we'll see, the latter may require a bit extra work to obtain). Start with Ross's Puzzle. Even in FR, the Ross inference is not a priori valid: Op 6▷DMÅUR O(p or q). A counterexample is given in Figure 6. Let us say a model M a priori satisfies φ, written M ▷ φ, if M , z , z , φ for all z P W . (Thus, Γ ▷C φ iff for every M P C, if M ▷ Γ, then M ▷ φ.) The deontic matrix model in Figure 6 a priori satisfies the premise Op. But it fails to a priori satisfy the conclusion O(p or q). The witness for this failure is 〈wq , wq〉: at 〈wq , wq〉, the disjunction p or q expresses the proposition that q. But q is not obligatory (or even permissible) at 〈wq , wq〉. To get a better sense of why failures of the Ross inference arise, it helps to contrast this counterexample with a model where O(p or q) is a priori satisfied, such as in Figure 7. We can see that q-the actually true answer to pp, or q?q in wq-is obligatory from the point of view of 〈wq , wq〉.26 We can also see that p-the actually true answer to pp, or q?q in wp-is obligatory from the point of view of 〈wp , wp〉, the other diagonal, or 'a priori', point in the model. If it is q which the agent actually does-settling that she is at the diagonal point on the second row of the matrix-then it is q she ought to do. But the reverse is true if it is p which she actually does. Since the choice of actuality is up to her, what is deontically 24An anonymous reviewer asks whether consequence over DMU can be axiomatized by just adding U to D2D. Alternatively, one might conjecture that we can drop Å from F. The answer is negative on both counts, since U is not sound for the class of (even rigid) uniform deontic matrix models that are not Åqvist. A counterexample is given in § B (Figure 11). 25These logics hence preserve the one-dimensional notion of "good arguments" from Gregory 2001 and Williamson 2013 (glossed in footnote 19). 26To simplify the visuals, p and q are represented in Figures 6–7 asmutually exclusive and jointly exhaustive. 21 wp wq wp wq Figure 6: A counterexample to Ross's Puzzle in FR, where for any α, β, γ P Prop, 〈wβ , wγ〉 P V(α) iff γ  α. ideal is also up to her: she has a deontically free choice in Figure 7 which she lacks in Figure 6.27 wp wq wp wq Figure 7: A model where O(p or q) is a priori satisfied. 4.3 Permissibility and Free Choice Permission Let's turn now to FCP. Before we explain how two-dimensional deontic logic captures this principle, we add a qualification. The FCP inference seems to be licensed for both disjuncts only when it's possible for the agent to make each disjunct true without the other. For instance, suppose you are told you may take 27It is worth noting that a proposal for understanding FCP which is similar in spirit but quite different in implementation to the present approach, is independently developed in the linguistics literature by Kaufmann (2016). Unfortunately, we lack the space to compare the similarities and differences of the views at length here. 22 an apple or a pear, but it turns out (say, for practical reasons) that you can only take the pear if you also take the apple. In that case, it seems as though you are not, in general, permitted to take the pear. In light of this, the form of free choice we will be interested in makes explicit that it's possible for each disjunct to be true without the other:28 FCP*. M(φ or ψ),◇(φ ^ ¬ψ),◇(ψ ^ ¬φ) ( Mφ ^Mψ We'll start with deontically rigid models, since the explanation of FCP* over this class is