Combinations and completeness transfer for quantified modal logics

Logic Journal of IGPL Advance Access published January 20, 2010

Combinations and completeness transfer for quantified modal logics Gerhard Schurz, Institute for Philosophy, University of D¨ usseldorf, D¨ usseldorf, Germany. E-mail: [email protected] Abstract This paper focuses on three research questions which are connected with combinations of modal logics: (i) Under which conditions can (frame-)completeness (and related properties) be transferred from a propositional modal logic (PML) to its quantificational counterpart (QML)? (ii) Does (frame-) completeness generally transfer from monomodal QMLs to their multimodal combination? (iii) Can completeness be transferred from QMLs with rigid designators to those with non-rigid designators? The paper reports some recent results on these questions and provides some new results. Keywords: Logic combination, fusion, completeness transfer, quantified modal logic, (non-)rigid designator, ambiguous language, worldline semantics

1 Introduction1 Combinations of modal logics are of special importance in philosophical applications. One example is Hume’s is-ought thesis, i.e., the thesis that normative conclusions cannot be derived from non-normative premises. Ethical naturalists argue that normative conclusions can be derived from non-normative premises about necessary features of human nature or society. The formalization of their arguments requires at least a combined bimodal logic with a deontic obligation operator O and an alethic necessity operator 2. To be as comprehensive as possible, Schurz [34, 36] has investigated Hume’s is-ought thesis in as many bimodal and multi-modal logics as possible. It turned out that central theorems about the validity of Hume’s thesis were dependent on central properties of the underlying logics, such as frame-completeness, interpolation or Halld´en-completeness (cf. [36], theorems 2–5; see p. 303). At that time most of these properties were only known for monomodal logics but not for their multimodal combinations. So the question arose whether and to which extent various logical properties can be transferred from monomodal logics to their multimodal combination. In a paper together with Kit Fine which was written up in 1990 but whose publication was delayed until 1996, transfer theorems have been proved for all of the mentioned properties of modal logics, and similar results were independently established by Kracht and Wolter [28]. Meanwhile the investigation of combinations of (modal) logics and the study of transfer problems has become an increasing field in modal logics (cf. [29]). Besides simple combinations of modal logics, which are also called fusions or joins (cf. [29], 155; [11], 170), logicians have investigated more complicated kinds of combinations such as fibring which is 1

I am indebted to two unknown referees for valuable comments which helped to improve this paper.

© The Author 2010. Published by Oxford University Press. All rights reserved. For Permissions, please email: [email protected] doi:10.1093/jigpal/jzp085

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2 Combinations and completeness transfer for quantified modal logics a generalization of fusion (cf. [44], [38], [5]), or products (cf. [14]). In this paper I confine my attention to logic combinations in the sense of fusions. The philosophical need of multimodal combinations is illustrated by many examples. E.g., Aqvist’s system PIE [1] of erotetic logic requires an epistemic-deontic logic, Costa-Leite’s [8] investigation of Fitch’s paradox requires an alethic-epistemic logic, my investigation of Weber’s thesis requires a doxastic-evaluative logic ([36], theorem 7, p. 150), and my generalization of Hume’s thesis requires a multimodal logic with countably many modal operators ([36], prop. 23, p. 170). However, the above-mentioned transfer results are not sufficient to satisfy the applied philosopher’s demands, because they were established only for propositional (multi)modal logics, in short: PMLs, while for interesting philosophical applications one needs (1st order) quantified (multi)modal logics, in short: QMLs. Until the late 1990ies only a few results on combination and transfer of QMLs have been established (e.g. [41]; [36], chs. 2.7, 10), but in the meantime a variety of further results on QMLs have been reached (cf. [38], [2], [15]). In sections 3–5 of my paper I will present some basic questions and results on the following three research questions concerning combination and transfer for QMLs. Question 1 (sec. 3): Surprisingly completeness does not generally transfer from PMLs to their QML-counterparts. Under which conditions does it transfer? Question 2 (sec. 4): The transfer from monomodal to multimodal QMLs is much more difficult than for PMLs. How can it be proved? Question 3 (sec. 5): There are many different kinds of QMLs concerning such choices as rigid versus non-rigid designators or fixed versus varying domains (cf. [17], 267). Is it possible to establish transfer theorems which transfer completeness (and other properties) from one to another kind of QML? Not many results on these research questions have been reached in the literature so far. In the following I will report some of these results, with a focus on their philosophical importance, and I will present some new results. Results which are at least partly new will be presented in the form of ‘‘theorems’’ and ‘‘proofs’’, while results which have been established elsewhere will be reported informally in the text of by way of examples.

2 Combination and Transfer: Basic Framework and Results Terminological Conventions: Capital Latin letters A,B,... will vary over formulas of the object language and capital Greek letters
,,... over sets of them. F will always denote a frame, M a model, F a set of frames and M a set of models, W a possible world set, R the accessibility relation, and V a valuation function. The letters w, u, v range over possible worlds (all symbols may also be used in an indexed way). L is the language of PML (propositional modal logic). It contains as nonlogical symbols a denumerably infinite set of propositional variables P, and as primitive logical symbols the truth-functional connectives ¬ (negation), ∨ (disjunction) and the necessity operator 2, with the standard formation rules for (well-formed) formulas; L is identified with the set of its formulas. The other truthfunctional connectives ∧ (conjunction), ⊃ (material implication), ≡ (material equivalence), ⊥ (falsum),  (verum), and the possibility operator 3 are defined as usual. A frame is a pair F = W ,R where W = ∅ (a non-empty set of ‘possible worlds’) and R ⊆ W ×W (the accessibility relation; uRv abbreviates u,v ∈ R). A model for L based

