The Logic of Sequence Frames Fabio Lampert University of California, Irvine Preprint of September, 2018 Forthcoming in The Review of Symbolic Logic Abstract This paper investigates and develops generalizations of two-dimensional modal logics to any finite dimension. These logics are natural extensions of multidimensional systems known from the literature on logics for a priori knowledge. We prove a completeness theorem for propositional n-dimensional modal logics and show them to be decidable by means of a systematic tableau construction. Keywords: Modal Logic * Multidimensional Modal Logic * Epistemic Logic * A Priori * Actuality * Necessity * Completeness * Decidability Introduction Two-dimensional modal logics have been the object of increasing philosophical interest given intuitive interpretations assigned to the modal operators. In particular, that such logics provide us with a logical analysis of metaphysical necessity, actuality, and a priori knowledge is now an important facet of prominent philosophical views in philosophy of language, epistemology, and metaphysics. Logics developed with the purpose of shedding light on a priori reasoning and its relation with the modal notions of necessity and actuality arguably originated with Davies and Humberstone (1980), and have recently been investigated by Restall (2012), Fritz (2013, 2014), Fusco (forthcoming), and others. Furthermore, the semantic treatment of epistemic and indexical terms in the works of Evans (1979), Kaplan (1989), and Chalmers (2004, 1 2014) make essential use of elements present in two-dimensional modal logics, while Weatherson (2001) and Wehmeier (2013) adopt similar two-dimensional semantics to provide formal treatments of both subjunctive and indicative conditionals. The goal of this paper is to generalize two-dimensional modal logics with actuality operators to any finite dimension. To be more precise, the logics investigated here are generalizations of the logic for epistemic two-dimensional semantics studied by Fritz (2014) under the name 2Dg, which in a certain sense is the same logic as those investigated proof-theoretically in Restall (2012) and Lampert (2018)-although the logics in Lampert (2018) contain first-order quantifiers. The formal language defined here contains several modal operators, one for each dimension, including indexed boxes, 2i, as well as actuality operators, @i. As will be seen below, the resulting logics can be characterized as logics of generalized diagonal sequences, and therefore as natural generalizations of two-dimensional modal logics in which formulas in the scope of certain operators are evaluated at the diagonal points of square models, that is, models based on frames containing ordered pairs of worlds. Notwithstanding the formal character and scope of this paper, whose principal concern is with modeland proof-theoretic investigations of certain classes of frames and models for modal logics, the development of n-dimensional modal logics is in fact philosophically motivated and arises from considerations regarding the expressive power of modal languages for necessity, actuality, and apriority. In what follows we briefly describe some of these motivations as well as the plan for the rest of the paper. Background Crossley and Humberstone (1977) introduced the actuality operator, @, to remedy an expressive deficit in the first-order modal language,1 i.e. the language of first-order logic augmented by the modal operators 2 and 3. What they observed was that there is no formula in that language that expresses the following truth condition: (1) ∃w∀x(Rx-at-z → Sx-at-w), where z is the "actual world" of the underlying frame.2 According to (1), 1See also Hazen (1976). 2It is common to notate the actual world as w∗ (see, for instance, Crossley and Humberstone (1977) and Davies and Humberstone (1980)), or even as @, if another symbol is used 2 there is a world w such that every object that is R in the actual world z is S in w. As Crossley and Humberstone (1977: 12) point out, there are readings of English sentences with truth conditions corresponding to (1), such as "It is possible for every red thing to be shiny". But the most obvious attempts to represent (1) in the object language all fail: 3∀x(Rx → Sx) expresses the truth condition ∃w∀x(Rx-at-w → Sx-at-w), ∀x3(Rx → Sx) the truth condition ∀x∃w(Rx-at-w → Sx-at-w) and, if we assume that truth in a model is defined as real-world truth,3 i.e. truth at the actual world of the model-in which case formulas are initially evaluated at z-∀x(Rx → 3Sx) expresses the truth condition ∀x(Rx-at-z → ∃w(Sx-at-w)). None of these truth conditions are equivalent to (1). Yet, in the presence of the actuality operator (1) can be expressed by the formula 3∀x(@Rx → Sx). This is so because the semantic evaluation clause for the actuality operator invariably takes back the evaluation of a formula in its scope to the actual world of the frame. Let M = (W, z,R,D, V ) be a model for the first-order modal language containing @, where W is a set of possible worlds, z ∈ W , R ⊆ W ×W , D is a domain of objects, and V is a valuation function defined as usual.4 For any formula φ, M, w  @φ⇐⇒M, z  φ.5 So even if the formula @Rx occurs within the scope of 3, the semantic entry for @ makes us consider the objects in the domain of the world introduced by 3 that are R in the actual world z, thereby, as it were, temporarily suspending the scope of the 3 operator. for the actuality operator (see, for instance, Wehmeier (2014)). 3This terminology comes from Crossley and Humberstone (1977: 15), where real-world validity is defined as truth at the actual world of every model, and general validity as truth at every world of every model. The distinction is not without a difference: the formula @φ→ φ for example, being real-world but not generally valid. Since, however, there can be worlds at which @φ → φ is false, real-world validity gives rise to contingent logical truths, a point explored by Zalta (1988) in his defense of real-world validity over general validity as the correct generalization of the Tarskian notion of logical truth for modal languages. A reply to Zalta can be found in Hanson (2006), but we also direct the reader to Nelson and Zalta (2012), French (2012), Hanson (2014), and Wehmeier (2014). 4We shall not be concerned with first-order models in this paper apart from this section. For simplicity, though, one can just assume that the constant symbols in the first-order modal languages mentioned here are interpreted rigidly, even though this undermines somewhat the purpose of extending two-dimensional modal languages with the first-order quantifiers, where it is possible to distinguish, say, the metaphysical rigidity of proper names, subsumed under the constant symbols of the language, from their epistemic non-rigidity. 5For simplicity, we leave the assignment function implicit here. 3 There is, however, a subtle difference in the semantic clause for @ when we move to a full-blooded two-dimensional modal logic.6 In this case, formulas are evaluated against pairs of possible worlds, (w, v), where w is taken intuitively as a counterfactual world and v as a world considered as actual (or an epistemic scenario, such as in Fritz (2014)), and so the set of evaluation points W in the frames can now be defined as the Cartesian product of some non-empty underlying set of worlds, say, S. This means that rather than having a single fixed actual world as a part of the frame, any world can be actual as long as it occupies the second coordinate of the pair of worlds relative to which a formula is evaluated. That is, to use a metaphor from Humberstone (2004: 26), we first dethrone the single actual world from the identity of the frames or models, and formulate the semantic clause for @ as M, (w, v)  @φ⇐⇒M, (v, v)  φ,7 where v can be any element of S. Then the truth conditions for 21φ, where 21 is now used for the metaphysical necessity operator, can be given with the aid of its corresponding accessibility relation, R21 , which relates a pair (w, v) to any pair (x, y) such that y = v. That is, the actual world is held fixed while the counterfactual world varies. Additionally, two-dimensional modal languages are equipped with a diagonal necessity operator, interpreted intuitively as an apriority operator, which is sometimes taken as a primitive operator in the language (Restall (2012), Fritz (2014), Lampert (2018)), or defined in terms of other operators (Davies and Humberstone (1980)). Here we notate it as 22, and its corresponding accessibility relation will be R22 , which in turn relates a pair (w, v) to any pair (x, y) such that x = y. Intuitively, the semantic entry for the apriority operator then says that a formula is true a priori if and only if, no matter which world turns out to be actual, that formula remains true in that world. The reason why we call this a diagonal operator is because when models are defined with a domain of pairs of possible worlds the operator 22 projects precisely along the pairs consisting of identical coordinates. This can be illustrated by distributing the pairs constructed out of possible worlds w, v, u in a 2-D matrix, in which the X axis contains counterfactual worlds 6We distinguish the number of dimensions based on the evaluation points in the frames: if formulas are evaluated against a possible world w, the semantics is one-dimensional. By contrast, if formulas are evaluated against a pair (w, v) of possible worlds, we say that the semantics is two-dimensional; similarly for triples (w, v, u), and so on. 7This can be compared with Davies and Humberstone (1980: 4), although they use a different notation. 4 and the Y axis worlds taken as actual. 