simpler. Three ingredients are required. First, M is the dual of O, i.e., M B Ô. Second, the deontic accessibility relation is uniform. And third, the formulas involved in the inference are Åqvist, i.e., their propositional content does not vary with the world-as-actual. Thus, where the Å operator is defined as in § 2.4, we have the following free choice theorem:29 , 30 Theorem 5. M(φ or ψ),◇(φ ^ ¬ψ),Åφ,Åψ ▷DMÅUR Mφ. Proof. Let M be a Åqvist, uniform, and rigid deontic matrix model such that: (i) M ▷ M(φ or ψ) (ii) M ▷ ◇(φ ^ ¬ψ) (iii) M ▷ Åφ ^ Åψ Let w P W ; we will show that M , w , w , Mφ. By (ii), there exists a v P W such that M , w , v , φ ^ ¬ψ. By (iii), M , w , v , φ ^ ¬ψ iff M , w , v , :φ ^ ¬:ψ, which holds iff M , v , v , φ ^ ¬ψ. Hence, Ans(v , φ, ψ)  {φ}. By (i), M , v , v , M(φ or ψ). So there exists a u P W such that 〈v , v〉RO 〈v , u〉 andM , v , u , φ or ψ. Since Ans(v , φ, ψ)  {φ}, that means M , v , u , φ. By (iii), M , w , u , φ. But because RO is uniform, 〈w , v〉RO 〈w , u〉. And because RO is rigid, that means 〈w , w〉RO 〈w , u〉. Hence,M , w , w , Mφ.  28Empirical data suggests these this caveat is connected to the exclusivity data from footnote 1, especially in the case where there are more than two disjuncts. In the simplest case, suppose that for the disjunction pp1 or p2 or . . . or pnq it holds that ◻(pi Ą pi+1). Then it is impossible to make p1 true without making ∧ i pi true. In this case, FCP, but not FCP, would hold that M(p1 or . . . or pn) entails M( ∧ i pi), contrary to EX from footnote 1. Recent experimental work suggests this entailment is not licensed in these cases (Fusco, 2020a). 29In fact, Theorem 5 holds even for the class of uniform and rigid deontic matrix models-that is, we don't have to require valuations to be Åqvist. But the completeness proof of F(R) requires the deontic matrix models be Åqvist, so we've stated the theorem in terms of Åqvist models. 30An anonymous reviewer points out that ▷DMÅUR may be too strong to plausibly formalize natural language entailment. For instance, p ▷DMÅUR Op and p ▷DMÅUR ◻p. This is because the atoms are Åqvist, so if they're true at every diagonal, they're true everywhere. (More generally, φ,Åφ ▷DM ◻φ and φ,Åφ ▷DM Oφ.) This is related to the "If p, Ought p" problem, which information-sensitive theories of 'ought' tend to face. See Carr 2014 for discussion. We do not propose a solution to this problem, though we suspect it will require a better understanding of how conditionals relate to disjunction (observe, e.g., that 6▷DMÅUR ¬p or Op). 23 We can illustrate this using by contrasting Figure 8 and Figure 6. In Figure 8, M(p or q) is a priori satisfied because no matter which world is actual, the actual answer to pp, or q?q is permissible. By contrast, in Figure 6, q is not permissible, and so M(p or q) is not a priori satisfied-specifically, it's not satisfied at 〈wq , wq〉. wp wq wp wq Figure 8: M ▷ M(p or q) The addition of premises Åφ and Åψ is new relative to Fusco 2015. There, Fusco states the free choice inference (using our notation) as: M(φ or ψ),◇(φ ^ ¬ψ),◇(¬φ ^ ψ) ▷ Mφ ^Mψ where φ and ψ are both disjunction-free and non-modal. In that framework, Fusco implicitly assumes that all atomics areÅqvist, which entails that any such formulas are Åqvist. Thus, our theorem is a generalization of Fusco's result: her free choice inference can be extended to any φ and ψ so long as their content does not vary from row-to-row. There are cases involving non-Åqvist disjuncts, however, where free choice does not hold in this semantics. Here is a counterexample. Our matrix model M will consist of four worlds wpq , wpq , wpq , wp q . Our (Åqvist) valuation will be the obvious one: 〈wαβ , wγδ〉 P V(p) ô γ  p 〈wαβ , wγδ〉 P V(q) ô δ  q. The deontic accessibility relation will say all and only p-worlds are deontically ideal: 〈wαβ , wγδ〉RO 〈wαβ , wγ1δ1〉 ô γ1  p Note, RO is both uniform and rigid. This model is summarized in Figure 9. 