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on the frame W ,R is a triple M = W ,R,V ; V : P → Pow(W ) is the valuation function which assigns to each propositional variable p ∈ P the set of worlds V (p) ⊆ W at which p is true (‘Pow’ for ‘power set’). The assertion ‘‘formula A is true at world w in model M ’’ (where w ∈ W M ) is abbreviated as (M ,w) |= A and recursively defined as usual; in particular (M ,w) |= 2A iff for all u ∈ W such that wRu, (M ,u) |= A. A formula A ∈ L is said to be valid in model M , in short: M |= A, iff A is true at all worlds of M ; A is valid on a frame F , in short F |= A, iff A is valid in all models based on F ; A is valid w.r.t. (with respect to) a class of models M, in short M |= A (or w.r.t. a class of frames F, in short F |= A) iff A is valid in all M ∈ M (or on all F ∈ F, respectively). A formula set
⊆ L is said to be (simultaneously) satisfiable in a model M (or on some frame F ) iff all formulas in
are true at some world in M (or at some world in some M based on F ). Analogously, a formula set
is valid in a model M , M |=
, iff all formulas in
are valid in M (and analogously for validity of
on F , w.r.t. M, and w.r.t. F). This paper investigates normal modal logics. The minimal normal modal logic K (for ‘Kripke’) is semantically defined as the set of modal formulas which are valid on all frames. It is axiomatically characterized as the smallest set of L-formulas which contains all instances of the axiom schemata Taut (all tautologies) and K: 2(A ⊃ B) ⊃ (A ⊃ B), and which is closed under the rule of Modus Ponens (MP) and under the rule of necessitation N:A/2A. In general, a normal PML (i.e., a normal extension of K) is defined as any subset L ⊆ L which contains K and is closed under the rules MP, N and under the rule of substitution (Subst). Every normal PML L is representable as L = KX, where X is some set of additional axiom schemata (cf. [36], p. 50, lemma 4); if X is recursively enumerable (resp., finite), then KX is recursively (resp., finitely) axiomatizable (cf. [6], 495f). Instead of K{X1 ,X2 } we sometimes also write K+X1 +X2 . The notion of a proof of formula A in an axiomatized
PML L, L A is defined as usual. The derivability relation is defined by
L A iff L
f ⊃ A for finite
f ⊆
. This definition (which is often used in modal logic, cf. e.g. [7], 47) entails the standard deduction theorem (whose validity for QMLs requires that free variables receive interpretations; see below). Since validity on frames is closed under the K-rules MP, N and Subst, every class of frames F defines a normal MPL L = L(F) = {A: F |= A}. Formula A is said to be a valid consequence of
, in short
|= A, iff for all worlds w in all models M based on some frame, (M ,w) |=
implies (M ,w) |= A. An additional axiom schema X of a normal PML is said to correspond to a structural condition CX over frames iff for every frame F , F |= X iff F satisfies CX . In this sense, the axiom schema T: 2A ⊃ A, corresponds to the frame-condition that R is reflexive – many more correspondence results have been discovered (cf. [42], [37]). M(F) denotes the class of all models based on some frame in frame-class F. A model class M is frame-based iff M = M(F) for some F. Frame classes are defined by purely structural conditions, i.e. conditions solely on R which do not constrain the valuation functions. In contrast, not-frame-based model classes (such as the ‘general frames’) are defined by restrictions on the valuation function. Prima facie, a modal logic should admit all possible valuations of its nonlogical symbols. Therefore, frame-classes and frame-based model classes are the philosophically more important means to characterize modal logics, as compared to not-frame-based model-classes. Let M(L) and F(L) denote the class of models or frames validating all theorems of a normal PML L, respectively (so, L is correct w.r.t. M(L) and F(L) by definition). If L = K{X1 ,...,Xn } and each Xi corresponds to frame-condition Ci , then F(L) = the class of frames satisfying all of the Ci . The major property of (axiomatized) normal PMLs is their completeness. A normal

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4 Combinations and completeness transfer for quantified modal logics PML L is called model-complete iff (for all
⊆ L and A ∈ L)
|=M(L) A implies
L A; it is called (strongly) frame-complete iff
|=F(L) A implies
L A; and it is called weakly framecomplete iff |=F(L) A implies L A. According to the (equivalent) consistency-formulation of completeness, a logic L is (weakly, resp.) frame-complete iff every L-consistent formula set
(or formula A, resp.) is satisfiable in F(L) (and likewise for model-completeness). Every normal PML is model-complete. Model-completeness implies completeness for socalled general frames of the form G = W ,R,Prop, where Prop ⊆ Pow(W ) is a set of ‘valuation-admissible’ subsets of W closed under intersection, relative complement, and under the operation ‘2: W → W ’ defined by 2X = {w: ∀u(wRu ⊃ u ∈ X)}. To every model M = W ,R,V  there corresponds a minimal general frame GM defined as W ,R,PropM  with PropM = the set of W -subsets which are the value of some L-formula under V ([6], p. 237). It follows that every normal PML L which is valid in a model M must also be valid in GM . As a result, model-completeness implies its general-frame-completeness; thus general-frame-completeness is not essentially stronger than ordinary model-completeness. The models of a logic M(L) are frame-based iff L is weakly frame-complete, i.e. L = L(F) for some F. Frame-completeness is philosophically more important than model-completeness, but it is also much more difficult to achieve it. Some important frame-completeness results for PMLs were early established, e.g., that K{T} is frame-complete for reflexive frames (for many such results cf. [21], [6], [24]). Later on a variety of PMLs was discovered which are frameincomplete – a simple example is van Benthem’s logic K{VB}, where VB = 32⊥∨2(2(2A ⊃ A) ⊃ A) (cf. [21], 57ff). The standard technique of proving completeness of a normal PML L is based on the construction of the so-called canonical model Mc (L) = Wc ,Rc ,Vc  of L, which is defined as follows: (1) Wc is the class of all maximally L-consistent formula sets, (2) ∀u,v ∈ Wc : uRc v iff {A: 2A ∈ u} ⊆ v, and (3) for all p ⊆ P and w ∈ Wc , w ∈ Vc (p) iff p ∈ w. It is shown that every L-consistent
⊆ L can be extended to a maximally consistent formula set, i.e. world w in Mc (L) (‘Lindenbaum lemma’), and that for all A ∈ L and w ∈ Mc (L), A ∈ w iff (Mc (L),w) |= A; this entails that every normal PML is model-complete. To prove by the canonical method that L is frame-complete one must show in addition that the frame of Mc (L) is a frame for L; a normal PML which satisfies this condition is called canonical. Canonicity implies frame-completeness; whether the reverse direction holds is an open question (though it is well-known that weak frame-completeness does not imply canonicity). A multimodal language LI contains a countable set {2i : i ∈ I } of modal operators (I = {1,2,...} is an index set). A normal multimodal PML in LI is any LI -formula set containing all Ki -axioms for each 2i and being closed under MP, Subst, and the rules Ni : A/2i A (for all i ∈ I ). A multimodal PML is called a combination (or fusion, join) of normal monomodal 2i -logics {Li : i ∈ I } iff L is the smallest normal PML in LI containing every Li ; we write L = ⊕{Li : i ∈ I } in this case. Axiomatically, ⊕{Li : i ∈ I } is obtained from the Li := Ki Xi (i ∈ I ) by joining their representative axiom sets Xi , and closing them under the rules MP, Subst ∈ I ) in the combined language LI . Hence, ⊕{Li : i ∈ I } is representable as and Ni (for all i KI XI with XI = i ∈ I Xi . For example, K{1,2} {D1 ,T2 } is the logic K1 +K2 +D1 : A → 31 A+ T2 : 22 A → A. Frames for multimodal PMLs have the form W ,{Ri : i ∈ I }, and they are understood as semantic combinations (fusions, joins) of the monomodal frames; thus we define ⊕{W ,Ri : i ∈ I } := W ,{Ri : i ∈ I }. According to the multimodal frame lemma, a join of frames ⊕{W ,Ri : i ∈ I } is a frame for a join of logics ⊕{Li : i ∈ I } iff each single frame W ,Ri  is a frame for Li (cf. prop. 1, p. 174, and prop. 1∗ , p. 198, in [11]). A property ψ transfers from a class of monomodal logics Li (i ∈ I ) to their combination ⊕{Li : i ∈ I } iff