22φ is then true if φ holds along the diagonal points of the matrix, as seen in Figure 1 below. w v u w φ − − v − φ − u − − φ  Figure 1: 2-D matrix It is possible to subsume the semantic clauses for 21 and 22 under a single schema as follows. Let σ and τ be sequences (of length n) of possible worlds. Then: (GB) M, σ  2iφ⇐⇒ for every τ ∈ W , if σR2iτ , then M, τ  φ, where σR2iτ if and only if the first i coordinates in τ are all identical and σ and τ are identical beyond i, i.e. for all j > i, the jth coordinate of σ is the same as the jth coordinate of τ .8 It is simple to check that (GB) delivers the correct semantic entries for both 21and 22-formulas. And since there is no single distinguished point in the underlying frames anymore, we can add to them a set D = {(z, z) | z ∈ S} of diagonal points, and define truth in a model relative to the elements of D, in which case D is the set of distinguished points of the underlying frame. This is by and large the approach used by Fritz (2014) to define a class of what he calls matrix frames with distinguished elements for the propositional two-dimensional language. Of a first-order two-dimensional framework we may ask whether the following relative of the truth condition in (1) is expressible in it: (1*) ∃(w, z)∀x(Rx-at-(z, z)→ Sx-at-(w, z)). The answer is positive if @ is present in the language, for (1*) is expressed by the formula 31∀x(@Rx → Sx). In fact, it would seem that standard proofs for the analogous inexpressibility result in the one-dimensional setting would transfer, more or less directly, to the two-dimensional setting if the latter did not contain @ in the language. This can be done with Wehmeier's (2003) proof, for example, under the obvious generalization of the models for the 8Such sequences are defined more carefully in Definition 1.2. 5 two-dimensional case. Now, assuming the availability of @, and given that 32 quantifies over diagonal pairs of worlds in W , it seems reasonable to expect that the truth condition in (2) be expressible, too: (2) ∃(w,w)∀x(Rx-at-(z, z)→ Sx-at-(w,w)). But it is not obvious whether (2) can be expressed by a formula in this language; the most obvious candidate, 32∀x(@Rx → Sx), expresses the truth condition ∃(w,w)∀x(Rx-at-(w,w) → Sx-at-(w,w)). This is so because within the scope of 32, @ is idle. Thus, the expressive deficit of the first-order (onedimensional) modal language that motivated the introduction of @ in the first place seems to reoccur in the two-dimensional case: while @ is able to, as it were, temporarily suspend the scope of 31, there is apparently no analogous device in the two-dimensional language that has the same effect on the scope of 32. Furthermore, there are readings of English sentences such as "It is not a priori that not every red thing is shiny" whose truth conditions correspond to (2). That is, besides the apparent inability to express truth conditions that, on the face of it, should be expressible with the two-dimensional operators, it does not seem to be possible to formalize relevant portions of the kind of natural-language discourse targeted by two-dimensional semantics. One way to fix this is to add a single distinguished point (pair of worlds) back in the frames and a new operator, say, A, that forces formulas in its scope to be evaluated at that distinguished point. In other words, A does just what @ did in the one-dimensional case (before it was dethroned). If we let this point be (z, z), the semantic clause for A will be: M, (w, v)  Aφ⇐⇒M, (z, z)  φ. Now the truth condition (2) can be expressed by the formula32∀x(ARx→ Sx). Alternatively, we can take a point z from the underlying set S as distinguished, and instead of adding A to the language, add an operator D with the following semantic clause: M, (w, v)  Dφ⇐⇒M, (w, z)  φ. The truth condition (2) can then be expressed by the formula 32∀x(D@Rx→ Sx), and A can be simulated in this language by the compound operator D@. Yet another option in a similar spirit is to dethrone the single distinguished element of S and thereby move to three-dimensional frames, just as the original actual world was dethroned in the introduction of two-dimensional frames. 6 In this setting, formulas are evaluated relative to triples of worlds, that is, W = S3, and a set of distinguished elements is defined in the frames as D = {(z, z, z) | z ∈ S}. In this new language we can have three 2i operators whose semantic entries can be fully specified by the (GB) clause above, this time for i ∈ {1, 2, 3}. Accordingly, 21φ is true at a triple (w, v, u) if and only if φ is true at any R21-related triple (x, y, z), where y = v and z = u; 22φ is true at a triple (w, v, u) if and only if φ is true at any R22-related triple (x, y, z), where x = y and z = u (i.e. at any triple (x, x, u)); and, finally, 23φ is true at a triple (w, v, u) if and only if φ is true at any R23-related triple (x, y, z), where x = y = z (i.e. at any triple (x, x, x)). Now, just as Fritz (2014) adds an accessibility relation for the actuality operator in the two-dimensional frames,9 we can relabel the actuality operator @ as @2, add a new actuality operator, @3, to the language, and accessibility relations to describe their semantic clauses schematically as follows: (GA) M, σ  @iφ⇐⇒ for every τ ∈ W , if σR@iτ , then M, τ  φ, where σR@iτ if and only if the first i coordinates in τ are all identical and σ and τ are identical beyond i − 1. Thus, according to (GA), @2φ is true at a triple (w, v, u) if and only if φ is true at any R@2-related triple (x, y, z), where x = y = v and z = u (i.e. at the triple (v, v, u)); and @3φ is true at a triple (w, v, u) if and only if φ is true at any R@3-related triple (x, y, z), where x = y = z = u (i.e. at the triple (u, u, u)). Additionally, (GA) gives us an @1 operator which turns out to be redundant, for @1φ is true at a triple (w, v, u) if and only if φ is true at anyR@1-related triple (x, y, z), where x = w, y = v, and z = u (i.e. at the triple (w, v, u) itself). If we now adapt the truth condition (2) to the three-dimensional environment we obtain (2*) ∃(w,w, z)∀x(Rx-at-(z, z, z)→ Sx-at-(w,w, z)), which is expressible by the formula 32∀x(@3Rx→ Sx). What motivated the introduction of the original actuality operator, and the subsequent move to two dimensions, is really the same thing that motivates the introduction of the new actuality operator into the two-dimensional language, and the subsequent move to three dimensions. And even though it is not obvious what natural language correspondents for operators such as 23 and @3 9There are many advantages in doing this, also illustrated by the fact that the axioms involving actuality operators can then be defined as Sahlqvist formulas, and so the Sahlqvist completeness theorem can be applied (see §1.3). 7 might be, it is clear that the original expressive deficit in the first-order modal language, which reoccurred in the two-dimensional case, seems to appear once again in the three-dimensional setting since @3 does not seem able to reset the point of evaluation within the scope of 33. That is to say, the following truth condition is apparently not expressible by any formula in the three-dimensional language under consideration: (3) ∃(w,w,w)∀x(Rx-at-(z, z, z)→ Sx-at-(w,w,w)). The most obvious candidate, 33∀x(@3Rx→ Sx), is unsuccessful for the same reason 32∀x(@Rx→ Sx) fails to express the truth condition in (2). It is now evident that analogous cases of expressive deficit will seem to occur for any finite dimension n. To be more specific, where n ≥ 2, a first-order ndimensional language contains operators 2i and @i for each i ∈ {1, ..., n}, and a frame for this language is defined by a set W = Sn, where S is a non-empty set of worlds, relations R2i and R@i for each i ∈ {1, ..., n}, a set D = {s | s ∈ S} of distinguished points, where s is that sequence σ ∈ W for which σi = s ∈ S for all 1 ≤ i ≤ n, besides a domain of objects D. The truth conditions for 2iand @i-formulas are set by (GB) and (GA). Then, where σ = s and τ = t, the following truth condition would not seem to be expressible by any formula in the n-dimensional language: (n) ∃σ∀x(Rx-at-τ → Sx-at-σ). In Lampert (2018), some of the philosophical ramifications of this general case for two-dimensional semantics were explored. In particular, the move from two to three-dimensions was motivated by the fact that some of the other solutions to the expressive deficit of the first-order two-dimensional language- namely, the ones involving the addition of the operators A or D-would result in the frames validating Aφ → 22Aφ or D@φ → 22D@φ, respectively. But according to the intuitive interpretation we assign to these operators, such formulas should not be valid: if φ is true in the real actual world, it does not follow intuitively that it is a priori that it is true in that world. However, a similar issue ultimately arises in the three-dimensional language, as @3φ → 22@3φ, too, would be valid in its respective class of frames. In this paper, however, we will not be concerned with philosophical implications of employing multidimensional frameworks to model a priori knowledge of finite or ideal agents. Rather, we shall focus on the logical generalization to higher-dimensions of logics for necessity, actuality, and a priori knowledge, which do raise a variety of questions from a purely logical point of view. The 8 n-dimensional frames with distinguished elements defined earlier are natural generalizations of the two-dimensional matrix frames with distinguished elements defined in Fritz (2014). Fritz proves that such frames, which he defines for a propositional two-dimensional language, admit of a finite axiomatization through a more general class of matrix frames, which do not contain distinguished elements. Fritz calls the logic of such frames 2Dg. It is natural to ask, in light of the constructions above, whether 2Dg is a special case of a more general framework, and whether similar results transfer to n-dimensional frames defined for n-dimensional propositional languages.10 Are these frames finitely axiomatizable? Are the logics decidable? We settle these and more questions in the course of this paper. In §1 we work towards proving a completeness theorem for the logic of n-dimensions relative to n-dimensional frames without distinguished elements, and then we use this result to show that n-dimensional frames with distinguished elements are also complete. In §2 we prove that the resulting logics are decidable by means of n-dimensional tableaux and a systematic procedure guaranteeing termination for every tableau so constructed. At the end of the paper we suggest some additional questions to be investigated in future work. 1 The logic of sequence frames 1.1 Syntax and semantics Definition 1.1 (Multidimensional language) Let PROP be a denumerable set of propositional variables p, q, .... The language L@n is recursively generated by the following grammar: φ ::= p | ¬φ | (φ ∧ φ) | 2iφ | @iφ for all 1 ≤ i ≤ n, where n ≥ 2. The other Boolean connectives are defined as usual, and 3iφ := ¬2i¬φ. Finally, let O ∈ {2,@}. Definition 1.2 (Sequences) Let S be a non-empty set and W = Sn, i.e. W is the n-fold Cartesian product of S, so thatW contains sequences σ = (s1, ..., sn) of elements s1, ..., sn ∈ S. When σ = (s1, ..., sn), we write σi for si. For s ∈ S, s is that sequence σ for which σi = s for all 1 ≤ i ≤ n. Moreover, we say that 10It is reasonable to investigate the propositional languages first, especially because the whole technology consisting of finitely many 2i and @i operators is already present at the propositional level. 