24 wpq wpq wpq wp q wpq wpq wpq wp q pq pq pq pq Figure 9: Counterexample to free choice with non-Åqvist disjuncts. The diagram on the right is a visualization of the accessibility relations on each row. Then letting φ  @p Ø p and ψ  @q Ø q, we have:31 M ▷ M((@p Ø p) or (@q Ø q)) M ▷ ◇((@p Ø p)^ ¬(@q Ø q)) M ▷ ◇(¬(@p Ø p)^ (@q Ø q)) M 6▷ M(@p Ø p) Thus, FCP* does not hold for all non-Åqvist cases in Fusco's semantics. With that said, we think this is not a serious cost to Fusco's approach, for two reasons. First, these sorts of counterexamples are quite strange. It is hard to see how we could have robust and reliable intuitions about their analogues in natural language. This is especially complicated by evidence that the actually operator @ is not be an adequate formalization of the English word 'actually'.32 So these consequences of Fusco's semantics would likely be hard to test empirically. Second, there is an alternative formulation of free choice available that avoids 31To seewhy, note that for each v P W , Ans(v ,@pØp ,@qØq)  {@pØp ,@qØq}. This is because @φ Ø φ is a diagonal validity. Thus, for each v P W , we haveM , v , v , M((@p Ø p) or (@q Ø q)) iff ∃δ P {q , q} : M , v , wpδ , (@p Ø p)_ (@q Ø q). It is easily verified this holds for all v, since some deontically idealworld agreeswith v on q. Moreover, the possibility premises are satisfied on every row, since every combination of truth values to p and q is realized. But whileM ▷ M(@q Ø q), we do not have M(@p Ø p) satisfied at every diagonal point. Specifically, M , wpq , wpq . M(@p Ø p) andM , wpq , wpq . M(@p Ø p). 32For data showing that English 'actually' is more complex, see Yalcin (2015). 25 these complications. We can drop the premises Åφ and Åψ from the inference so long as we diagonalize the disjuncts first. Theorem 6. M(:φ or :ψ),◇(:φ ^ ¬:ψ) ▷DMÅUR M:φ Proof. Immediate from Theorem 5 and the fact that (DM Å:φ.  In fact, Theorem 5 can be seen as a special instance of this more general principle, sinceÅφ ( ◻(φØ:φ). We often diagonalizematerial in the scope ofmodals as part of a broader reinterpretation strategy. For instance, while (8) is not metaphysically necessary, and thus (9) is false so-interpreted, there does seem to be a reading of (9) on which it is true, viz., one where the flavor of necessity is a priority. (8) I am here now. (9) Necessarily, I am here now. a. False: ◻(I am here now) b. True: ◻:(I am here now) It would therefore be unsurprising if diagonalization occurred in the scope of disjunction, given that disjunction is itself interpreted as a kind of binary modal. This concludes the explanation of free choice when the accessibility relation is deontically rigid. Let us now turn to the case where it is not. In fact, Fusco (2015) does not endorse deontic rigidity because of cases like the following: Choosing Childbearing. You face a choice between conceiving a child early in your life [ p] or a different child significantly later [ q]. You believe that your values will be transformed by the choice