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⊕{Li : i ∈ I } has ψ whenever all of the Li have ψ. Transfer theorems have been successfully proved for multimodal PMLs: EXAMPLE 2.1 (Transfer for multimodal PMLs) (2.1.1) weak and strong frame-completeness, canonicity and f.m.p. transfer from the Li (i ∈ I ) to ⊕{Li : i ∈ I }; and (2.1.2) decidability, interpolation and Halld´en-completeness transfer from the Li to ⊕{Li : i ∈ I } under presupposition of weak frame-completeness of the Li . (see [11]). Often one is interested in multimodal logic which are not simple combinations but contain additional interactive axioms which relate distinct modalities. A direct transfer theorem for interactive multimodal logics is possible only in certain cases. For example, a transfer theorem applies to loop-free dynamic logics, and another one to bimodal logics of the form (L1 ⊕L2 )+2k2 2∞ 1 X (cf. [11], 210f). In cases where such a general transfer theorem is not possible, it is usually easy to extend the completeness result from a multimodal combination ⊕{Li : i ∈ I } to a multi-modal logic ⊕{Li : i ∈ I }+Int including an interactive axiom schema Int with a corresponding frame-condition CInt simply by proving that the canonical frame Fc (⊕{Li : i ∈ I }+Int) satisfies CInt (results of this sort are proved in [36], 129). Turning to quantified modal logics (QMLs), we let LQ be the modal quantified language, which contains the following countably infinite sets of symbols: a set V of individual variables (x,y,...), in short: variables, a set C of individual constants (a, b, ...), in short: constants, and for each n ≥ 0, a set Rn of n-ary relation symbols (F , G, Q, ...; 0-ary relation symbols are propositional variables). For reasons of simplicity we omit function symbols; thus singular terms, denoted by t, t1 , t2 , ..., are constants or variables; T (the set of all terms) := V ∪C. The new primitive logical symbols are the universal quantifier ∀ and the identity symbol =, with the standard formation rules for (well-formed) formulas; the existential quantifier ∃ is defined as usual. A[t/x] denotes the result of the correct substitution of term t for variable x in A and is defined as the result of the replacement of every free occurrence of x in A∗ by t; where A∗ is the first alphabetic variant of A (according to a given formula enumeration) in which x does not occur in the scope of a quantifier binding t. A[t1−n /x1−n ] denotes the result of the correct (simultaneous) substitution of ti for xi in A (for 1 ≤ i ≤ n; the xi are pairwise distinct). ∀x1−n abbreviates ∀x1 ... ∀xn . For all kinds of QML-semantics considered here, the notion of a frame is the same as for PMLs. The simplest way of extending models based on frames to quantified modal languages are rQ-models: they assume that all terms are rigid (‘‘r’’), i.e. denote the same object in all worlds, and that each world has the same fixed domain of objects D. Hence, a rQ-model is a quadruple M = W ,R,D,V  where W ,R is a frame, D = ∅ is a nonempty domain of individuals (or objects), and the valuation function V is defined as follows: (1) V : T → D, i.e. ∀t ∈ T : V (t) ∈ D, and (2) for all n ≥ 0, V : W ×Rn → D n , i.e. for all Q ∈ Rn , V (w,Q) := Vw (Q) ⊆ D n . The value Vw (Q) is called the extension of Q at world w, and the partial function V (Q): W → D n is called Q’s intension. Each world has its local nonmodal model D,Vw . Hence the model W ,R,D,V  can be considered as the combination of the modal frame W ,R  with the local models D,Vw : ⊕{W ,R}∪{D,Vw : w ∈ W } := W ,R,D,V . Our semantical treatment of free variables and open formulas (cf., e.g., [33], p. 151f) – which differs from the closure-interpretation of open formulas in model theory – simplifies matters concerning deduction theorem. Since free variables receive semantic values, open formulas (Fx) are interpreted like closed formulas (Fa), the deduction theorem (which syntactically holds by definition) is semantically valid (in particular, Fx |= ∀xFx is semantically as invalid as Fa |= ∀xFx).

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6 Combinations and completeness transfer for quantified modal logics M [x: d] denotes a model which is like M except that V M assigns d ∈ D to x; and similar for M [x1−n : d1−n ]. The definition of ‘(M ,w) |= A’ is as usual; in particular for quantified formulas: (M ,w) |= ∀xA iff ∀d ∈ D, (M [x: d],w) |= A. The so-defined rQ-models validate the following axiom schemata and rules which go beyond normal PMLs (for all A ∈ L, x ∈ V, t ∈ T ): Universal instantiation UI: ∀xA ⊃ A[t/x] Barcan formula BF: ∀x2A ⊃ 2∀xA ∀1:∀x(A ⊃ B) ⊃ (∀xA ⊃ ∀xB) ∀2: A ⊃ ∀xA, provided x is not free in A. Identity I: t = t Rigid identity-substitution rISub: t1 = t2 ⊃ (A[t1 /x] ⊃ A[t2 /x]) Rigid non-identity rnI: ¬t1 = t2 ⊃ 2¬t1 = t2 Rule of universal generalization UG: A/∀xA. The minimal rQ-logic is abbreviated as rQK and defined as the smallest set of LQ-formulas containing all K-axioms and all instances of the above axiom schemata and being closed under the K-rules MP, N and the rule UG. A rigid quantified modal logic, in short a rQML, is defined as any LQ-formula set containing all rQK-axioms, being closed under all rQKrules, and under the rule of substitution for predicates. The latter rule was first described by Kleene ([23], pp. 155-162) and is described in more detail in Schurz ([35] and [36], 45-52). Every normal rQML is representable (but not necessarily axiomatizable) as rQKX with X as a set of additional (possibly quantified) axiom schemata. Every PML KX has an rQ-counterpart rQKX. If L is the rQ-counterpart of the PML then L is propositionally representable, i.e. iff L = rQKX for some X consisting solely of propositional axiom schemata. The rQ-counterpart rQKX of a PML KX can be considered as the (axiomatic) combination of KX with the nonmodal quantified logic Q plus the interactive axioms BF and rnI which result from the assumption of constant domain and rigid designators. Thus we may write rQKX = ⊕{KX,Q}+{BF,rnI}. Not every (interesting) QML is propositionally representable: in the next section it will be seen that canonical PMLs may have frame incomplete rQ-counterparts which can be made canonical by adding additional LQ-axiom schemata. The canonical method of proving canonicity (frame-completeness, model-completeness) for rQMLs is more complicated than for PMLs. A formula set
⊆ LQ is called ω-complete iff for every A ∈ LQ:
L ∀xA iff
L A[t/x] for every t ∈ T ;
is called L-saturated iff it is both maximally L-consistent and ω-complete. The canonical model Mc (L,) = Wc ,Rc ,Dc ,Vc  of a normal rQML L is explicitly relativized to a saturated formula set  ⊇
in an extended language LQ∗ which arises from LQ by the addition of a countably infinite set V ∗ of new constants; it is shown that the given L-consistent formula set
in a language LQ can be extended to such a saturated formula set  ⊇
in an extended language LQ∗ . Mc (L,) is defined as follows: (1.) Wc is the set of all L-saturated LQ∗ -formula sets w which preserve the -identities; i.e. for all t1 ,t2 ∈ T and w ∈ Wc : t1 = t2 ∈ w iff t1 = t2 ∈  (this ensures constant domain and rigid designators); (2.) Rc is as in the propositional case; (3) for all t ∈ T , Vc (t) = {t  : t = t  ∈ } and for all Q ∈ Rn and w ∈ Wc , Vw (Q) = {Vc (t1 ),...,Vc (tn ): Qt1 ...tn ∈ w}; (4.) D = {Vc (t): t ∈ T }. Recall the canonical method of proving model-completeness for rQMLs (for an overview cf. [17], ch. 2.2.1): to prove the truth lemma (for all w ∈ Wc and A ∈ LQ∗ : A ∈ w iff (Mc ,w) |= A), one has to establish the following rQ-Lemma: if w ∈ Wc (L) and ¬2B ∈ w, then {A: 2A ∈ w}∪{¬B} is L-consistent and can be extended to a saturated formula set ∗ in the same language LQ ∗ which satisfies the same term identities in ; for this purpose one needs classical quantifier principles, Barcan formula, and rnI.