9 a tuple σ is i-diagonal just in case σ1 = σ2 = ... = σi, and for any sequence σ ∈ W , let σiz be that sequence τ for which τ1 = τ2 = ... = τi = z and for j > i, τj = σj, so that σiz is always i -diagonal. We say that σ and τ are identical beyond i if for all j > i, σj = τj.11 Now we define sequence frames for n-dimensional modal logics. The number of dimensions dictates the length of the sequences in the frame, which can be thought of as sequences of possible worlds. Strictly speaking, all that is needed to fully specify the desired frames are pairs (S, n), where S is a nonempty set. Yet, it will be helpful to add the relations R2i and R@i to the official definition of the frames, as these play an essential role in the Sahlqvist completeness theorem proved in §1.3, besides making several definitions easier to articulate. Thus the official definition of n-dimensional sequence frames is the following: Definition 1.3 (n-dimensional sequence frame) Let n ≥ 2. An n-dimensional sequence frame for L@n is a triple, F = (W, (R2i)1≤i≤n, (R@i)1≤i≤n), such that W = Sn for some non-empty set S, R2i ⊆ W ×W and R@i ⊆ W ×W for all 1 ≤ i ≤ n, such that for any sequences σ, τ ∈ W , • σR2iτ iff (i) τ is i-diagonal, and (ii) σ and τ are identical beyond i. • σR@iτ iff (i) τ is i-diagonal, and (ii) σ and τ are identical beyond i− 1. Additionally, let F be the class of n-dimensional sequence frames. To make the behaviour of each accessibility relation more explicit, we register the main characteristic properties that hold in every n-dimensional sequence frame. We begin with the properties involving only the R2i relations, moving in turn to the properties involving only the R@i relations, and then to the properties involving both families of relations. The proofs of these facts are straightforward, and so we omit them. Proposition 1.1 Let 1 ≤ j ≤ i ≤ n and F = (W, (R2i)1≤i≤n, (R@i)1≤i≤n) be an n-dimensional sequence frame. Then: (1) R21 is an equivalence relation. (2) R2i is serial, i.e. for all sequences σ there is a τ such that σR2iτ . 11To illustrate some of the items defined above, in the case of basic modal logic σ is always a one-place sequence, while in a two-dimensional modal logic σ is an ordered pair. Also, if, for example, σ = (x, u, z, w), then σ3u = (u, u, u, w). 10 (3) (Upward transitivity) If σR2jτ and τR2iυ, then σR2iυ. (4) (Upward Euclideanity) If σR2jτ and σR2iυ, then τR2iυ. (5) (Downward weak density) If σR2iτ , there is a υ such that σR2jυ and υR2iτ . (6) (Downward shift reflexivity) If σR2iτ , then τR2jτ . (7) (Strictly decreasing weak density) When 1 < i ≤ n, if σR2i−1τ , there is a υ such that σR2iυ and υR2i−1τ . σ τ υ R2j R2i R2i σ τ υ R2j R2i R2i σ τ υ R2i R2i R2j σ τ R2i R2j σ τ υ R2i−1 R2i−1 R2i Figure 2: Upward transitivity, upward Euclideanity, downward weak density, downward shift reflexivity, and strictly decreasing weak density properties. Proposition 1.2 Let 1 ≤ i ≤ n and F = (W, (R2i)1≤i≤n, (R@i)1≤i≤n) be an n-dimensional sequence frame. Then: (1) R@1 is reflexive. (2) R@i is serial. (3) R@i is functional, i.e. if σR@iτ and σR@iυ, then τ = υ. (4) (Upward-downward transitivity) When 1 < i ≤ n, if σR@iτ and τR@i−1υ, then σR@iυ. 11 σ τ υ R@i R@i−1 R@i Figure 3: Upward-downward transitivity. Remark 1.1 By Proposition 1.2(3), it follows that each R@i is a function. Additionally, because R@1 is reflexive, R@1 is the identity function on the frame. Proposition 1.3 Let 1 ≤ j < i ≤ n and F = (W, (R2i)1≤i≤n, (R@i)1≤i≤n) be an n-dimensional sequence frame. Then: (1) (Strictly decreasing act-box) When 1 < i ≤ n, if σR@iτ , then σR2i−1τ . (2) (Act-box) If σR@iτ , then σR2iτ . (3) (Mixed upward transitivity) If σR2jτ and τR@iυ, then σR@iυ. (4) (Mixed shift reflexivity) If σR2iτ , then τR@iτ . σ τ R@i R2i−1 σ τ R@i R2i σ τ υ R2j R@i R@i σ τ R2i R@i Figure 4: Strictly decreasing act-box, act-box, mixed upward transitivity, and mixed shift reflexivity properties. Note that downward shift reflexivity now follows immediately from mixed shift reflexivity together with act-box and strictly decreasing act-box. Similarly, the properties (5) and (7) in Proposition 1.1 are provable from the other properties involving the R@i relations. With respect to (3) and (4), that is, upward transitivity and upward Euclideanity, they can also be derived from the other properties alongside (simple) transitivity and the Euclidean property. Thus, in order to characterize sequence frames syntactically we only need 12 to add the modal axioms for transitivity and the Euclidean property alongside the other axioms listed below. Moreover, from the properties listed in Proposition 1.1 one can show that sequence frames have the property of left commutativity as well as the Church-Rosser property, that is: Corollary 1.1 Let 1 ≤ j ≤ i ≤ n, and F = (W, (R2i)1≤i≤n, (R@i)1≤i≤n) be an n-dimensional sequence frame. Then: (1) (Left commutativity) If σR2jτ and τR2iυ, then there is a σ′ such that σR2iσ′ and σ′R2jυ. (2) (Church-Rosser) If σR2jτ and σR2iυ, then there is a σ′ such that τR2iσ′ and υR2jσ′. Proof. (1). If σR2jτ and τR2iυ, it follows that σR2iυ, by upward transitivity. But then υR2jυ, by downward shift reflexivity. Therefore, there is a sequence σ′, namely, υ itself, such that σR2iσ′ and σ′R2jυ. (2). If σR2jτ and σR2iυ, then τR2iυ, by the upward Euclidean property. But then υR2jυ, by downward shift reflexivity. Therefore, there is a sequence σ′, namely, υ itself, such that τR2iσ′ and υR2jσ′. σ τ υ σ′ R2j R2i R2i R2j σ τ σ′ υ R2j R2i R2i R2j Figure 5: Left commutativity and Church-Rosser properties. These properties are known for playing an important role in characterizing product frames for product logics (see Gabbay et al (2003: 222)). Note, however, that the property of right commutativity, which holds in product frames, does not hold in general for n-dimensional sequence frames.12 Consider, for instance, the two-dimensional sequence frame F = (W,R21 ,R22 ,R@1 ,R@2) displayed in Figure 6, where W = S2 and S = {w, v}. Then each of the properties listed in Proposition 1.1, 1.2, and 1.3 hold in that frame, but while (v, w)R22(v, v) and (v, v)R21(w, v), there is no pair σ′ such that (v, w)R21σ′ and σ′R22(w, v). 12Right commutativity in our framework corresponds to the property that for all σ, τ, υ, if σR2iτ and τR2jυ, then there is a σ′ such that σR2jσ′ and σ′R2iυ, for 1 ≤ j ≤ i ≤ n. 13 (w,w)(v,w) (v, v) (w, v) F R21 R22 R22 R22 R22 R21 R22 R21,2 R21 R21,2 R21 Figure 6: Two-dimensional sequence frame displaying only the R2i relations falsifying right commutativity. Even though no accessibility relation except for R21 is in general reflexive or symmetric, there is at least a weak sense in which n-dimensional sequence frames can be seen as generalizations to n dimensions of S5-frames, a sense that emerges when we restrict attention to the relations between sequences that are i-diagonal for some i. Consider the following definition: Definition 1.4 Let S be non-empty and W = Sn as before. R2i is pseudoreflexive if it is reflexive on W i; it is pseudo-symmetric if for all σ, τ ∈ W i, σR2iτ implies τR2iσ; it is pseudo-transitive if for all σ, τ, υ ∈ W i, σR2iτ and τR2iυ implies σR2iυ; and pseudo-Euclidean if for all σ, τ, υ ∈ W i, σR2iτ and σR2iυ implies τR2iυ. Finally, R2i is a pseudo-equivalence relation if it is pseudo-reflexive and pseudo-Euclidean. Note that if n = 1 (and the language is one-dimensional), the pseudo notions coincide with the original notions, so they are legitimate generalizations of the latter. Now it can be shown that every n-dimensional sequence frame contains a subframe on which every relation R2i is an equivalence relation: Proposition 1.4 Let F = (W, (R2i)1≤i≤n, (R@i)1≤i≤n) be an n-dimensional sequence frame. For each 1 ≤ i ≤ n, R2i is an equivalence relation on W i. In other words, for each 1 ≤ i ≤ n, R2i is a pseudo-equivalence relation on W . Proof. Obviously, R2i is pseudo-reflexive, since any member of W i is idiagonal and identical with itself beyond i. Moreover, it is easy to see that R2i is pseudo-Euclidean, for if σR2iτ and σR2iυ, then σ, τ , and υ are all identical beyond i, so that τR2iυ as long as υ ∈ W i. 14 In addition, observe that any n-dimensional sequence frame has a certain kind of universality property governing its last accessibility relation, R2n , in the sense that every point inW is R2n-related to every n-diagonal point. This property, which we may call shift universality, plays an important role in the completeness theorem below, for we use the fact that if a point is n-diagonal, then it isR2n-accessible to any point in the frame. Yet, shift universality is not modally definable. To verify this, let F1 = (W1,R1) be a usual Kripke frame with points w, u, v such that R1 = {(w, v), (v, v), (u, v)}, and F2 = (W2,R2) a frame with points a, b, c such that R2 = {(a, b), (b, b), (c, b)}. Then the firstorder formula expressing shift universality, that is, ∀x∃y(yRx → ∀z zRx), is valid in both F1 and F2, but not in their disjoint union, as illustrated in Figure 7. Therefore, by the Goldblatt-Thomason theorem-which states that a firstorder definable class of frames is modally definable if and only if it is closed under taking disjoint unions, generated subframes, bounded morphic images, and reflects ultrafilter extensions-shift universality is not modally definable.13 w v u F1 a b c F2 w v u a b c ⊎ i∈{1,2} Fi Figure 7: Non-definability of shift universality. Next we define concepts such as models based on n-dimensional sequence frames, truth of a formula at a sequence in a model, as well as the core logical notions of validity and consequence. 13The Goldblatt-Thomason theorem is proved in Blackburn et al. (2001: 180-183), §3.8, Theorem 3.19. The reader can also find definitions for the notions of disjoint unions, generated subframes, bounded morphic images, and ultrafilter extensions in Blackburn et al. (2001), §3.3. The notions of generated subframes and bounded morphic images will be used later in the proof of Lemma 1.5. 