you make. In particular, because the choice you make will be a necessary condition for the existence of a person youwill love, youwill affirm that choice over any other. There is no single psychological standpoint that values both of these potential persons to equal degree. (Parfit, 1984, pg. 360–361; Paul, 2014, Ch. 3; Paul, 2015.) Nice Choices at the Spa. Aromatherapy [ p] or body-wrap [ q]- which is it to be? You believe that, whichever you choose, you will be very glad you chose it. Mid-aromatherapy, the aromatherapywill seem self-evidently superior [to the body-wrap]. Mid-body-wrap, the bodywrap will seem self-evidently superior [to the aromatherapy]. (Hare and Hedden, 2016, pg. 3) Suppose "nice choices" like the ones described above are possible. Then worlds where the agent makes different choices may have different deontic points of view. This would require abandoning Deontic Rigidity as a constraint on the deontic accessibility relation. 26 p w1 q w2 Figure 10: A one-dimensional nice choice To accommodate free choice inferences in this more general framework, Fusco proposes to capture FCP* by reconceiving of the connection between obligation and permission. In brief, where the standard view is that M is just Ô, i.e., the dual of O, Fusco proposes to analyze M in terms of a weaker condition, viz.,◇Ô. Thus, the revised truth conditions for M are as follows: (M2) y , x , Mφ iff there is some x1 such that x1RO x1 and y , x1 , φ Redefining M in this way, we obtain free choice once again: Theorem 7. Where M B ◇Ô: (a) M(φ or ψ),◇(φ ^ ¬ψ),Åφ,Åψ ▷DMÅU Mφ (b) M(:φ or :ψ),◇(:φ ^ ¬:ψ) ▷DMÅU M:φ Proof. The proof of (a) is the same as that of Theorem5up towherewe inferred that M , v , v , M(φ or ψ). From there, it follows that there exists a u P W such that for some u1 P W , 〈v , u〉RO 〈v , u1〉 and M , v , u1 , φ or ψ. Since Ans(v , φ, ψ)  {φ}, that means M , v , u1 , φ. By (iii), M , w , u1 , φ. But because RO is uniform, 〈w , u〉RO 〈w , u1〉. Hence,M , w , u , Ôφ. SoM , w , w , ◇Ôφ  Mφ. The proof of (b) is immediate as before.  Thus, one does not need to commit oneself to the claim that eachworldmust agree with every other about which worlds are deontically ideal in order to capture free choice, so long as we take a revisionary stance on the relationship between 'may' and 'ought'. Note that, in deontically rigid models, this definition of M is equivalent to Ô: Theorem 8. (DMR ÔφØ◇Ôφ. Thus, one could in principle adopt this definition (M B ◇Ô) in FR as well; it's just that doing so would not be a genuine departure from the orthodox view that 'may' is the dual of 'ought'. 27 5 Conclusion In this paper, we explored several ways of axiomatizing ◻, :, @, and the deontic operators O and M, with an eye to FCP and Ross's Puzzle, following the path sketched by Fusco (2015, 2019). The language also provides a way of connecting the apriori (A) operator to Stalnaker's "Assertion", via the idea that a natural notion of global or informational consequence is in fact a diagonalized one as well. We axiomatized Fusco's logic of disjunction using rules that collapse into the standard disjunction introduction/elimination rules in the one-dimensional setting. We then explored several different formulations of two-dimensional deontic logic and how these different choice points relate to the deontic puzzles in § 1. While our focus has primarily been ondeontic logic, the various choice points in § 4 highlight a broader lesson for the study of two-dimensional