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3 Combining MPL and Q: From Propositional to Quantified Modal Logic 3.1 Standard QML-Semantics: Constant Domain and Rigid Designators To prove that a rQML L stronger than rQK is canonical requires to show that the frame of L’s canonical model is an L-frame. Does canonicity always transfer from a PML to its rQ-counterpart? This question was stated as an open problem in Hughes/Cresswell ([21], p. 183f) and was wrongly answered in the positive by Garson ([16], 276, 2nd paragraph). Quite astonishingly, general canonicity transfer fails. An example of a canonical PML with a frameincomplete rQ-counterpart is the logic S4.1 = S4+(0.1): 23A ⊃ 32A (0.1 is also called ‘‘M’’ for ‘‘McKinsey’s axiom’’). The S4.1-axioms correspond to reflexive and transitive frames in which every world reaches a ‘dead end’ (∀u∃v: uRv ∧∀w(vRw ⊃ v = w)). EXAMPLE 3.1 (Counterexample to transfer from PMLs to QMLs) (3.1.1) S4.1 is canonical ([30], 75; [42], p. 202). (3.1.2) rQS4.1 is frame-incomplete ([36], 292f, prop. 4.2; [22], 264–270). (3.1.3) rQS4.1+(32∃xA ⊃ 3∃x2A) is canonical ([36], 293–5, prop. 4.3). The reason why the proof of canonicity works for S4.1 but not for rQS4.1 is that the dead-end frame condition corresponding to S4.1 contains an existential quantifier. This means in the propositional case that it has to be shown that a certain formula set has a maximally consistent extension, while in the quantified case it has to be shown that this formula set has a maximally consistent and ω-complete extension in the same language; but this is only guaranteed if the additional axiom schema (32∃xA ⊃ 3∃x2A) is available. Another axiomatic completion of rQS4.1, which is rQS4.1-equivalent with (32∃xA ⊃ 3∃x2A), is the axiom schema ‘‘FINAX’’: 3∀x1 ...xn (A ⊃ 2A), with {x1 ,...,xn } = Vfree (A), which is introduced in Cresswell ([9], 160). Another example of a canonical PML with frameincomplete Qr-counterpart mentioned in Hughes/Cresswell ([22], 271) is S4.2 = K{4,0.2}, where 0.2 := 32A ⊃ 23A corresponds to frames satisfying the following convergence condition for R: ∀u,v: ∃w(wRu ∧wRv) ⊃ ∃w  (uRw  ∧vRw  ). An axiomatic completion of rQS4.2 has so far not been found (see [9], 170f). A general transfer theorem from PMLs to their rQ-counterparts can be proved under the restriction that the frames of the given PML L are closed under sub-frames, i.e. if W ,R ∈ F(L), then for every ∅ = U ⊆ W ,U ,R ∩U 2  ∈ F(L). A similar result for intermediate logics was obtained by Shimura ([39], 36), and a mere sketch of the proof for QMLs was given in Schurz ([36], 295). The full proof is given below. It utilizes an important technique for combination and transfer results, namely the technique of ambiguous languages ([11], 175) which works as follows: if L is a language and L∗ a sublanguage of L in which certain logical symbols of L are missing, then L can also be viewed as an L∗ -language with an extended set of primitive symbols, namely all those complex expressions of L which are elementary in L∗ . Thus, for example, if L12 is a bimodal (21 ,22 )-language, then L12 can be considered as a monomodal 21 -language having as propositional variables the extended (but still countably infinite) set P ∪{22 A: A ∈ L12 }. Likewise, the language LQ can also be viewed as a PML-language L having (for each n ∈ ω) as n-ary predicates the set Rn ∪{∀xA: A ∈ Form n+1 (LQ), x free in A}, where Form n (LQ) is the set of LQ’s formulas with n distinct free variables.

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8 Combinations and completeness transfer for quantified modal logics THEOREM 3.2 (3.2.1) If F(KX) of a normal PML KX is closed under subframes, then canonicity transfers from KX to rQKX. (3.2.2) If KX’s frames are definable by purely universal first order formulas, then F(KX) is closed under subframes. PROOF. Theorem 3.2.1: Let  be the saturated extension of the given rQKX-consistent formula set
in the language LQ∗ . Consider the propositional canonical model Mc (0) := Mc (KX) in the language LQ∗ -viewed-as-L. The quantified canonical model Mc (1) := Mc (rQKX,) in the language LQ∗ is a submodel of Mc (0), because of the following facts: (a) Wc (1) is a proper subset of Wc (0), since every w ∈ Wc (1) is rQKX-consistent and hence also KX-consistent (while the inverse direction fails); (b) Rc (1) coincides with Rc (0) for all worlds in Wc (1) (since Rc ’s definition in the propositional and quantified case is the same), and (c) Vc (0) and Vv (1) coincide for all C ∈ Rn ∪{∀xA: ∀xA ∈ Form n+1 (LQ∗ ), x free in A}; this is ensured by the truth lemma for quantified model-completeness which says that for every A ∈ LQ∗ and w ∈ Wc (1): (Mc (1),w) |= A iff A ∈ w (cf. [16], 275f; [21], 84, 176). It follows that the frame of Mc (1) is a sub-frame of Mc (0)’s frame. Because KX is frame-complete, the frame of Mc (0) is a frame for KX, and because the frames of KX are closed under subframes, also the frame of Mc (1) must be a frame for KX; so rQKX is canonical. Theorem 3.2.2: Frames can be considered as nonmodal models of a 1st order language with the binary predicate R and domain W . By the Los-Tarski-theorem the models of purely universal formulas are closed under submodels (cf. [40], 76); so F(KX) is closed under subframes. Theorem 3.2 covers the axiom schemata T (R reflexive), B (R symmetric), 4 (R transitive), 5 (R euclidean), as well as Altn : 2A1 ∨2(A1 ⊃ A2 )∨...∨2(A1 ∧...∧An ⊃ An+1 ) (every world reaches at most n distinct worlds), Ver: 2⊥ (∀u,v: ¬uRv), Triv: 2A ≡ A (∀u,v: uRv ⊃ u = v), 0.3: 2(2A ⊃ B)∨2(2B ⊃ A) (∀u,v,w: wRu ∧wRv ⊃ uRv ∨vRu), because their corresponding frame-conditions in brackets are universal 1st order formulas. Theorem 3.2 also covers all subframe logics in the sense of Fine ([10], p. 624; cf. [6], 380ff). It is an open question whether canonicity-transfer holds for larger classes of normal Q1MLs; so I formulate this as a problem: PROBLEM 3.3 Generalize theorem 3.2 for more comprehensive classes of PMLs than the PMLs whose frames are closed under subframes. The failure of general transfer for QMLs with the simple rQ-semantics has motivated logicians to investigate the transfer problem in alternative QML-semantics. Important successes in this directions are reported in sections 3.2–3.3. In spite of these successes, what one would also like to have are just ordinary frame-complete rQMLs. As we have seen in example 3.1, a natural way to obtain them even in the case where frame-completeness does not transfer is axiomatic completion. Thus, another kind of solution of the frame-completeness problem of rQMLs, and maybe the philosophically more desirable one, were achieved if one could design a general recipe for the construction of additional axioms which make the rQ-counterpart of a canonical PML canonical, when its frame-condition does not satisfy the conditions of theorem 3.2 but involves existential quantifiers.

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PROBLEM 3.4 Define a (large and interesting) class  of canonical PMLs, and a computable function φ which associates with each frame-condition CX for a canonical PML KX an (possibly empty) additional quantified axiom schema Y := φ(CX ) such that rQKX+φ(CX ) is canonical.