15 Definition 1.5 (n-dimensional sequence models) An n-dimensional sequence model is a quadruple, M = (W, (R2i)1≤i≤n, (R@i)1≤i≤n, V ), where V is a function assigning to each p ∈ PROP a subset V (p) ⊆ W . We say that M is based on an n-dimensional sequence frame, F = (W, (R2i)1≤i≤n, (R@i)1≤i≤n). Definition 1.6 (Truth) We define 'φ is true at σ in a model M', written M, σ  φ, by recursion on φ. For a sequence σ ∈ W , and a valuation V in M, M, σ  p ⇐⇒ σ ∈ V (p) M, σ  ¬φ ⇐⇒ M, σ 2 φ M, σ  (φ ∧ ψ) ⇐⇒ M, σ  φ and M, σ  ψ M, σ  2iφ ⇐⇒ for every τ ∈ W , if σR2iτ , then M, τ  φ M, σ  @iφ ⇐⇒ for every τ ∈ W , if σR@iτ , then M, τ  φ Definition 1.7 (Logical notions) A sentence φ is satisfiable in a model M iff there is a sequence σ inM such thatM, σ  φ, and satisfiable in a frame F iff it is satisfiable in a model based on F. A sentence φ is valid in an n-dimensional sequence model M, written M  φ, iff M, σ  φ for every σ ∈ W . A sentence φ is valid at a point σ in a frame F, written F, σ  φ, iff φ is true at σ in every model M based on F, and φ is valid in a frame F, written F  φ, iff it is valid at every point in F. A sentence φ is valid on a class of frames C, written C  φ, iff φ is valid in every F ∈ C. A sentence φ is a logical consequence of a set of sentences Γ over a class of frames C if and only if for every M in C and sequence σ in M, if M, σ  γ for all γ ∈ Γ, then M, σ  φ. 1.2 Axiomatization In this section we axiomatize n-dimensional sequence frames. The logic of n-dimensional sequence frames is called S@n: Definition 1.8 (The logic S@n) Let 1 ≤ i ≤ n, where n ≥ 2. S@n is the 16 classical normal modal logic defined by the following axioms: (T21) 21p→ p. (42i) 2ip→ 2i2ip. (52i) 3ip→ 2i3ip. (T@1) @1p→ p. (D@i) @ip→ ¬@i¬p. (F@i) ¬@i¬p→ @ip. (Ai) 2i−1p→ @ip, for 1 < i ≤ n. (ATi) 2ip→ @ip. (A4i) @ip→ 2j@ip, for 1 ≤ j < i ≤ n. (ARi) 2i(@ip→ p). Where `n denotes the provability relation in S@n, Γ `n φ if and only if there are formulas ψ1, ..., ψn in Γ such that `n ψ1 ∧ ... ∧ ψn → φ. As derived rules we can also add the rule of regularity and the rule of congruentiality, similarly to the basic modal case: (RR) If `n p→ q, then `n 2ip→ 2iq. (RC) If `n p↔ q, then `n 2ip↔ 2iq. Moreover, left commutativity 2i2jp → 2j2ip and the Church-Rosser axiom 3j2ip→ 2i3jp are both derivable from the axioms above. With the exception of T@1 , and by setting n = 2, the axioms above are just the ones found in Fritz (2014: 391), for the logic 2Dg. Note that by setting n = 2 there are two box-like operators in the language, namely, 21, corresponding to the (metaphysical) necessity operator 2 in 2Dg, and the final box-like operator 22, corresponding to the a priori operator A in 2Dg. The reason why T@1 is not found in 2Dg is because its language has a single actuality operator which, provided n = 2, corresponds to @2 in L@2 . Having a single actuality operator in the language is, after all, expected since 2Dg is designed to be a logic of necessity, actuality, and the a priori, and hence @2 is just the traditional actuality operator found in other two-dimensional modal logics, such as Davies and Humberstone's (1980) logic for deep necessity, for example. As mentioned before, we have added @1 to the language for the sake of generality, so that there are n boxes and actuality operators. Nevertheless, it is now easy to see that p→ @1p is derivable from the axioms above, and hence 17 p ↔ @1p is also derivable, by T@1 , which in turn means that @1 is directly eliminable from the language L@2 (in fact, L@n ) without loss of expressive power. Therefore, by discounting the @1 operator, S@n is a generalization of 2Dg to n dimensions with many actuality operators. 1.3 Completeness The axioms listed above are all Sahlqvist formulas, and hence we can compute their locally corresponding frame conditions by the general Sahlqvist algorithm for converting modal formulas into first-order formulas (see Blackburn et al. (2001, chapter 3.6) for a general description), from which it follows that S@n is strongly complete with respect to the class of frames it defines. Informally, we say that a class of frames C is defined by a set of formulas Γ (respectively, formula φ) if for every frame F, F ∈ C just in case F  Γ (respectively, F  φ). Additionally, let ψ(x) be a formula of first-order logic where x is the only free variable in ψ. Then ψ(x) locally corresponds to a modal formula φ if ψ(x) expresses a condition satisfiable on a point w in a first-order structure F just in case φ is valid at w in F, where F is taken as a Kripke frame. That is, for any frame F and point w in F, F, w  φ just in case F, α[x/w]  ψ, where α[x/w] is the assignment function that sends x to w. Then the following first-order formulas locally correspond to the axioms of S@n: T21 21p→ p wR21w 42i 2ip→ 2i2ip ∀vz((wR2iv ∧ vR2iz)→ wR2iz) 52i 3ip→ 2i3ip ∀vz((wR2iv ∧ wR2iz)→ vR2iz) T@1 @1p→ p wR@1w D@i @ip→ ¬@i¬p ∃v(wR@iv) F@i ¬@i¬p→ @ip ∀vz((wR@iv ∧ wR@iz)→ v = z) Ai 2i−1p→ @ip ∀v(wR@iv → wR2i−1v), for 1 < i ≤ n ATi 2ip→ @ip ∀v(wR@iv → wR2iv) A4i @ip→ 2j@ip ∀vz((wR2jv ∧ vR@iz)→ wR@iz), for 1 ≤ j < i ≤ n ARi 2i(@ip→ p) ∀v(wR2iv → vR@iv) Theorem 1.1 (Sahlqvist completeness) S@n is sound and strongly complete with respect to the class of frames FS@n (that is, the class of first-order frames defined by S@n). 18 Proof. By Theorem 4.42 in Blackburn et al. (2001), the proof of which can be found in chapter 5. Let an S@n-frame be any frame in FS@n . It is clear that the axioms above are sound, too, with respect to the class of all n-dimensional sequence frames, given their correspondence to the first-order properties of these frames. An important question is whether the axioms above are also sufficient, that is, whether they settle the completeness problem for S@n with respect to ndimensional sequence frames. This question is settled here by generalizing the approach in Fritz (2014), in which a completeness theorem for 2Dg is derived by first passing through an intermediate class of frames-in that case, the frames defined in Restall (2012). But rather than using an intermediate class of frames such as Restall's, we take the point-generated subframe of a frame in FS@n and construct a bounded morphism from a sequence frame onto this point-generated subframe. Still, the proof strategy extends the main ideas employed in Fritz (2014), Lemmas 2.5-8. First we prove a series of lemmas in order to establish the main completeness result for S@n. In particular, Lemmas 1.1 to 1.4 are instrumental in the proof of Lemma 1.5, which is in turn the principal lemma for the completeness theorem, namely, Theorem 1.3. Lemma 1.1 The following Sahlqvist formulas are derivable in S@n. Let 1 ≤ j < i ≤ n: (1) (Seriality) `n 2ip→ 3ip. (2) (Upward transitivity) `n 2ip→ 2j2ip. This formula corresponds to the frame condition ∀vz((wR2jv ∧ vR2iz)→ wR2iz). (3) (Upward Euclideanity) `n 3ip → 2j3ip. This formula corresponds to the frame condition ∀vz((wR2jv ∧ wR2iz)→ vR2iz). Proof. (1). This is derivable from ATi and its dual, @ip→ 3ip. 19 (2). Consider the following derivation: 1. `n 3ip→ 2i3ip 52i 2. `n 2i3ip→ @i3ip ATi 3. `n 3ip→ @i3ip 1, 2 4. `n ¬@i¬2i¬p→ 2i¬p 3 5. `n @i2ip→ 2ip D@i , 4 6. `n 2j@i2ip→ 2j2ip Necj,K2j , 5 7. `n @i2ip→ 2j@i2ip A4i 8. `n @i2ip→ 2j2ip 6, 7 9. `n 2ip→ 2i2ip 42i 10. `n 2i2ip→ @i2ip ATi 11. `n 2ip→ @i2ip 9, 10 12. `n 2ip→ 2j2ip 8, 11 (3). Consider the following derivation: 1. `n 2i3ip→ 3i3ip (1) 2. `n 3i3ip→ 3ip Dual of 42i 3. `n 2i3ip→ 3ip 1, 2 4. `n 2j2i3ip→ 2j3ip Necj, 3,K2j 5. `n 2i3ip→ 2j2i3ip (2) 6. `n 3ip→ 2i3ip 52i 7. `n 3ip→ 2j3ip 4, 5, 6 For the next lemmas, ROi [X] is the image of the set X under the relation ROi , and for each function R@i , R@i(w) is the unique v such that wR@iv, and im(R@i) is the image of the function R@i . Lemma 1.2 Let F = (W, (R2i)1≤i≤n, (R@i)1≤i≤n) be an S@n-frame, w ∈ W , and Fw = (Ww, (Rw2i)1≤i≤n, (R w @i )1≤i≤n) the subframe of F generated by w. Then: (1) Ww = Rw21 [R w 22 [...[Rw2i [...[R w 2n [{w}]]]]]]. (2) vRw2nu if and only if u ∈ im(R w @n ). 20 Proof. (1). One inclusion is clear. For the other direction, let Z = Rw21 [R w 22 [...[Rw2i [...[R w 2n [{w}]]]]]]. We first prove that (1a) w ∈ Z, and then that (1b) Z is closed under every relation RwOi in the frame F w. (1a). Let z be the element in Ww such that Rw@n(w) = z, and for each i such that 1 ≤ i < n, let Rw@i(w) = zi and R w @i (zi) = vi. Then wRw2nz, by ATn, and both wRw2n−1zn−1 and zn−1R w 2n−1vn−1, by ATn−1. So wR w 2n−1vn−1, by transitivity of Rw2n−1 . But wR w 2n−1z follows by An, and so zR w 2n−1vn−1, since Rw2n−1 is Euclidean. Then wRw2nzR w 2n−1vn−1. Moreover, both wRw2n−2zn−2 and zn−2R w 2n−2vn−2 follow by ATn−2. So wRw2n−2vn−2, by transitivity of R w 2n−2 . Also, both wR w 2n−2zn−1 and zn−1Rw2n−2vn−1, by An−1. So wR w 2n−2vn−1, again by transitivity of R w 2n−2 . Hence vn−1Rw2n−2vn−2, as R w 2n−2 is Euclidean. Then wRw2nzR w 2n−1vn−1R w 2n−2vn−2. By repeating this argument multiple times, it follows that there is a chain wRw2nzR w 2n−1vn−1R w 2n−2vn−2...R w 23 v3Rw22v2. Now, by A2, both wRw21z2 and z2R w 21 v2, whence wRw21v2 and so v2R w 21 w, by transitivity and symmetry of Rw21 , respectively, thereby extending the chain above to (∗) wRw2nzR w 2n−1vn−1R w 2n−2vn−2...R w 23 v3Rw22v2R w 21 w. Therefore, w ∈ Z. (1b). For Rw21 , suppose that v ∈ R w 21 [Z]. Then there is a u ∈ Z such that uRw21v. But since u ∈ Z, it follows by an argument similar to that which established (∗) in (1a) that there are zn, ..., z2 ∈ Ww such that wRw2nznR w 2n−1zn−1R w 2n−2zn−2...R w 23 z3Rw22z2R w 21 u. By transitivity of Rw21 , it then follows that z2R w 21 v, and so v ∈ Z. For Rw2i , where 1 < i ≤ n, suppose that v ∈ R w 2i [Z]. Then there is a u ∈ Z such that uRw2iv. But since u ∈ Z, it follows by an argument similar to the one establishing (∗) in (1a), that there are zn, ..., z2 ∈ Ww such that wRw2nznR w 2n−1zn−1R w 2n−2zn−2...R w 23 z3Rw22z2R w 21 u. 21 By upward transitivity, it follows that z2Rw2iv. Now, if i = 2, then z3R w 22 v, by transitivity of Rw22 , from which it follows that v ∈ Z, as R w 21 is reflexive. If, on the other hand, 2 < i < n, since z3Rw22z2, it follows that z3R w 2i v, by upward transitivity; since z4Rw23z3 and z3R w 2i v, it follows that z4Rw2iv by upward transitivity; after multiple repetitions of this argument, since ziRw2i−1zi−1 and zi−1Rw2iv, it follows that ziR w 2i v by upward transitivity. Then vRw@iv, by ARi, whence