logics. When lifting a one-dimensional system into two dimensions, one needs to keep in perspective the originalmotivations for doing so. For some applications, itwill be desirable not to impose uniformity on accessibility relations, thus preserving the full generality of the two-dimensional framework. But for other applications to natural language, especially in the context of a Stalnakerian picture of assertion, uniformity may be motivated by more than mere convenience. We leave further discussion of these issues for another time. A Axiomatizing D2D In this appendix, we prove Theorem 3: D2D(R) is sound and complete for the class DM(R) of (rigid) deontic matrix models. The proof is an extension of the completeness results in Fritz 2014; Fusco 2020b for two-dimensional languages without deontic operators. Below, we have omitted the proofs of all the "Facts", as they can be found in Fusco 2020b (or are straightforward extensions of facts therein). The proof strategy is in three steps. We identify three classes of frames: 1. FrD2D(R), the class of D2D(R)-frames; 2. RD(R), an intermediate class of (rigid) deontic Restall frames; 3. MFD(R), the class of (rigid) deontic matrix frames. The first step is to establish the soundness and completeness of D2D(R) over FrD2D(R). This follows immediately from Sahlqvist's theorem, since all the axioms are Sahlqvist formulas.33 33In fact, as an anonymous reviewer notes, this is the only place where we appeal to DO and OTO . Thus, the proof strategy generalizes to any set of axioms governing O that are canonical in 28 Theorem 9. D2D(R) is sound and strongly complete with respect to FrD2D(R). Proof. By Sahlqvist's Theorem (Blackburn et al., 2002, Ch. 4).  The second step is to show that FrD2D(R) is modally equivalent to an intermediate class of frames RD(R), viz., (rigid) deontic Restall frames. This is established by showing that RD(R) is the class of point-generated subframes of FrD2D(R). The final step is then to show that every deontic Restall frame is equivalent to a matrix frameby showinghow to construct a boundedmorphism froman arbitrary deontic Restall frame into a deontic matrix frame. FrD2D(R) RD(R) MFD(R) gen. submodel bounded morphism We start with some facts about D2D, which are all left as an exercise to the reader: Fact 1. The following are all provable in 2D (Fusco, 2020b): X :(@φØ φ) Y @(:φØ φ) :◇: :◇:φØ◇:φ Red: ::φØ :φ Red@ @@φØ@φ A@ AφÑ@φ In addition, the following is provable in D2D: @5Ô Ô@φÑ@φ Definition 12. A Kripke frame is a tuple F  〈W, R◻, R@, R:, RO〉 where W is a set of points and each R△ is a binary relation on W . Fact 2. The axioms of D2D(R) have the following (global) first-order correspondents in Kripke frames: T◻ R◻ is reflexive 5◻ R◻ is Euclidean @5◇ ∀w , v , u : (wR◻v & wR@u)ñ vR@u G R@ Ď R◻ R@ R@ is a function R: R: is a function XÑ ∀w , v , u : the sense of Blackburn et al. 2002, p. 206. However, the same is not true for the proof of Theorem 4 in § B (since RYO may not satisfy the appropriate constraints). 