3.2 Simple PML-Q-Combinations and Functor Semantics One might conjecture that the problem of non-transfer from PMLs to rQMLs is due to the fact that a rQML is not a simple combination of a PML with Q but contains the interactive axioms BF and rnI. This is an error. Let us call QML-logics which are simple axiomatic combinations of a PML with Q as sQMLs (‘‘s’’ for ‘‘simple’’); hence for each sQML sQKX it holds that sQKX = ⊕{KX,Q}. Since the models of sQMLs, in short: sQ-models, need not satisfy the constant domain condition, they contain for each world u ∈ W a local domain Du . Hence sQ-models have the form W ,R,D,Df ,V  where Df : W → D is the domain function (Dw := Df (w)) and D is the total domain of all objects at arbitrary different worlds. But since the converse Barcan formula cBF: 2∀xA → ∀x2A is a theorem of every normal QML with classical quantifier principles, the sQ-models must satisfy the nested domain condition: uRv implies Du ⊆ Dv . The designators of sQMLs are not necessarily rigid (the rigid non-identity axiom rnI is missing in sQMLs), whence sQ-models have world-relative extensions Vu (t) of the terms t ∈ T . These world-relativized term-extensions encode a counterpart relation between the objects of different worlds as follows: for each u ∈ W , d ∈ Du and v with uRv, {d  ∈ Dv : ∃t ∈ T (Vu (t) = d) and Vv (t) = d  } is the set of the counterparts of d ∈ Du in world v. However, the rigid identity axiom t1 = t2 ⊃ 2(t1 = t2 ) is a consequence of the classical axiom of substitution rSub of identicals (this is the reason why we speak here of ‘‘rigid’’ substitution). Therefore the counterpart relation of sQMLs must satisfy the functionality constraint: for each w ∈ W , every object d ∈ Dw can have only one counterpart d  ∈ Dv in every R-accessible world v, though two objects in w may have the same counterpart in v. Thus, the counterpart relation has to be a function c: Du → Dv for uRv. The resulting sQ-semantics is called functor semantics and was investigated by Ghilardi [18]. Interestingly he obtains much stronger non-transfer results for frame-completeness than for rQML logics. For example, sQAltn is frame-incomplete in sQ-semantics, although rQAltn is canonical in rQ-semantics. That sQAltn is frame-incomplete follows from corollary 7.5 of Ghilardi ([18], 537) which implies, among other things, that all sQMLs lying (properly) between sQS4.3 and sQS5 are frame-incomplete. What one can learn from Ghilardi’s result is twofold. First, completeness does not always transfer from two logics two their (simple) combination – sQMLs are a drastic counterexample. Second, transfer of completeness does not always become more difficult if interactive axioms are added; sometimes it may becomes easier – this is exemplified by rQMLs in comparison to sQMLs.

3.3 Alternative QML-Semantics Functor Semantics is a kind of counterpart semantics. Counterpart relations are needed in models for normal QMLs with nonrigid designators. Their valuation function for terms is world-relative, i.e.V : W ×T → D, where D is the total domain of objects (comprising all objects in all worlds of the model). For each term t ∈ T , the function V (t): W → D, with V (t)(w) = V (w,t) is called the intension of term t, and Vw (t) = V (w,t) is called the

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10 Combinations and completeness transfer for quantified modal logics extension of t at world w. The debate between Kripke and Lewis, whether individuals in different worlds are strictly identical (Kripke) or merely counterparts of each other (Lewis), is logically less decisive than one might think. The rigid designator axioms (rSub, rnI, BF, cBF) of rQ-logics are also adequately characterized by unique counterpart models, in which every individual possesses a unique counterpart in every possible world (cf. also [12], 60ff). The important point of a semantics for nonrigid designators which does not validate the rigid designator axioms is the assumption that the counterpart relation is not unique. The need of a counterpart relation for nonrigid designators can be seen from the semantical interpretation of quantified de-re formulas. Take, e.g., (M ,w) |= ∃x2Fx. This means that there exists d ∈ Dw (where Dw is the set of objects which are extensions of terms at world w) such that for all w-accessible worlds u: V [x: d](u,x) ∈ Vu (F ). But how do we define the x-variation V [x: d]: W ×T → D of V M ? The most natural way is to assume a counterpart relation which specifies for each d ∈ Dw and u ∈ W the u-counterparts of d in w. This account has been developed by Lewis [32], though not within the framework of modal logic but within that of ordinary 1st order logic. Unfortunately, Lewis’ counterpart theory, if taken as a semantics for QMLs, is logically not well-behaved. For example, it is not closed under substitution. More drastically, Wollaston [43] has shown that Lewis’ semantics invalidates the modal axioms K and M : 2(A∧B) ⊃ 2A∧2B, and even the nonmodal axiom UI. A well-behaved counterpart-semantics which generalizes Ghilardi’s functor semantics has been developed by Kracht and Kutz [25]. General frame-completeness transfer fails in this counterpart semantics just as in functor semantics. Since proving canonicity for each QML separately (cf. [13] and [3]) is cumbersome, more complicated QML-semantics have been developed for which transfer is easier to obtain. An nice overview on alternative QMLsemantics is found in Kracht and Kutz [27] (see also [4]). Skvortsov and Shehtman [41] have introduced metaframe semantics for QMLs, which is a generalization of Ghilardi’s functor semantics. They are able to show that completeness w.r.t. metaframes generally transfers for all PMLs containing KS4 to their quantified sQcounterpart. Technically, this is a great success. However, metaframe semantics is not based on domains of individuals, but on domains of abstract n-tuples which are not reducible to the nth Cartesian product of an ordinary domain. It would be desirable to have a philosophically transparent interpretation for the technically powerful metaframe semantics of Skvortsov and Shehtman, but so far, no one has provided such an interpretation. Fibring is a generalization of the fusion operation which has been applied in Zanardo et al. [44] to a general class of propositional logics, including MPLs. In Sernadas et al. ([38], 420), the fibring technique is extended to QMLs endowed with a novel semantics, in which the Barcan formula is invalid although the domain of individuals is constant. It is proved that for a large class of QMLs, completeness for models of their semantics transfers from component QMLs to their fibring (ibid., p.449). The investigation of the consequences of this result for our three research questions is an important task for the future. More recently Goldblatt and Mares [19] have introduced a non-classical semantics for QMLs with constant domain D which employs general frames of the form W ,R,D,Prop,PropFun, where Prop is a set of valuation-admissible propositions and PropFun is a set of propositional functions which are the admissible valuations of predicates (models add to these general frames an admissible valuation function V ). PropFun contains functions of the form φ: D ω → Prop with D ω being the set of all variableassignments f : ω → D. PropFun and Prop are not closed under the semantic operation corresponding to the classical quantifier ∀: even if for all d ∈ D, the proposition

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|Fx: d|M := {w ∈ W : (M [x: d]) |= Fx} ∈ Prop, the proposition |∀xFx|M := {w ∈ W : (M ,w) |= ∀xFx} = d∈D |Fx: d|M need not be in Prop. Instead, Goldblatt and Mares adopt a nonclassical semantics (|=G/M ) which interprets the quantifier as follows: (M ,w) |=G/M ∀xA iff there is a proposition U ∈ Prop with w ∈ U which entails (but may be stronger than) |∀xFx|M . Goldblatt and Mares are able to prove the transfer of frame-completeness for all QMLs with and without BF (ibid., theorem 6, corollary 9), but on the cost that their models W ,R,D,Prop,PropFun,V  based on frames W ,R interpret quantifiers in a non-classical way.