vRw2i−1v, by Ai, and so vR w @i−1 v, by ARi−1, whence vRw2i−2v, by Ai−1, and so on. After multiple repetitions of this argument, vRw@3v, by AR3, whence vRw22v, by A3. Also vR w 21 v, since Rw21 is reflexive, and so v ∈ Z. In case i = n, since wRw2nzn, and znR w 2n v follows as in the argument above, it then follows that wRw2nv by transitivity of R w 2n . The rest of the argument for the descending chain is exactly as the case 2 < i < n above. For Rw@1 , suppose that v ∈ R w @1 [Z] = Z, since Rw@1 is the identity function. So v ∈ Z. For Rw@i , where 1 < i ≤ n, suppose that v ∈ R w @i [Z]. Then there is a u ∈ Z such that uRw@iv. So uR w 2i v, by ATi. That v ∈ Z now follows from the argument for Rw2i . (2). For the left-to-right direction, suppose that vRw2nu. By ARn, uR w @n u, so u ∈ im(Rw@n). For the converse, assume that u ∈ im(R w @n ), and let z be the element in Ww such that Rw@n(w) = z. Now, consider any point v ∈ W w. We show that (2a) vRw2nz, and then that (2b) zR w 2n u, from which vRw2nu follows by transitivity of Rw2n , thereby proving the lemma. (2a) Since v ∈ Ww, it follows by an argument similar to the one establishing (∗) in (1a) that there are zn, ..., z2 ∈ Ww such that wRw2nznR w 2n−1zn−1R w 2n−2zn−2...R w 23 z3Rw22z2R w 21 v. By symmetry of Rw21 , it follows that vR w 21 z2. Now consider z′3 ∈ Rw@2(z3). By AT2, it follows that z3Rw22z ′ 3. So z3Rw22z2 and z3R w 22 z′3, from which z2Rw22z ′ 3 follows from the Euclidean property of Rw22 . So vR w 21 z2 and z2Rw22z ′ 3, hence vRw22z ′ 3, by upward transitivity. Now consider z′4 = Rw@3(z4). By AT3, it follows that z4Rw23z ′ 4. So z4Rw23z3 and z4R w 23 z′4, from which z3Rw23z ′ 4 follows from the Euclidean property of Rw23 . Since z3R w 22 z′3, it follows that z′3Rw23z ′ 4, by upward Euclideanity. And since vRw22z ′ 3, it follows that vRw23z ′ 4, by upward transitivity. By repeating this argument multiple times, it follows that vRw2n−1z ′ n. Now consider z = Rw@n(w). By ATn, it follows that wR w 2n z. So wRw2nzn and wR w 2n z, whence znRw2nz, by the Euclidean property ofR w 2n . Then znRw2n−1z ′ n (by a previous iteration of the argument) and znRw2nz, so z ′ nRw2nz follows by upward Euclideanity. Therefore, vRw2n−1z ′ n and z′nRw2nz, from which 22 vRw2nz follows by upward transitivity. (2b). Since u ∈ Ww, by hypothesis, it follows by an argument similar to the one establishing (∗) in (1a) that there are zn, ..., z2 ∈ Ww such that wRw2nznR w 2n−1zn−1R w 2n−2zn−2...R w 23 z3Rw22z2R w 21 u. Consider z = Rw@n(w). Then wR w 2n z, by ATn. Since Rw2n is Euclidean, it follows that zRw2nzn. Since u ∈ im(R w @n ), there is a un ∈ Ww such that unRw@nu. By ATn, unR w 2n u, whence uRw@nu, by ARn, and so uR w 2n u, by ATn. By An, in turn, it also follows that unRw2n−1u, so uR w @n−1 u by ARn−1, whence uRw2n−2u by An−1. Then uR w @n−2 u by ARn−1, whence uRw2n−3u by An−2. By repeating this argument multiple times, it follows that uRw23u by A4 and, similarly, that uRw22u by A3. Now, since z2R w 21 u and uRw22u, z2R w 22 u follows by upward transitivity. And because z3Rw22z2 and z2R w 22 u, z3Rw22u follows by transitivity of Rw22 . But since z3R w 22 u and uRw23u, z3R w 23 u follows by upward transitivity, and so z4Rw23u follows by transitivity of R w 23 . By repeating this argument multiple times it follows that znRw2n−1u. But since uR w 2n u, it follows that znRw2nu by upward transitivity. And since zR w 2n zn, it follows that zRw2nu by transitivity of Rw2n , as desired. Lemma 1.3 Let F = (W, (R2i)1≤i≤n, (R@i)1≤i≤n) be an S@n-frame, and v ∈ W . Then R@i(R@i+1(v)) = R@i+1(v), for 1 ≤ i < n. Proof. Let R@i+1(v) = z. Then vR2i+1z, by ATi+1, and so zR@i+1z follows by ARi+1. Now zR2iz follows by Ai+1, and zR@iz follows by ARi. So vR@i+1zR@iz, and hence R@i(R@i+1(v)) = R@i(z) = z = R@i+1(v), as desired. In order to establish the main completeness result of this section we now want to show the existence and relevant properties of certain families of surjective functions, which are essential to construct the relevant bounded morphism from a sequence frame onto an S@n-frame (or, more precisely, a point-generated subframe of an S@n-frame). In Fritz's (2014), Lemma 2.8, completeness proof for the two-dimensional case, this is done as follows. Let F = (W,R21 ,R22 ,R@1 ,R@2) be an S@2-frame, and Fw = (Ww,Rw21 ,R w 22 ,Rw@1 ,R w @2 ) the subframe of F generated by w. Validity is preserved by taking generated subframes,14 and hence the S@2 axioms hold 14This is proved in Blackburn et al. (2001: 140), Theorem 3.14(ii). 23 in Fw. Now, for every element w ∈ Ww there is a surjective function g1w : W w → Rw21 [g 0 ∅(w)] such that g 1 w(w) = Rw@2(g 0 ∅(w)), where g 0 ∅ = IdWw . The family g1 of surjections exists since for every w ∈ Ww, Rw21 [g 0 ∅(w)] ⊆ Ww, and also becauseRw@2 is a function such thatR w @2 ⊆ Rw21 , byA2. We then construct a two-dimensional sequence frame, F′ = (W ′,R′21 ,R ′ 22 ,R′@1 ,R ′ @2 ), where W ′ = Ww×Ww, and define a map f from W ′ onto Ww by f((w, v)) = g1v(w), which is a bounded morphism from F′ onto Fw-this can be verified by a standard back-and-forth argument. Since we know by Sahlqvist completeness (Theorem 1.1) that S@2 is strongly complete with respect to S@2-frames, completeness of S@2 relative to the class of two-dimensional sequence frames now follows since modal satisfaction is invariant under taking bounded morphisms between models, and the fact that every formula in the language is satisfiable in an S@2-model if and only if it is satisfiable in a point-generated submodel of it.15 This is by and large Fritz's argument except for minor notational differences, the absence of anR@i relation, and the fact that he uses an intermediate class of frames from Restall (2012) indirectly instead of the point-generated subframes Fw. To see how this method can be generalized for n dimensions it is instructive first to briefly mention the adaptation of the argument above for three dimensions, as the definition of the bounded morphism in this case will need multiple families of surjections which are constructed in stages, just as in the n case. So, let F be an S@3-frame, and Fw the subframe of F generated by w. In order to prove that S@3 is complete relative to three-dimensional sequence frames, we construct a three-dimensional sequence frame, F′ = (W ′, (R′2i)1≤i≤n, (R ′ @i )1≤i≤n), where W ′ = Ww×Ww×Ww, and a bounded morphism from F′ onto Fw. This is done as follows. Let g0∅ = IdWw . Then for every w ∈ Ww there is a surjective function g1w : Ww → Rw22 [g 0 ∅(w)] such that (i) g1w(w) = Rw@3(g 0 ∅(w)). The family g1 of surjections exists since for any w ∈ Ww, Rw22 [g 0 ∅(w)] ⊆ Ww, and also because Rw@3 is a function such that R w @3 ⊆ Rw22 , by A3. Additionally, for every w, v ∈ Ww there is a surjective function g2w,v : Ww → Rw21 [g 1 v(w)] such that (ii) g2w,v(w) = Rw@2(g 1 v(w)). 15See Blackburn et al. (2001), §2.1, for the definitions of bounded morphisms between models and generated submodels, as well as the relevant invariance results. 24 The family g2 of surjections exists since for any w, v ∈ Ww, Rw21 [g 1 v(w)] ⊆ Ww, and also because Rw@2 is a function such that R w @2 ⊆ Rw21 , by A2. Then for every w ∈ Ww, since g2w,w(w) = Rw@2(g 1 w(w)), by (ii), g1w(w) = Rw@3(g 0 ∅(w)), by (i), and Rw@2(R w @3 (w)) = Rw@3(w), by Lemma 1.3, it follows that (iii) g2w,w(w) = Rw@3(w). The map f from W ′ onto Ww defined by f((w, v, z)) = g2v,z(w) is then a bounded morphism from F′ onto Fw, which can again be verified by a standard back-and-forth argument. The aim of the next two lemmas is to register the existence of similar surjections in the general n-dimensional case as well as to construct the desired bounded morphism for the completeness proof. Lemma 1.4 Let F = (W, (R2i)1≤i≤n, (R@i)1≤i≤n) be an S@n-frame. Then: (1) There is a sequence g0, ..., gn−1 of families of surjective functions defined on W , where the family gi is indexed by W i, g0∅ = IdW , and for each 1 ≤ i ≤ (n− 1) and every w1, ..., wi ∈ W , giw1,...,wi : W → R2n−i [g i−1 w2,...,wi (w1)] such that giw1,...,wi(w1) = R@n−(i−1)(g i−1 w2,...,wi (w1)). (2) If an initial segment of w1, ..., wn−1 is such that w1 = ... = wj = z, for 1 ≤ j ≤ (n− 1), and z ∈ W , then gn−1w1,...,wn−1(z) = R@j+1(g n−(j+1) wj+1,...,wn−1(wj)). Proof. (1). The proof is by induction on 1 ≤ i ≤ (n − 1). For i = 1, since for any w1 ∈ W , R2n−1 [g0∅(w1)] ⊆ W , and R@n is a function such that R@n ⊆ R2n−1 , by An, there is a family of surjective functions g1w1 : W → R2n−1 [g 0 ∅(w1)] such that g1w1(w1) = R@n(g 0 ∅(w1)). Assume the induction hypothesis, for i = k, where 1 ≤ k < (n− 1), that there is family of surjective functions 25 gkw1,...,wk : W → R2n−k [g k−1 w2,...,wk (w1)] such that gkw1,...,wk(w1) = R@n−(k−1)(g k−1 w2,...,wk (w1)). We show that the lemma holds for i = k + 1. Since, by the induction hypothesis, there are functions gkw1,...,wk , and both R2n−(k+1) [g k w2,...,wk+1 (w1)] ⊆ W and R@n−k ⊆ R2n−(k+1) , by An−k, it follows that there is a family of surjective functions gk+1w1,...,wk+1 : W → R2n−(k+1) [g k w2,...,wk+1 (w1)] such that gk+1w1,...,wk+1(w1) = R@n−k(g k w2,...,wk+1 (w1)), as desired. (2). The proof is by induction on j. For j = 1, it follows from (1) that gn−1w1,...,wn−1(z) = R@2(g n−2 w2,...,wn−1(w1)). Now let j = k, where 1 ≤ k < (n− 1), and assume that gn−1w1,...,wn−1(z) = R@k+1(g n−(k+1) wk+1,...,wn−1(wk)). Suppose that w1 = ... = wk+1 = z. Then the family of surjections g n−(k+1) wk+1,...,wn−1 : W → R2n−(n−(k+1)) [g (n−(k+1))−1 wk+2,...,wn−1(wk+1)] is such that g n−(k+1) wk+1,...,wn−1(wk) = R@n−((n−(k+1))−1)(g (n−(k+1))−1 wk+2,...,wn−1(wk+1)), by (1), and hence gn−1w1,...,wn−1(z) = R@k+1(R@n−((n−(k+1))−1)(g (n−(k+1))−1 wk+2,...,wn−1(wk+1))). Note, moreover, that n− ((n− (k + 1))− 1) = k + 2, and hence gn−1w1,...,wn−1(z) = R@n−((n−(k+1))−1)(g (n−(k+1))−1 wk+2,...,wn−1(wk+1)), by Lemma 1.3, as desired. Lemma 1.5 Let F = (W, (R2i)1≤i≤n, (R@i)1≤i≤n) be an S@n-frame, w ∈ W , and Fw = (Ww, (Rw2i)1≤i≤n, (R w @i )1≤i≤n) the subframe of F generated by w. Then Fw is a bounded morphic image of a sequence frame. 