29 (wR:v & vR@u)ñ v  u YÑ ∀w , v , u : (wR@v & vR:u)ñ v  u 4A R◻   R: is transitive 5A ∀x , y , a , b : (xR◻y & xR◻a & aR:b)ñ ∃c(bR◻c & ∀d(cR:d ñ yR:d)) MC RO Ď R◻ DO RO is serial UO RO is shift-reflexive DR ∀w , v , u : (wR◻v & wRO u)ñ vRO u The correspondents for X and Y in Fact 1 are as follows: X Within Im g(R:), R@ is the identity relation Y Within Im g(R@), R: is the identity relation The following facts hold of any D2D-frame: Fact 3. Im g(R@)  Im g(R:). Fact 4. ∀d1, d2 P Im g(R@): if d1(R◻   R:)d2, then d2(R◻   R:)d1. Fact 5. R◻   R: an equivalence relation on Im g(R@). Fact 6. Suppose wR@d1 and wR:d2. Then ∃w1 s.t. w1R@d2 and w1R:d1. Before we show FrD2D(R) is modally equivalent to RD(R), it will help to prove the following lemma. (Notation: if R△ is a function, we use "R△(x)" for the unique y such that xR△y) Lemma 1. For any point-generated subframe Fw  〈W 1, R1◻, R1:, R 1 @, R 1 O〉 P FrD2D: W 1  R◻[R:[R◻[{w}]]]. Proof. We let X B R◻[R:[R◻[{w}]]]. It suffices to show (1) w P X and (2) that for each operator O that RO[X] Ď X. These are proven in Fusco 2020b except for the O case for (2), which is easy since RO Ď R◻ and we have T◻.  Definition 13. A (rigid) deontic Restall Frame is a frame R  〈W, R◻, R@, R:, RO〉 such that 30 1. R◻ is an equivalence relation 2. R@ is a function such that (a) wR@v Ñ wR◻v (b) R@ maps any two R◻-related worlds to the same point 3. R: is a function such that: (a) for any w: R:[R◻[{w}]]  Im g(R@) (b) R: is reflexive over Im g(R@) 4. RO is a (rigid,) serial, shift-reflexive subrelation of R◻. Let RD(R) be the class of (rigid) deontic Restall frames. Lemma 2. RD(R) Ă FrD2D(R). Proof. To verify this, it suffices to go through the first-order correspondents of D2D(R) from Fact 2.  Lemma3. Everypoint-generated subframeFw P FrD2D(R) is a (rigid) deontic Restall frame. Proof. Where Fw  〈W 1, R1@, R1:, R 1 ◻ , R1O〉 is a w-generated subframe in FrD2D, let v be an arbitrarily chosen world in W 1. We want to establish that Fw satisfies conditions 1–4 of deontic Restall frames. Conditions 1–3 are established in Fusco 2020b. Condition 4 follows from the correspondences in Fact 2.  Theorem 10. FrD2D(R) and RD(R) are modally equivalent. Proof. By Lemma 2, any formula falsifiable in R P RD(R) is falsifiable in some F P FrD2D(R). By Lemma 3, any formula falsifiable in a point-generated subframe Fw P FrD2D(R) is falsifiable in some R P RD(R). But any formula falsifiable in a F P FrD2D(R) is falsifiable in a point-generated subframe (Blackburn et al., 2002, Proposition 2.6.).  Now we just need to reduce RD(R) to MFD(R). Some notation: • We write R[Y] for the image of a set Y under R. • Let D  Im g(R@). We will use i , j . . . as indices over D. • Where R is a deontic Restall frame, let C be the set of R◻-cells in R. Since there is one d P D in each such cell, C is the cardinality of D. 31 Fact 7. ⋃iPD ci  W . Fact 8. There is a unique R@-fixed point in each R◻-cell ci Ă W . Notation: call this R@-fixed point R@(ci). Fact 9. For any w and any c j Ď W , ∃!v P c j s.t. wR:v. More notation: where ci P C, let c j i be the set {w P ci | ∃v P c j : wR:v}. Let R:(c ji ) be the unique v P c j such that ∀w P c ji , wR:v. That R:(c ji ) is a fixed point of R: follows from Red:. Corollary 1. Each ci can be partitioned into {c ji | j P D}, where c j i is a nonempty subset of ci such that∀w P c ji , wR:(R@(c j)). Hence ⋃ jPD c j i  ci , and ⋃ iPD( ⋃ jPD c j i )  W . Theorem 11. RD(R) and MFD(R) are modally equivalent. Proof. Since MFD(R) Ă RD(R), every (rigid) deontic matrix frame is equivalent to some (rigid)deonticRestall frame. For the converse, givenadeonticRestall frameR with domain W and Im g(R@)  D, we will build a matrix frameM and construct, row-by-row, a surjective bounded morphism f from M to R, from which modal equivalence follows (Blackburn et al., 2002, Proposition 2.14). The points of our matrix frameM will be (D W) (D W). For any i , j P D and y P W , let A ji ,y be the set {〈i , y〉}  {〈 j, x〉 : x P W}. Note that |A j i ,y |  |W |. For each i , j P D and y P W , fix some surjective g ji ,y : W Ñ c j i such that if i  j and y  x, then g ji ,y(x)  R@(ci). (This condition maps the "diagonal" points of the matrix frame into D in the deontic Restall frame.) We know such surjective functions exist since |A ji ,y | ě |c j i |. The accessibility relations RM ◻ , RM@ , and R M : are already determined by the definition of a deontic matrix frame. To define RMO , first define a function F : (D  W) (D W)ÑW as follows: F(〈〈i , y〉, 〈 j, x〉〉)  g ji ,y(x). We define R M O in terms of F: 〈〈i , y〉, 〈 j, x〉〉RMO 〈〈i , y〉, 〈 j 1, x1〉〉 ô F(〈〈i , y〉, 〈 j, x〉〉)ROF(〈〈i , y〉, 〈 j1, x1〉〉). Wewill now show that F is a bounded morphism fromM to R.34 To do this, it 34F is a bounded morphism if it satisfies the following conditions for each △ P {◻,@, :,O} (Blackburn et al., 2002, pg. 59): 1. w and F(w) satisfy the same proposition letters; 2. if wRM △ v then F(w)R△F(v) (the Forth condition); 3. if F(w)R△v1 then there exists some v such that wRM△v and F(v)  v 1 (the Back condition). 32 suffices to go through the Back and Forth conditions for△ P {◻,@, :,O} for each point. The only new case not in the proof from Fusco 2020b is the O case. (O, Forth) Immediate by the definition of RMO . (O, Back) Suppose F(〈〈i , y〉, 〈 j, x〉〉)RO v1. We want to show that there is some 〈〈i , y〉, 〈 j1, x1〉〉 such that 〈〈i , y〉, 〈 j, x〉〉RXO 〈〈i , y〉, 〈 j 1, x1〉〉 and also that F(〈〈i , y〉, 〈 j1, x1〉〉)  v1. To witness this existential, we can choose any 〈〈i , y〉, 〈 j1, x1〉〉 such that F(〈〈i , y〉, 〈 j1, x1〉〉)  v1 (we know one exists since F is surjective). Hence, F is a bounded morphism, and soM is modally equivalent to R.  B Axiomatizing Deontic Åqvist Logic In this appendix, we prove Theorem 4: F is sound and complete for the class of Åqvist uniform deontic matrix models DMÅU. (The proof that FR is sound and complete for DMÅUR is analogous.) Soundness is straightforward and left to the reader. For completeness, the strategy will be to bootstrap off the completeness result for D2D in § A. First, we need to prove some lemmas about the expressive power ofD2D. Lemma 4. For any LD2D-formula φ, the following formulas are strictly valid over DM: (a) ◻(@φ _ ψ)Ø (@φ _ ◻ψ) (b) O(@φ _ ψ)Ø (@φ _ Oψ). Recall the definition of explicitly 1D formulas (Definition 10). Lemma 5. For any deontic matrix model M  〈W  W, R◻, R@, R:, RO ,V〉, if M satisfies Å and U everywhere, then for any w , w1, v P W , any explicitly 1D LD2D-formula φ: M , w , v , φ ô M , w1, v , φ. Proof. By induction.  Definition 14. An @-atom is a LD2D-formula that is either explicitly 1D, or is of the form @φ where φ is explicitly 1D. The following is an extension of a result for languages without : or O proven in Hazen et al. 2013: 33 Lemma 6. Every LD2D-formula is strictly equivalent to a boolean combination of @-atoms over DMÅU. Proof. By induction on the structure of LD2D-formulas. Throughout, we'll use "φ " ψ" to mean φ and ψ are strictly equivalent over DMÅU. The atomic and boolean cases are trivial. We'll present the other cases. Assume for inductive hypothesis that the claim holds of φ. In particular, assume that: φ " k∧ i1 ni∨ j1 αi , j where each αi , j is an @-atom. (◻) Since ◻ commutes with conjunction: ◻φ " ◻ k∧ i1 ni∨ j1 αi , j " k∧ i1 ◻ ni∨ j1 αi , j So it suffices to show that each ◻∨nij1 αi , j is equivalent to a boolean combination of @-atoms. First, write ∨ni j1 αi , j as: @βi ,1 _       _@βi ,m _ γi ,m+1 _       _ γi , j where each βi ,x and γi ,y are explicitly 1D. Then by Lemma 4(a): ◻ ni∨ j1 αi , j " ◻(@βi ,1 _       _@βi ,m _ γi ,m+1 _       _ γi , j) " @βi ,1 _       _@βi ,m _ ◻(γi ,m+1 _       _ γi , j) But now all these terms are @-atoms. (O) Similar to the (◻) case. (@) Since @ commutes with booleans: @φ " @ k∧ i1 ni∨ j1 αi , j " k∧ i1 ni∨ j1 @αi , j Now, either αi , j is explicitly 1D, in which case @αi , j is an @-atom, or αi , j  @βi , j where βi , j is explicitly 1D, in which case @αi , j " @@βi , j " @βi , j " αi , j , so we can replace @αi , j with αi , j . The result is therefore a boolean combination of @-atoms. 34 (:) Since : commutes with booleans: :φ " : k∧ i1 ni∨ j1 αi , j " k∧ i1 ni∨ j1 :αi , j Now, either αi , j is explicitly 1Dor αi , j  @βi , j where βi , j is explicitly 1D. In the former case, :αi , j " αi , j by Lemma 5. In the latter case, :αi , j " :@βi , j " :βi , j , which is equivalent to βi , j by Lemma 5 again. So either way, :αi , j " αi , j , which means φ is strictly equivalent to :φ already.  Theorem 12. F is sound and complete for DMÅU. Proof. Let Γ be a F-consistent set of formulas. Take a maximal F-consistent extension Γ+ Ě Γ (the proof that one exists is standard). By Theorem 3, there is a deontic matrix model M and some y , x P W such that M , y , x , Γ+. We first show that the valuation V is already Åqvist. Then we show how to transform M into an equivalent model whose deontic accessibility relation is uniform. First, V is Åqvist: since ◻:◻(p Ø :p) P Γ+ for all p P Prop, it follows that M , w , v , p Ø :p for all w , v P X. Henec, for all w , w1, v P X: 〈w , v〉 P V(p) ô M , w , v , p ô M , w , v , :p ô M , v , v , p ô M , w1, v , :p ô M , w1, v , p ô 〈w1, v〉 P V(p). Next, we show how to transform M into an equivalent one whose deontic accessibility relation is uniform. Define RYO as follows: 〈w , v〉RYO 〈w , u〉 ô ∃z : 〈z , v〉RO 〈z , u〉 Define MY  〈W  W, R◻, R@, R:, RYO ,V〉. Clearly, R Y O is uniform, serial, and shift-reflexive. (Note also that if RO is rigid, so is RYO .) Lemma 7. For all w , v P W and all explicitly 1D LD2D-formulas φ: M , w , v , φ ô MY, w , v , φ. Proof. By induction on the complexity of φ. We only present the O case (the others are straightforward). Suppose for inductive hypothesis that the claim holds of φ. 35 Clearly, ifMY, w , v , Oφ, thenM , w , v , Oφ since RO Ď RYO . So we just need to establish the converse. Suppose MY, w , v . Oφ. Thus, for some u P W , we have 〈w , v〉RYO 〈w , u〉 and MY, w , u . φ. By definition of RYO , for some z P W , 〈z , v〉RO 〈z , u〉. And by inductive hypothesis, M , w , u . φ. Hence, by Lemma 5, M , z , u . φ. So M , z , v . Oφ. But again by Lemma 5,M , w , v . Oφ.  Lemma 8. For all w , v P W and all @-atoms φ: M , w , v , φ ô MY, w , v , φ. Proof. If φ is an explicitly 1D LD2D-formula, then this is ensured by Lemma 7. If φ  @ψ where ψ is explicitly 1D, then: M , w , v , @ψ ô M , w , w , ψ ô MY, w , w , ψ (Lemma 7) ô MY, w , v , @ψ.  Hence, by Lemma 6, for any LD2D-formula φ and any w , v P W : M , w , v , φ iff MY, w , v , φ. SoMY, y , x , Γ+.  Note that one cannot axiomatize consequence over DMU simply by droppingÅ from the axiomatization. The inductive step for O in the proof of Lemma 7 relies on Lemma 5, which in turn relies on the Å axiom for the base case. This use of Å is ineliminable: for U is not valid over the general class of (even rigid) uniform deontic matrix models. 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