4 Combining rQMLs: From Monomodal to Multimodal Quantified Logics Our second research question from sec. 1 concerns the transfer problem from mono- to multimodal QMLs. Because frame-completeness does not always transfer from a PML to its rQ-counterpart, the general transfer theorem from mono- to multimodal PMLs explained in example 2.1 of sec. 2 does not automatically generalize to modal quantified logics. For obtaining generalization of these results to rQMLs, one would have to extend the prooftechnique introduced in Fine/Schurz [11] to rQMLs (or other kinds of QMLs). I state this here as a further open problem: PROBLEM 4.1 Extend the proof-technique of Fine/Schurz [11] which establishes generalized transfer for PMLs to QMLs. So far I was only able to prove the transfer of canonicity from mono- to multimodal rQMLs, by using the technique of ambiguous languages as follows: if LQI is a given multimodal LQlanguage, then LQI -viewed-as-LQ i is the 2i -monomodal LQ-language with the extended  set of n-ary predicates Rn ∪ r∈I ,r =i {2r A: A ∈ Form n (LQI )}. THEOREM 4.2 Canonicity transfers from every countable set {Li : i ∈ I } of normal ( 2i -monomodal) rQMLs to their (multimodal) join L := ⊕{Li : i ∈ I }. PROOF. Assume an L-consistent formula set
, and extend it to a saturated formula set  in the extended language LQ∗ (recall the last paragraph of sec. 2). For each i ∈ I , the canonical model Mc (i) = Wc (i),Rc (i),Dc (i),Vc (i) of  viewed as an LQi -formula set contains as worlds all maximally Li -consistent and ω-complete formula sets of LQ∗ -viewed-as-LQ∗i which contain all term identities of . Since Li is canonical, the frame of Mc (i) is a frame for Li (for each i ∈ I ). Let Mc (I ) = Wc (I ),{Rc (i): i ∈ I },Dc (I ),Vc (I ) be the canonical model of  viewed as an LQI -formula set and consider (for any given i ∈ I ) its i-reduct Mc (I : i) := Wc (I ),Rc (i),Dc (I ),Vc (I ). Mc (I : i) is a LQi -model, and it is a submodel of Mc (i), because: (a) Wc (I ) is a proper subset of Wc (i), since every L-consistent formula set is Li -consistent, but not vice versa; (b) Rc (i) of Mc (I : i) is defined exactly in the same way as Rc (i) of Mc (i), but restricted to Wc (I ), (c) Dc (I ) = Dc (i) = {[t] : t ∈ T }, since all worlds in Mc (I ) and in Mc (i) contain the term identities of , and finally (d)  Vc (I ) coincides with Vc (i) on the set of all n-ary predicates of LQ-viewed-as LQi , Rn ∪ r∈I ,r =i {2r A: A ∈ Form n (LI )}; this follows from the ‘truth-lemma’ for rQML-completeness (see the last paragraph of sec. 2).

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12 Combinations and completeness transfer for quantified modal logics We show now that Mc (I : i) is also a generated submodel of Mc (i) (i.e., Wc (I ) is closed under Rc (i)-chains in Mc (i)). Assume, for reductio, that Mc (I : i) is not a generated submodel of Mc (i); then there must exist worlds u ∈ Wc (I ) and v ∈ Wc (i) with uRc (i)v such that v ∈ Wc (I ). So v must be L-inconsistent; i.e. v contains some formula A such that ¬A ∈ L and hence 2¬A ∈ L. But then (i) 3A must be in u (because uRc (i)v) as well as (ii) ¬3A must be in u, because ¬3A (equivalent with 2¬A) is a L-theorem; a contradiction. Because Mc (I : i) is a generated submodel of Mc (i), the frame of Mc (I : i) is a generated subframe of Mc (i). Since frame-validity is closed under generated subframes, the frame of Mc (I : i) must be a frame for Li . Therefore the frame of Mc (I ) is a join of frames for Li : Fc (I ) = ⊕{Fc (I : i): i ∈ I }, and therefore (by the multimodal frame lemma explained in Section 2) Fc (I ) is a frame for L, whence L is canonical.

5 Worldline Semantics: Transfer from Rigid to Nonrigid QMLs Our third research questions concerns the transfer problem from standard rQMLs to QMLs with varying domains and non-rigid designators. Can general transfer theorems be proved in that case? This is indeed the case if the QMLs are endowed with a semantics in which the quantifiers do not range over objects (term extensions), but over functions from worlds into objects (term intensions). This so-called substantial interpretation of quantifiers has been suggested by Hughes/Cresswell ([20], 198ff) and is elaborated in Garson ([17], sec. 1.4). Schurz ([36], ch. 10.8) shows that with some modifications, Garson’s semantics can be reinterpreted from the objectual view as a certain kind of counterpart semantics: the socalled world-line semantics. A world-line (or ‘substance’) is a function l: W → D (in analogy to wordlines in Minkowski’s space-time worldlines). The new component of wQ-models is a set L of worldlines. These worldlines are assigned to the terms of LQ (as their intensions). A worldline l ∈ L lands an object d at world w if l(w) = d. The set of worldlines L determines a four-place counterpart relation ‘‘object d1 in world u has d2 as a counterpart in world v’’ through the following condition: ‘‘there exists a worldline in L which lands d1 at u and d2 at v’’. For each world w in W , the local domain Dw of w is the set of objects in D landed by some wordline at w, (i.e., the set of all w-counterparts of arbitrary objects in arbitrary worlds). Predicate extensions at w are taken from Dw . Thus, a wQ-model for LQ (‘‘w’’ for ‘‘worldline’’) is a quintuple W ,R,D,L,V  where W ,R is a frame, D = ∅ is the total domain of objects, and L = ∅ is a set of worldlines l: W → D. The valuation function V assigns to each t ∈ T its term intension, V (t) ∈ L, where Vw (t) := V (t)(w) is t’s extension at world w; and V assigns to each n-ary predicate R its predicate intension V (R): W → Pow(D) such that for each w ∈ W , Vw (R) := V (R)(w) ⊆ (Dw )n , where Dw = {d ∈ D: ∃l ∈ L: l(w) = d}. M [x: l] denotes a model which is like M except that V [x: l] assigns the wordline l to x. The truth clauses are as follows: (i) (M ,w) |= Qt1 ...tn iff Vw (t1 ),...,Vw (tn ) ∈ Vw (Q); (M ,w) |= t1 = t2 iff Vw (t1 ) = Vw (t2 ); for propositional operators they are as usual; and for the quantifier: (M ,w) |= ∀xA iff for all l ∈ L, (M [x: l],w) |= A. Worldline semantics is equivalent to the semantics of coherence frames introduced in Kracht and Kutz ([26, 27], 973f). The essential difference between worldline semantics and counterpart semantics in the sense of Lewis [32] and Kracht/Kutz [25] is that in worldline semantics, quantification over counterparts is governed by the quantifiers of a formula, while in counterpart semantics it is governed by the modal operator. The de-re formulas ∀x2Fx