26 Proof. Let F, w ∈ W , and Fw be as in the hypothesis. Because generated subframes preserve validity between frames, all of the S@n axioms hold in Fw. So in what follows we construct a sequence frame, F′ = (W ′, (R′2i)1≤i≤n, (R ′ @i )1≤i≤n), such that W ′ = (Ww)n, where (Ww)n is the n-fold Cartesian product of Ww, and a surjective bounded morphism f : F′  Fw. Then, where σ ∈ (Ww)n, let f : (Ww)n → Ww be defined as f(σ) = gn−1σ2,...,σn(σ1), where the sequence g0, ..., gn−1 of families of surjective functions is defined on Ww. We prove that f so defined is a bounded morphism from F′ onto Fw by checking the back and forth conditions for each accessibility relation as follows: [R21 ] Suppose that σR′21σ ′. Then σ and σ′ are identical beyond 1. So f(σ) = gn−1σ2,...,σn(σ1) ∈ R w 21 [gn−2σ3,...,σn(σ2)], and f(σ ′) = gn−1σ′2,...,σ′n (σ′1) ∈ Rw21 [g n−2 σ3,...,σn (σ2)], from which it follows that f(σ)Rw21f(σ ′). Conversely, suppose that f(σ)Rw21z. Since f(σ) = gn−1σ2,...,σn(σ1) ∈ R w 21 [gn−2σ3,...,σn(σ2)], it follows that z ∈ Rw21 [g n−2 σ3,...,σn (σ2)]. And since gn−1σ2,...,σn is a surjection, there is a v ∈ Ww such that gn−1σ2,...,σn(v) = z, whence f(σ 1 v) = g n−1 σ2,...,σn (v) = z. Furthermore, σR′21σ 1 v . [R2i , 1 < i < n] Suppose that σR′2iσ ′. Then σ′ is i-diagonal and identical with σ beyond i. So f(σ′) = gn−1σ′2,...,σ′n(σ ′ 1) = Rw@i(g n−i σ′i+1,...,σ ′ n (σ′i)), by Lemma 1.4(2), and f(σ) = gn−1σ2,...,σn(σ1) ∈ R w 21 [gn−2σ3,...,σn(σ2)]. Since gn−iσ′i+1,...,σ′n (σ′i)Rw@if(σ ′), it follows that gn−iσ′i+1,...,σ′n(σ ′ i)Rw2if(σ ′), by ATi. Also, gn−iσ′i+1,...,σ′n (σ′i) ∈ Rw2i [g (n−i)−1 σi+2,...,σn(σi+1)], since the sequences σ and σ′ are identical beyond i, whence g(n−i)−1σi+2,...,σn(σi+1)Rw2if(σ ′), by transitivity of Rw2i . But gn−2σ3,...,σn(σ2) ∈ R w 22 [gn−3σ4,...,σn(σ3)], gn−3σ4,...,σn(σ3) ∈ R w 23 [gn−4σ5,...,σn(σ4)], ..., gn−iσi+1,...,σn(σi) ∈ R w 2i [g (n−i)−1 σi+2,...,σn(σi+1)]. So gn−iσi+1,...,σn(σi)R w 2i f(σ′), by the Euclidean property of Rw2i . Now, g n−(i−1) σi,...,σn (σi−1) ∈ Rw2i−1 [g n−i σi+1,...,σn (σi)]. So g n−(i−1) σi,...,σn (σi−1)Rw2if(σ ′), by upward Euclideanity. By an analogous argument, since gn−(i−2)σi−1,...,σn(σi−2) ∈ Rw2i−2 [g n−(i−1) σi,...,σn (σi−1)], it follows that g n−(i−2) σi−1,...,σn(σi−2)Rw2if(σ ′), by upward 27 Euclideanity. By multiple repetitions of this argument, it follows that gn−2σ3,...,σn(σ2)R w 2i f(σ′). Since gn−1σ2,...,σn(σ1) ∈ R w 21 [gn−2σ3,...,σn(σ2)], by upward Euclideanity, again, it follows that gn−1σ2,...,σn(σ1)R w 2i f(σ′), and therefore f(σ)Rw2if(σ ′). Conversely, suppose that f(σ)Rw2iz. Then f(σ) = g n−1 σ2,...,σn (σ1) ∈ Rw21 [g n−2 σ3,...,σn (σ2)]. So gn−2σ3,...,σn(σ2)R w 21 f(σ)Rw2iz, from which gn−2σ3,...,σn(σ2)R w 2i z follows by upward transitivity. Now, gn−2σ3,...,σn(σ2) ∈ Rw22 [g n−3 σ4,...,σn (σ3)], whence gn−3σ4,...,σn(σ3)R w 22 gn−2σ3,...,σn(σ2)R w 2i z, from which gn−3σ4,...,σn(σ3)R w 2i z follows by upward transitivity. Similarly, gn−3σ4,...,σn(σ3) ∈ R w 23 [gn−4σ5,...,σn(σ4)], whence gn−4σ5,...,σn(σ4)R w 23 gn−3σ4,...,σn(σ3)R w 2i z, from which gn−4σ5,...,σn(σ4)R w 2i z follows by upward transitivity. By repeating this argument, gn−iσi+1,...,σn(σi)R w 2i z, but since gn−iσi+1,...,σn(σi) ∈ R w 2i [g (n−i)−1 σi+2,...,σn(σi+1)], it follows that g (n−i)−1 σi+2,...,σn(σi+1)Rw2ig n−i σi+1,...,σn (σi)Rw2iz, and so g(n−i)−1σi+2...σn(σi+1)Rw2iz follows by transitivity of R w 2i . Then z ∈ Rw2i [g (n−i)−1 σi+2,...,σn(σi+1)]. Now, since gn−iσi+1,...,σn is a surjective function, it follows that there is a v ∈ Ww such that (∗) gn−iσi+1...σn(v) = z. So consider the i-diagonal sequence (v, ..., v, σi+1, ..., σn) ∈ (Ww)n, which is identical with σ beyond i. Then σR′2i(v, ..., v, σi+1, ..., σn), and so: f((v, ..., v, σi+1, ..., σn)) = g n−1 v,...,v,σi+1,...,σn (v) [by def. of f ] = Rw@i(g n−i σi+1,...,σn (v)) [by Lemma 1.4(2)] = Rw@i(z). [by (∗)] So zRw@if((v, ..., v, σi+1, ..., σn)), whence zR w 2i f((v, ..., v, σi+1, ..., σn)), by ATi. Then f(σ)Rw2if((v, ..., v, σi+1, ..., σn)) follows by transitivity of R w 2i . Furthermore, zRw@iz follows from the original hypothesis and ARi, in which case both zRw@if((v, ..., v, σi+1, ..., σn)) and zR w @i z, from which f((v, ..., v, σi+1, ..., σn)) = z follows by F@i , as desired. 28 [R2n ] Suppose that σR′2nσ ′. So σ′ is n-diagonal. Then f(σ′) = gn−1σ′2,...,σ′n (σ′1) = Rw@n(σ ′ n), by Lemma 1.4(2), whence f(σ′) ∈ im(Rw@n). It follows that f(σ)Rw2nf(σ ′), by Lemma 1.2(2). Conversely, suppose that f(σ)Rw2nz. So zRw@nz, by ARn, in which case there is a v ∈ W w, namely, u, such that (?) Rw@n(v) = z, as Rw@n is a function. Now consider the sequence σ n v ∈ (Ww)n. Then: f(σnv ) = g n−1 v,...,v(v) [by def. of f ] = Rw@n(v) [by Lemma 1.4(2)] = z. [by (?)] Additionally, σR′2nσ n v . [R@1 ] Suppose that σR′@1σ ′. Then σ = σ′. Since Rw@1 is reflexive, it follows that f(σ)Rw@1f(σ ′). Now suppose that f(σ)Rw@1z. Because R w @1 is reflexive, f(σ)Rw@1f(σ), hence f(σ) = z, as R w @1 is a function. Also, σR′@1σ. [R@i , 1 < i < n] Suppose that σR′@iσ ′. Then σ′ is i-diagonal and identical with σ beyond i− 1. So f(σ′) = gn−1σ′2,...,σ′n (σ′1) = Rw@i(g n−i σ′i+1,...,σ ′ n (σ′i)), by Lemma 1.4(2), and f(σ) = gn−1σ2,...,σn(σ1) ∈ R w 21 [gn−2σ3,...,σn(σ2)]. Then f(σ)Rw21g n−2 σ3,...,σn (σ2), by symmetry of Rw21 . Also, gn−2σ3,...,σn(σ2) ∈ R w 22 [gn−3σ4,...,σn(σ3)], gn−3σ4,...,σn(σ3) ∈ R w 23 [gn−4σ5,...,σn(σ4)], ..., g n−(i−1) σi,...,σn (σi−1) ∈ Rw2i−1 [g n−i σi+1,...,σn (σi)]. Now, since σ′ is identical with σ beyond i−1, it also follows that gn−(i−1)σi,...,σn (σi−1) ∈ [gn−iσ′i+1,...,σ′n(σ ′ i)]Rw2i−1 . And since g n−i σ′i+1,...,σ ′ n (σ′i)Rw@if(σ ′), it follows that gn−iσ′i+1,...,σ′n (σ′i)Rw2i−1f(σ ′), by Ai, and so g n−(i−1) σi,...,σn (σi−1)Rw2i−1f(σ ′), as Rw2i−1 is Euclidean. But since g n−(i−2) σi−1,...,σn(σi−2) ∈ Rw2i−2 [g n−(i−1) σi,...,σn (σi−1)], it follows that gn−(i−2)σi−1,...,σn(σi−2)Rw2i−1f(σ ′), by upward Euclideanity. And since g n−(i−3) σi−2,...,σn(σi−3) ∈ Rw2i−3 [g n−(i−2) σi−1...σn(σi−2)], it follows that 29 g n−(i−3) σi−2,...,σn(σi−3)Rw2i−1f(σ ′), again by upward Euclideanity. By repeating this argument, it follows that gn−2σ3,...,σn(σ2)R w 2i−1 f(σ′), by upward Euclideanity. Then f(σ)Rw2i−1f(σ ′), follows from upward Euclideanity, too. But since gn−iσ′i+1,...,σ′n(σ ′ i)Rw@if(σ ′), it follows that f(σ′)Rw@if(σ ′), by ATi and ARi, respectively. So f(σ)Rw@if(σ ′), by A4i. Conversely, suppose that f(σ)Rw@iz. Consider the sequence σ ′ ∈ (Ww)n such that σ′ is i-diagonal and identical with σ beyond i− 1. Then σR′@iσ ′. So f(σ′) = gn−1σ′2,...,σ′n (σ′1) = Rw@i(g n−i σ′i+1,...,σ ′ n (σ′i)), by Lemma 1.4(2), and f(σ) = gn−1σ2,...,σn(σ1) ∈ R w 21 [gn−2σ3,...,σn(σ2)]. By the same argument as in the forth direction, it follows that f(σ)Rw@if(σ ′). But since f(σ)Rw@iz, by assumption, it follows that f(σ ′) = z, by F@i . [R@n ] Suppose that σR′@nσ ′. Then σ′ is n-diagonal and identical with σ beyond (n− 1). So f(σ′) = gn−1σ′2,...,σ′n (σ′1) = Rw@n(σ ′ n), by Lemma 1.4(2), and f(σ) = gn−1σ2,...,σn(σ1) ∈ R w 21 [gn−2σ3,...,σn(σ2)]. Since σn = σ ′ n, it follows that σnRw@nf(σ ′), and so both σnRw2n−1f(σ ′), by An, and f(σ′)Rw@nf(σ ′), by ARn. Now, g1σn(σn−1) ∈ R w 2n−1 [{σn}], so it follows that g1σn(σn−1)R w 2n−1f(σ ′), as Rw2n−1 is Euclidean. Also, g 2 σn−1,σn(σn−2) ∈ Rw2n−2 [g 1 σn(σn−1)]. So g 2 σn−1,σn(σn−2)R w 2n−1f(σ ′) by upward Euclideanity. Analogously, g3σn−2,σn−1,σn(σn−3) ∈ R w 2n−3 [g 2 σn−1,σn(σn−2)]. So g3σn−2,σn−1,σn(σn−3)R w 2n−1f(σ ′), again by upward Euclideanity. By repetitions of this argument, it follows that gn−2σ3,...,σn(σ2)R w 2n−1f(σ ′), by upward Euclideanity. Since f(σ)Rw21g n−2 σ3,...,σn (σ2), by symmetry of Rw21 , it follows that f(σ)Rw2n−1f(σ ′), by upward transitivity. Therefore, f(σ)Rw@nf(σ ′), by A4n. Conversely, suppose that f(σ)Rw@nz. Now consider the sequence σ ′ ∈ (Ww)n such that σ′ is n-diagonal and identical with σ beyond n − 1. Also σR′@nσ ′. So f(σ′) = gn−1σ′2,...,σ′n (σ′1) = Rw@n(σ ′ n), by Lemma 1.4(2), and f(σ) = gn−1σ2,...,σn(σ1) ∈ R w 21 [gn−2σ3,...,σn(σ2)]. By the same argument as in the forth direction, it follows that f(σ)Rw@nf(σ ′). But since f(σ)Rw@nz, it follows that f(σ ′) = z, by F@n . 30 This concludes the back and forth cases. It remains to show that f is surjective. Consider any u ∈ Ww. By Lemma 1.2(1), u ∈ Rw21 [R w 22 [...[Rw2i [...[R w 2n [{w}]]]]]], so there are elements vn, ..., v2 ∈ Ww, and a chain wRw2nvnR w 2n−1vn−1R w 2n−2vn−2...R w 23 v3Rw22v2R w 21 u. Consider the mapping g1vn : W w → Rw2n−1 [{vn}]. As this is surjective, and vn−1 ∈ Rw2n−1 [{vn}], there is a zn−1 ∈ W w such that g1vn(zn−1) = vn−1. Now consider the mapping g2zn−1,vn : W w → Rw2n−2 [g 1 vn(zn−1)]. As this is surjective, and vn−2 ∈ Rw2n−2 [g 1 vn(zn−1)], there is a zn−2 ∈ W w such that g2zn−1,vn(zn−2) = vn−2. After multiple repetitions of this argument, consider the mapping gn−1z2,...,zn−1,vn : W w → Rw21 [g n−2 z3,...,zn−1,w(z2)]. By a previous iteration of the argument, gn−2z3,...,zn−1,vn(z2) = v2, and so u ∈ R w 21 [gn−2z3,...,zn−1,vn(z2)]. As gn−1z2,...,zn−1,vn is also surjective, there is a z1 ∈ W w such that gn−1z2,...,zn−1,vn(z1) = u. Therefore, there is a sequence (z1, z2, ..., zn−1, vn) ∈ (Ww)n such that f((z1, z2, ..., zn−1, vn)) = g n−1 z2,...,vn (z1) = u, as desired. This is sufficient to prove that S@n is the multidimensional logic of sequence frames: Theorem 1.2 (Soundness and completeness) S@n is sound and strongly complete with respect to the class F of sequence frames. Proof. Soundness is clear given the correspondence between the axioms of S@n and the properties of n-dimensional sequence frames. For strong completeness, by Theorem 1.1, S@n is strongly complete with respect to the class of S@nframes, that is, FS@n . So it suffices to prove that a set of formulas is satisfiable in a sequence frame if and only if it is satisfiable in an S@n-frame. Suppose that a set of formulas Γ is satisfiable in a sequence frame F ∈ F. Because every sequence frame is an S@n-frame, it follows that Γ is also satisfiable in an S@n-frame. Conversely, suppose that Γ is satisfiable in an S@n-frame, say, F. Then Γ is satisfiable in a model M = (W, (R2i)1≤i≤n, (R@i)1≤i≤n, V ) based on F. So there is a point w in M such that M, w  Γ. Let Mw = (Ww, (Rw2i)1≤i≤n, (R w @i )1≤i≤n, V w) be the submodel of M generated by w. Since modal satisfaction is invariant under taking generated submodels, and w ∈ Ww, it follows that Mw, w  Γ, and so Γ is satisfiable in the point-generated subframe Fw of F. But, by Lemma 1.5, Fw is a bounded 31 morphic image of a sequence frame, say, F′ = (W ′, (R′2i)1≤i≤n, (R ′ @i )1≤i≤n). Let f : F′  Fw be a surjective bounded morphism from F′ onto Fw, and M′ = (W ′, (R′2i)1≤i≤n, (R ′ @i )1≤i≤n, V ′) a model based on F′ such that for each p ∈ PROP, V ′(p) = {σ ∈ W ′ | f(σ) ∈ V w(p)}. Because f is surjective, and w ∈ Ww, let σ′ be a sequence inW ′ such