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and ∃x3Fx are evaluated in the same way, but the de-re formulas ∀x3Fx and ∃x2Fx are evaluated differently: in counterpart semantics, (M ,w) |= ∀x3Fx iff for every d ∈ Dw there exists some w-accessible world u such that some u-counterpart d  of d-in-w is in Vu (F ), while in worldline semantics, (M ,w) |= ∀x3Fx iff for every d ∈ Dw there exists some w-accessible world u such that every u-counterpart d  of d-in-w is in Vu (F ). Analogously for the formula ∃x2Fx. In the evaluation of singular formulas, counterpart semantics implicitly quantifies over the counterparts of term extensions, e.g. (M ,w) |= 2Ft iff in all w-accessible worlds u all u-counterparts of Vw (t) are F . In contrast, wordline semantics determines the counterparts of Vw (t) by the worldline V (t) assigned to t; i.e. (M ,w) |= 2Ft iff in all w-accessible worlds u the object Vu (t) (landed by worldline V (t) at u) is F . Worldline semantics fits nicely with philosophical intuitions. This can be seen from the following example. The term intensions of ‘‘morning star’’ and ‘‘evening star’’ are different worldlines, which land the same object in our world, namely Venus. In all worlds, ‘‘morning star’’ and ‘‘evening star’’ denote particular stars which send visible light to the earth in the morning, or in the evening, respectively. But in some possible worlds the stars denoted by these terms are different stars. Thus, according to worldline semantics, the statement (1) ‘‘the morning star shines necessarily in the morning’’ is true while (2) ‘‘the morning star shines necessarily in the evening’’ is false (and vice versa for the evening star). Moreover, (3) ‘‘there exists a star which necessarily shines in the morning’’ is worldline-semantically true (and likewise for the evening). In contrast, the statements (1) and (3) are false in counterpart semantics, because the morning star has the evening star as its counterpart, which in other possible worlds does not shine in the morning. In my view, philosophical intuitions are on the side of worldline semantics. A technical advantage of worldline semantics over counterpart semantics is its greater expressive power: counterpart-semantical modal operators, abbreviated as 2c and 3c , are definable within worldline semantics as follows (cf. [36], 222):
2c A[t1−n /x1−n ] := ∀x1−n ( {xi = ti : 1 ≤ i ≤ n} ⊃ 2A, where {x1 ,...,xn } = Vfree (A). In words: Formula A[t1−n /x1−n ] is counterpart-necessarily true at a given world w iff all n-tuples of worldlines l1 , ... ,ln which land the same objects at world w as the worldlines designated by t1 , ... ,tn satisfy the open formula
A. Likewise: 3c A[t1−n /x1−n ] := ∃x1−n ( {xi = ti : 1 ≤ i ≤ n}∧3A). The QMLs validated by worldline semantics are called wQMLs. In the axiomatization of the minimal wQML wQK, the axiom rnI of rQMLs is dropped, and the rigid principle of substitution of identicals rISub is restricted to nonmodal formulas as follows: ISub: t1 = t2 ⊃ (A[t1 /x] ⊃ A[t2 /x]), provided A does not contain ‘2’. As a result, the theorems of wQMLs which involve identity are no longer closed under the unrestricted rule of substitution for predicates. For example, t1 = t2 ⊃ (2Ft1 ⊃ 2Ft2 ) is not wQ-valid because t1 and t2 need not be identical in worlds differing from the given actual world. But the theorems of normal wQMLs are still closed under substitution of arbitrary nonmodal formulas for predicates (cf. [36], 221). A normal wQML is defined as any set of LQ-formulas which contains all instances of axioms of wQK and is closed under the rules of wQK and under the rule of substitution for predicates (non-modally restricted for formulas involving identity). Each wQML is representable as wQKX with X a set of additional axiom schemata. wQMLs satisfy the classical

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14 Combinations and completeness transfer for quantified modal logics quantifier principles of rQMLs, and they satisfy the Barcan-formula BF, because their quantifiers range over a fixed set of worldlines which is the same for all possible worlds. Also the converse BF, cBF: 2∀xA ⊃ ∀x2A, is valid because it is a theorem of all QMLs with classical quantifier principles. The great advantage of wQMLs is that model-completeness of rQ-logics can be directly transferred to wQ-logics, by the reduction of theoremhood and inference in wQMLs to theoremhood and inference in rQMLs without identity-axioms. The reduction method is again an instance of the technique of ambiguous languages, now applied to the identity predicate which can also be viewed as a non-logical predicate. For each rQML or wQML L, let L = be its identity-free sublogic in the same language, axiomatically obtained by dropping those axiom schemata of L which contain the identity sign. In L = the identity sign ‘‘=’’ is an ordinary nonlogical predicate. The reduction of wQMLs to rQMLs is possible because wQKX = and rQKX = have exactly the same axioms and rules; only their semantics is different. Let ∀2(I) be the ∀2-closure of all instances of the identity axioms I = {I,ISub} for wQ-logics (i.e., if A ∈ ∀2(I), then ∀xA and 2A ∈ ∀2(I)). In L = , ∀2(I) is a set of extralogical premises which characterize the identity predicate as an equality and congruence relation. From now on we represent a given wQML as wQKX+Z, where X is free of additional identity axiom schemata and Z is a set of additional identity axiom schemata (whose instances are restricted to nonmodal substitutions). First we establish the following: LEMMA 5.1 (Reduction-Lemma) (5.1.1)  ⊆ LQ is wQKX+Z consistent iff ∪∀2(I∪Z) is wQKX = -consistent iff ∪∀2(I∪Z) is rQKX = -consistent. (5.1.2) wQKX+Z A iff ∀2(I∪Z) wQKX = A. PROOF. Lemma 5.1.1: The first ‘‘iff’’ follows from lemma 7 in Schurz ([36], 53), and the second ‘‘iff’’ holds because rQKX = and wQKX = have the same axioms and rules. Lemma 5.1.2: This follows from lemma 7 in Schurz ([36], 53). The proof of the next theorem transfers model-completeness from rQMLs to wQMLs. THEOREM 5.2 Every normal wQ-logic is model-complete. PROOF. Assume
is a wQKX+Z-consistent formula set. Since
is wQKX+Z-consistent,
∪ ∀2(I∪Z) is rQKX = -consistent by the reduction lemma (5.1.1); and so
is true at a world w in an rQ-model M = W ,R,D,V . Let Mw = Ww ,Rw ,D,V   be the w-R-generated submodel of M . The truth of ∀2(I) at w in Mw guarantees that for every u ∈ Ww Vw (=) is an equivalence relation over D and a congruence relation w.r.t. the extension of predicates at w. For each u ∈ Ww and d ∈ D, let [d]u := {d  ∈ D: (d,d  ) ∈ Vu (=)} be the Vu (=)-equivalence class of d in u. We define a wQ-model Mw∗ = Ww ,Rw ,D ∗ ,L,V ∗  based on the same frame as that of Mw as follows: (a) D ∗ = {[d]u : d ∈ D,u ∈ Ww }, (b) L := {ld : W → D ∗ such that d ∈ D} where ld is such that ld (u) = [d]u , and (c) Vu∗ (t) = [V  (t)]u , and (d) Vu∗ (Q) = {[d1 ]u ,...,[dn ]u : d1 ,...,dn  ∈ Vu (Q)} for all n-ary predicates Q (for all u ∈ Ww ). By induction on formulas it is proved that for all models Mw , Mw and the corresponding model Mw∗ verify the same formulas at all of their worlds. (1) Atomic formulas: (Mw ,u) |= Qt1 ...tn iff V (t1 ),...,V (tn ) ∈ Vu (Q) iff [V (t1 )]u ,...,[V (tn )]u  ∈ Vu∗ (Q) (because Vu (=) is a congruence relation for predicate-extensions at u as explained above) iff V ∗ (t1 ),...,V ∗ (tn ) ∈ Vu∗ (Q) by definition. (2.) The induction steps for A = ¬B, A = B ∨C , and A = 2B are without