that f(σ′) = w. Then Mw, f(σ′)  Γ. But as Mw is a bounded morphic image of M′, this implies that M′, σ′  Γ, since modal satisfaction is invariant under bounded morphisms between models. Therefore, Γ is satisfiable in the sequence frame F′. Recall that in Fritz (2014: 386) a class of frames with distinguished elements is derived from matrix frames-or, according to our nomenclature, twodimensional sequence frames (again, for a language without @1). The logic of these frames is then derived syntactically from 2Dg, and a completeness proof is presented in Fritz (2014: 394). Given the completeness result established just above, we show that Fritz's argument can once more be generalized for the n-dimensional case. We say that an n-dimensional sequence frame with distinguished elements is a quadruple, FD = (W, (R2i)1≤i≤n, (R@i)1≤i≤n, D) where (W, (R2i)1≤i≤n, (R@i)1≤i≤n) is an n-dimensional sequence frame such that W = Sn, and D = {s | s ∈ S}. Let FD be the class of n-dimensional sequence frames with distinguished elements. Models based on such frames with distinguished elements are defined in the obvious way, and the logical notions of validity and consequence are now relativized to generalized diagonal points s, as expected. Even though, as pointed out by Fritz (2014: 286), such logics characterizing frames with distinguished elements are not, in general, normal, for they do not have to be closed under the rule of generalization (or necessitation) for each Oi, they can be defined syntactically from their normal counterparts, as it were, as follows: Definition 1.9 `Dn φ iff `n @nφ. Let S@Dn be the logic characterizing n-dimensional sequence frames with distinguished elements. Given the strong completeness of S@n relative to the class of n-dimensional sequence frames, the argument for the strong completeness of S@Dn relative to FD is then a simple adaptation of the argument in Fritz (2014: 394), Theorem 2.11. Theorem 1.3 (Soundness and completeness) S@Dn is sound and strongly complete with respect to FD. 32 2 n-dimensional tableaux It is very natural, as will be seen below, to define indexed tableau systems for S@n following the style of Melvin Fitting's prefixed tableaux for modal logics.16 In fact, generalizations of indexed tableau systems for a variety of two-dimensional modal logics appear in Lampert (2018) as well as in Gilbert (2016). In order to define indices and index-sequences for the tableaux, we apply similar conventions as found in Definition 1.2 for sequences of possible worlds. A nice feature of these tableaux is their simplicity, for we only need, in effect, a single rule for each modal operator in the language rather than multiple rules corresponding to multiple properties of the accessibility relations, as is usually the case. Next we define the notions of index-sequences, indexed formulas, and root of an n-dimensional tableau. These will consist in slight modifications of the notions appearing in Definition 1.2 for sequences of possible worlds. Definition 2.1 (Index-sequence, formula, root) An index is a natural number greater than 0. Let s = (x1, ..., xn) be an index-sequence, where each xi is an index. We will often write si for xi. An indexed formula is an expression, [φ]s, where s is an index-sequence and φ is a formula of L@n . All indexed formulas are enclosed within brackets. The root of an n-dimensional tableau always contains the negation of the formula we are attempting to prove indexed by (1, 2, ..., n), that is, an index-sequence with the natural numbers ordered from 1 to n. Definition 2.2 (Notation) Let siz be that index-sequence t for which t1 = t2 = ... = ti = z and for j > i, tj = sj. Additionally, for an index x, x is that sequence s for which si = x for all i ∈ {1, ..., n}. As per usual, a branch of a tableau is closed if for some formula, φ, and index-sequence s, both [φ]s and [¬φ]s occur on the branch. A tableau is closed just in case all of its branches are closed. Otherwise, the tableau is open. Finally, a tableau proof of a sentence, φ, is a closed tableau for [¬φ]s, where s is the index-sequence (1, 2, ..., n). The tableau rules for S@n are displayed in Figure 8 with some provisos for the modal rules explained below. The rules (¬¬), (∧), and (∨) comprise the portion from the classical propositional calculus. The necessity rules, (νi), are applied to any index z already occurring on the branch, and the possibility rules, (πi), may be used provided the index z is new to the branch. These are just the usual restrictions on 16These can be found in several papers as well as in Fitting and Mendelsohn (1998). 33 (¬¬) : [¬¬φ]s [φ]s (∧) : [φ ∧ ψ]s [φ]s [ψ]s [¬(φ ∨ ψ)]s [¬φ]s [¬ψ]s [¬(φ→ ψ)]s [φ]s [¬ψ]s [φ↔ ψ]s [φ→ ψ]s [ψ → φ]s (∨) : [φ ∨ ψ]s [φ]s | [ψ]s [¬(φ ∧ ψ)]s [¬φ]s | [¬ψ]s [φ→ ψ]s [¬φ]s | [ψ]s [¬(φ↔ ψ)]s [¬(φ→ ψ)]s | [¬(ψ → φ)]s (νi) : [2iφ]s [φ]siz [¬3iφ]s [¬φ]siz (πi) : [3iφ]s [φ]siz [¬2iφ]s [¬φ]siz (@i) : [@iφ]s [φ]sisi [¬@iφ]s [¬φ]sisi Figure 8: n-Dimensional Tableau Rules. indexed-tableau rules but generalized for n-dimensional modal logic. In the case of (@i), the rule says to take the ith element of the sequence in question and copy it (i − 1) times towards the very first element in the sequence. For instance, the following is a proof of 33p→ 2221@333p in a four-dimensional tableau: 1. [¬(33p→ 2221@333p)](1, 2, 3, 4) 2. [33p](1, 2, 3, 4) 3. [¬2221@333p](1, 2, 3, 4) 4. [p](5, 5, 5, 4) 5. [¬21@333p](6, 6, 3, 4) 6. [¬@333p](7, 6, 3, 4) 7. [¬33p](3, 3, 3, 4) 8. [¬p](5, 5, 5, 4) × Items 2 and 3 result from the rule (∧) applied to the negated conditional. Item 4 results from applying the (π3) rule to item 2, and so the index-sequence 34 (5, 5, 5, 4) appears on the tableau because the index 5 is new, that is, it had no previous occurrences. Items 5 and 6 result from applications of (π2) and (π1) to items 3 and 5, respectively. The index-sequence (6, 6, 3, 4) appears in 5 because we needed a new index for ¬22 from item 3, and the index-sequence (7, 6, 3, 4) occurs because we needed a new index for ¬21 from item 5. Finally, 7 results from 6 by an application of (@3), and so the index 3 is copied down in the index-sequence twice, and 8 results by applying (ν3), and so any index could have been chosen to compose its index-sequence. The index-sequence (5, 5, 5, 4), therefore, is chosen so that the tableau closes. 2.1 Soundness Definition 2.3 (Satisfiability) Let F be a set of indexed formulas. We say F is satisfiable in a model M = (W, (R2i)1≤i≤n, (R@i)1≤i≤n, V ), where W = Sn, if there is a function f assigning to each single index si occurring in a sequence s in F a possible world f(si) ∈ S, and, where g is a function such that g(s) = (f(s1), ..., f(sn)), • If [φ]s ∈ F , then φ is true at g(s), i.e. M, g(s)  φ. • If the index-sequences s and siz are in F , then g(s)R2ig(s)if(z). If, moreover, z = si, then also g(s)R@ig(s)if(z). Definition 2.4 A tableau branch b is satisfiable if the set of indexed formulas on it is satisfiable in some model, and a tableau is satisfiable if some branch of it is satisfiable. It follows immediately from the definitions that a closed tableau is not satisfiable. Then the following lemma can be established by induction on formulas: Lemma 2.1 If one of the rules is applied to a tableau that is satisfiable in an n-dimensional sequence model M = (W, (R2i)1≤i≤n, (R@i)1≤i≤n, V ), it results in another tableau satisfiable in M. Theorem 2.1 (Soundness) If φ has an n-dimensional tableau proof, then φ is valid on the class of n-dimensional sequence frames F. Proof. Suppose φ has a tableau proof, in which case there is a closed tableau, T , beginning with [¬φ]s, where s is the sequence (1, 2, ..., n). For a contradiction, assume that φ is not valid. Thus, there is an n-dimensional sequence model M = (W, (R2i)1≤i≤n, (R@i)1≤i≤n, V ), where W = Sn, for some 35 non-empty set S, such that M, σ 2 φ for some sequence σ ∈ W . Define a function f such that for each si ∈ s and σi ∈ σ, f(si) = σi, such that g(s) = (f(s1), ..., f(sn)) = σ. Then {[¬φ]s} is satisfiable in M. Moreover, since the one point tableau [¬φ]s is satisfiable in M, T is also satisfiable in M, by Lemma 2.1. Therefore, T is both closed and satisfiable, which is impossible, whence M, σ  φ, as desired. 2.2 Completeness We establish completeness by constructing a systematic tableau proof procedure in the style of Fitting (1983, ch. 8) producing a tableau proof in case there is one, and directing us to a countermodel otherwise. As it is shown below, this will also give us a decision procedure for the validities. 2.2.1 Systematic proof procedure For the purposes of a systematic proof procedure, we do not want to apply the rules for a single occurrence of an indexed formula more than one time, so we need to make sure that each occurrence of an indexed formula is used only once. This is not difficult to keep track of, as we can just introduce a device to declare formulas used-a check mark, for instance, as in Smullyan (1968). The procedure is defined by stages, and for stage n = 1, introduce [¬φ]s at the tableau's root, where s is the sequence (1, 2, ..., n). Next, suppose n stages of the procedure have been completed. If the tableau is closed, or every occurrence of an indexed formula is used, then stop. If, on the other hand, the tableau remains open, then we proceed to stage n + 1 as follows: take the highest occurrence of an indexed formula in the tree, say, [ψ]s, that is not yet used.17 Now for each open branch b through the occurrence of [ψ]s, do the following: I If [ψ]s is atomic or a negation thereof, then declare it used. This ends stage n+ 1. II If [ψ]s is [¬¬χ]t, add [χ]t to the end of b. III If [ψ]s is [(ζ ∧ χ)]t, add both [ζ]t and [χ]t to the end of b (analogously for the other conjunctive cases). 17If there are multiple occurrences of unused formulas at the same level, chose the unused formula occurring on the leftmost branch. 36 IV If [ψ]s is [(ζ∨χ)]t, split the end of b into b1 and b2, adding [ζ]t and [χ]t to the end of b1 and b2, respectively (analogously for the other disjunctive cases). V If [ψ]s is [3iχ]t, take the smallest integer, z, which is new to b, and add [χ]tiz to the end of b (analogously for ¬2i). VI If [ψ]s is [2iχ]t, for every index z occurring on the branch, add [χ]tiz to the end of b (analogously for ¬3i), and then add a fresh occurrence of [2iχ]t to the end of b. If, however, all possible [χ]tiz already occur on the branch, do nothing (not even checking off the original boxed formula). VII If [ψ]s is [@iχ]t, add [χ]titi to the end of b (analogously for ¬@i). Once this procedure is completed for each open branch b through [ψ]s, mark that formula as used. This completes stage n + 1. Now, either the systematic procedure resulted in a closed tableau, producing a proof; the procedure terminated, producing an open branch; or it does not terminate, producing a possibly infinite open branch. Let b be a complete open branch of a tableau T if any application of a rule to an indexed formula occurring on the branch would only introduce indexed formulas already occurring on it. In order to prove completeness we show that if b is any complete open branch of a tableau, then b is satisfiable in an n-dimensional sequence model constructed out of the formulas and indexsequences occurring on b. Once completeness is established we come back to the case where the systematic procedure does not terminate. Definition 2.5 We define a frame Fb = (W, (R2i)1≤i≤n, (R@i)1≤i≤n) induced by b as follows. Let the set of indices in b be S = {x | for some s ∈ b and some i, x = si}, and let W = Sn. For every s, siz ∈ W , set sR2isiz, and when z = si, set sR@isiz, too. A model Mb = (W, (R2i)1≤i≤n, (R@i)1≤i≤n, V ), based on Fb, is defined as follows: for each p ∈ PROP, let s ∈ V (p) provided that [p]s occurs on b; otherwise set s /∈ V (p). It is simple to verify that the frames defined above are in fact n-dimensional sequence frames. The following lemma is then established by induction on formulas: Lemma 2.2 (Truth lemma) Let b be a complete open branch of a tableau. Then, let Fb = (W, (R2i)1≤i≤n, (R@i)1≤i≤n) be a frame induced by b, and Mb = (W, (R2i)1≤i≤n, (R@i)1≤i≤n, V ) a model based on Fb. For every indexed formula, [φ]s, 37 [φ]s occurs on b⇔Mb, s  φ. Theorem 2.2 (Completeness) If φ is valid on the class of n-dimensional sequence frames F, then φ has an n-dimensional tableau proof. Proof. We prove the contrapositive. Suppose φ does not have a tableau proof, in which case any tableau T beginning with [¬φ]s remains open. Let b be a complete open branch of T such that [¬φ]s occurs on b. By Lemma 2.2, there is an n-dimensional sequence frame Fb = (W, (R2i)1≤i≤n, (R@i)1≤i≤n), induced by b, and index-sequence s ∈ W , such that for an n-dimensional sequence model, Mb = (W, (R2i)1≤i≤n, (R@i)1≤i≤n, V ), based on Fb, we have Mb, s 2 φ. Consequently, φ is not valid on the class of n-dimensional sequence frames F. n-dimensional tableaux can be easily adapted for n-dimensional sequence frames with distinguished elements. For an index x, let x be that indexsequence s = (x1, ..., xn) for which xi = x for all 1 ≤ i ≤ n. Then a tableau proof in this case for a sentence φ is a closed tableau for [¬φ]x. The soundness and completeness results above (as well as the decidability argument below) are also straightforward to adapt. 2.3 Decidability In this section we prove decidability for S@n by showing that the systematic tableau procedure used to prove completeness is in fact a decision procedure. For this purpose we can adapt the argument in Fitting (1983: 410-413). Say we are attempting a proof by way of the systematic procedure described above. However, we now modify the procedure so that no branch contains multiple occurrences of the same indexed formulas: if the procedure says we should add an indexed formula to the end of a branch that already contains an occurrence of it, refrain from adding the repeated occurrence of that indexed formula. This involves a simple modification to our step VI above-which is completely analogous to Fitting's (1983: 411). Now, we prove that the (modified) systematic procedure we have described guarantees termination for n-dimensional tableaux, whereby they either deliver a proof or a countermodel. Proposition 2.1 (Termination) Let φ be an L@n -formula. A systematic attempt to prove φ terminates. 38 Proof. For a contradiction, suppose a systematic attempt at proving φ never terminates. Then there is a sequence T0, T1, ... of tableaux which successively properly extend each other. Let T be their limit, i.e. T = ⋃ n≥0 Tn. Then T is a finitely branching infinite labeled tree. Hence, it follows by König's lemma that T has an infinite branch, say, b. Consider the set Σ = {s | [ψ]s ∈ b for some formula ψ}. Because b is infinite, the set Σ must be infinite, too. Otherwise, the set of indexed formulas, say, Φ, occurring on b would be finite, since Φ ⊆ sb(φ) × Σ, where sb(φ) is the set of subformulas of φ, which is itself finite. Moreover, because no multiple occurrences of the same indexed formulas are allowed, it is not possible for some index-sequence in Σ to occur infinitely many times postfixing a formula on b. Let the height of an index-sequence s, written height(s), be the greatest natural number occurring in s, and for every k ∈ N, define Σk = {s | s ∈ Σ and height(s) = k}. Then we let the set Σ of all index-sequences on b be the union of all sets Σk. There are two cases that could allow Σ to be infinite: (1) For some height j, card(Σj) = א0. But this is impossible. Consider the set Sj = {s | height(s) = j} of all possible sequences (with length n) of height j. Note that card(Sj) = jn − (j − 1)n. Since card(Sj) is finite, and Σj ⊆ Sj, there is no height j such that card(Σj) = א0.18 (2) For each height j, card(Σj) < א0. Because Σ is infinite, and Σ is the union of all Σk, each of which is of finite cardinality, there must be infinitely many heights j such that Σj 6= ∅. Let the modal degree of a formula occurring on b be the number of modal operators occurring in that formula. Because the modal degree of formulas gets smaller as the index-sequences increase in height, there is a height k such that any formula on the branch b postfixed by an index-sequence of that height has modal degree 0. And since the modal rules do not apply in this case, there cannot be further index-sequences on b with height > k, in which case there are not infinitely many heights j such that Σj 6= ∅, a contradiction. Corollary 2.1 (Decidability) S@n is decidable. Proof. Immediately from the fact that systematic tableau constructions always terminate in a finite number of steps. 18The original argument for (1) was much more complicated and, in fact, unnecessary, in light of card(Sj) being finite, as a referee very helpfully pointed out. 39 Conclusion We have developed and proved several results about n-dimensional sequence modal logics with actuality operators. These are natural generalizations of two-dimensional modal logics known in the philosophical literature as logics for a priori knowledge, necessity, and actuality. In particular, the structures investigated here were shown to be extensions to any finite dimension of the structures studied in Fritz (2014) for the system 2Dg. The completeness argument from Fritz was also generalized to any finite dimension, and it can now be seen as a special case of the completeness proof presented in §1.3 by just setting n = 2 (given that @1 is directly eliminable from the language). Additionally, we have developed sound and complete tableau calculi for the logics herein considered, and used these to show decidability by means of a systematic tableau construction. There are, of course, many questions left open by the present paper which can be settled by future research, namely, questions in proof-theory, model-theory, and even in complexity theory. Some examples of these include the following: • Fritz's axiom system 2Dg was naturally generalized for any arbitrary dimension, and the tableau calculi presented here also involved a natural generalization of tableaux for basic modal logic. Is there a natural generalization for n dimensions of the hypersequents developed in Restall (2012) for two-dimensional modal logic? What about different proof systems for modal logic such as natural deduction systems? • What are the n-dimensional modal logics generated with multiple actuality operators such as A and D? There are many possibilities here, including classes of frames defined with a single distinguished point z ∈ S, and so formulas in the scope of D@n would be evaluated relative to the generalized diagonal point (z, ..., z); or even frames with a distinguished point (z, ..., z) for a language containing A, and so any formula in the scope of A would be evaluated relative to that point. It is expected that the main theorems proved here, such as completeness and decidability, would transfer to these cases, although it would be interesting to compare the expressive power of such languages relative to distinct notions of validity. One relevant question would be under which circumstances these actuality operators are eliminable from their respective languages, that is, whether the eliminability of the actuality operator, for instance, from the basic (one-dimensional) modal language when real-world valid40 ity is assumed (see, for example, Hazen et al. (2013)) carries over to n-dimensions. • Antonelli and Thomason (2002) proved that adding propositional quantifiers to a modal logic with two S5 modalities results in a system that is (recursively) intertranslatable with full second-order logic. What about in this framework? Even though 21 is an equivalence relation, any 2i, for i > 2, is not. So, in case there is a translation, it has to be modified accordingly, for the procedure described by Antonelli and Thomason does not seem to generalize for S@2 (or S@n) with propositional quantification. Still, is there any recursive translation with full second-order logic, such as in Antonelli and Thomason's case, that shows second-order S@2 and hence S@n (1 < n) to be undecidable? • Even though the validity problem for n-dimensional sequence modal logics was shown to be decidable, what is its complexity? • Is the sequence modal logic of ω-dimensions complete or decidable? • Standefer (forthcoming) investigates relevant logics with the actuality operator. It would be interesting to see the relevant counterparts of multidimensional modal logics with, possibly, many actuality operators. Acknowledgments Portions of this paper were presented at the Society for Exact Philosophy Meeting in Storrs, Connecticut, in 2018, at the Department of Philosophy and at the Department of Linguistics at Goethe University Frankfurt, in 2018, and at C-ALPHA's Logic Seminar at the University of California, Irvine, in 2019. 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