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problems. (3.) Quantifier: (Mw ,u) |= ∀xB iff for all d ∈ D,(Mw [x: d],u) |= B iff for all for all d ∈ D, (Mw∗ [x: ld ],u) |= B by induction hypothesis (the ∗ -counterpart of Mw [x: d] assigns ld to x) iff for all l ∈ L, (Mw∗ [x: l],u) |= B (because L = {ld : d ∈ D}) iff (Mw∗ ,u) |= ∀xB. Finally (4.) Identity (as nonlogical predicate): (Mw [x: d],u) |= t1 = t2 iff [t1 ]u = [t2 ]u (because Vu (=) is an equivalence relation over D) iff (Mw∗ [x: ld ] |= t1 = t2 (by definition). It follows that
is true at w in the wQ-model Mw∗ , and Mw∗ validates the logic wQKX = as well as ∀2(I∪Z). Hence by the reduction lemma (5.1.2), Mw∗ validates all theorems of wQKX+Z. So, wQKX+Z is model-complete. With help of this reduction technique even weak and strong frame-completeness of rQlogics can be transferred to wQ-logics under the proviso that the additional axiom schemata in X do not contain the identity sign. THEOREM 5.3 Assume L = wQKX, where the axiom schemata in X do not contain ‘‘=’’. Then: (5.3.1) F(wQKX) = F(rQKX). (5.3.2) Weak and strong frame-completeness transfers from rQKX to wQKX. PROOF. Theorem 5.3.1: Direction F(rQKX) ⊆ F(wQKX): Assume, for reductio, that W ,R ∈ F(rQKX), but there exists a wQ-model M = W ,R,D,L,V  based on W ,R which falsifies an instance A of an axiom schema in X. Since A does not contain the identity sign, A is also a theorem of wQKX = and hence a theorem of rQKX. We show that there exists an rQ-model M ∗ = W ,R,D ∗ ,V ∗  based on W ,R which is pointwise equivalent with M for identity-free formulas. Thus M ∗ falsifies the rQKX-theorem A, contradicting the assumption that W ,R is a frame for rQKX. We obtain the model M ∗ by identifying D ∗ with L, by identifying V ∗ (t) with t’s term intension V (t): W → D in M , and by defining for every n-ary predicate Q and world u ∈ W , Vu∗ (Q) = {l1 ,...,ln  ∈ Ln : l1 (u),...,ln (u) ∈ Vu (R)}. By induction it is verified that M and M ∗ are indeed pointwise equivalent for identity-free formulas. (1.) Atomic case: (M ,w) |= Qt1 ...tn iff Vw (t1 ),...,Vw (tn ) ∈ Vw (Q) iff V (t1 )(w),...,V (tn )(w) ∈ Vw (Q) iff V ∗ (t1 ),...,V ∗ (tn ) ∈ Vw∗ (Q) (by definition) iff (M ∗ ,w) |= Qt1 ...tn . (2.) The induction steps for A = ¬B, B ∨C and 2B are without problems. (3.) (M ,w) |= ∀xB iff for all l ∈ L, (M [x: l],w) |= B iff for all d ∈ D ∗ = L, (M ∗ [x: d],w) |= B (by induction hypothesis) iff (M ∗ ,w) |= ∀xB. Direction F(wQKX) ⊆ F(rQKX): Assume, for reductio, that W ,R ∈ F(wQKX), but there exists an rQ-model M = W ,R,D,V  based on W ,R which falsifies an instance A of an axiom schema in X. Since A is also a theorem of rQKX = , it is also one of wQKX = . Again we show that there exists an wQ-model M ∗ = W ,R,D,L,V ∗  based on W ,R which is pointwise equivalent with M for identity-free formulas. Thus M ∗ falsifies the wQKX-theorem A, contradicting the assumption that W ,R is a frame for wQKX. We obtain the model M ∗ simply by letting L be the set {cld : d ∈ D} of all constant worldlines cld : W → {d}, and defining V ∗ (t) = clV (t) , and Vw∗ (Q) = {ld1 (w),...,ldn (w): d1 ,...,dn  ∈ Vw (Q)}. By simple induction on formulas it is shown that M and M ∗ verify the same identity-free formulas in all of their worlds. Theorem 5.3.2: Assume
is an wQKX-consistent formula set (for weak completeness we assume
to be finite). As for the proof of theorem 5.2 we conclude that
∪∀2{I} is rQKX = -consistent and, thus, is verified at some world w of an rQ-model M which is based on a frame W ,R for rQKX. Hence
∪∀2{I} is also verified in w of the w-Rgenerated submodel Mw = Ww ,Rw ,D,V   of M , and moreover, the frame Ww ,Rw  of Mw

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16 Combinations and completeness transfer for quantified modal logics must also be a frame for rQKX (since frame-validity is preserved under the formation of generated subframes). By the same technique as in the proof of theorem 5.2 we construct a pointwise equivalent wQ-model Mw∗ = Ww ,Rw ,D ∗ ,L,V ∗  which validates wQKX and is based on Ww ,Rw  ∈ F(rQKX). Theorem (5.3.1) implies that Ww ,Rw  ∈ F(wQKX), whence wQKX is (weakly/ strongly) frame-complete. The use of theorems 5.2 and 5.3 lies in the fact that they transfer all logical discoveries on (frame- and model-) completeness of rQMLs directly to wQMLs.

6 Concluding Remarks Several important results on the three major research questions of Section 1 have been reported in this paper, including new results concerning the transfer of frame-completeness from PMLs to rQMLs (theorem 3.2), the transfer of canonicity from monomodal QMLs to their multimodal combinations (theorem 4.2), and the transfer of model- and framecompleteness from rQMLs to wQMLs (Theorems 5.2 & 5.3). Many questions are still unanswered, and in this respect three open problems have been stated. In the last section we have worked out the philosophical and logical advantages of worldline semantics. Our version of worldline semantics assumes a constant domain of worldlines and, hence, validates the Barcan formula. However, for certain purposes one needs QMLs without Barcan formula and its converse. One possibility to get rid of BF within worldline semantics is to assume a word-relative set of worldlines Lw , for each w ∈ W . This suggestion has been elaborated by Garson ([17], sec. 1.4), and a variant of it is developed in Schurz ([36], ch. 10.4, 10.10), in which the world-relative sets Lw are defined by the extension of an ordinary existence predicate E. If the range of the quantifiers are restricted to world-relative sets of worldlines Lw , then the classical quantifier principles have to be replaced by the quantifier principles for free modal logics. For example, the axiom UI has to be replaced by its free version fUI: ∀xA ⊃ (Ey ⊃ A[t/x]), and similarly for the rule UG. Unfortunately, the simple transfer of (model- or frame-) completeness from rQMLs to free QMLs is not possible (cf. [36], 229). However, the expressive power of wQMLs is greater than that of free QMLs: the free quantifiers of free QMLs can be defined within wQMLs. Thus, there is no real need to leave the realm of wQMLs. The semantics of free quantifiers works as follows: assume E is an existence predicate whose intension V (E): W → Pow(D) associates with each world w the set of objects Vw (E) which actually exist at w. Thus every local domain Dw divides into an inner domain Vw (E) ⊆ Dw of actual objects (at w) and an outer domain (Dw −Vw (E)) of non-actual objects at w. The world-relative sets of worldlines of free QMLs are defined as Lw := {l ∈ L: l(w) ∈ Vw (E)}, i.e. Lw contains those world-lines which land some actual object in w. Then the free quantifiers ∀f and ∃f of free QMLs can be simply defined within the framework of wQMLs as follows (cf. [36], 201): ∀f xA := ∀x(Ex → A), and ∃f xA := ∃x(Ex ∧A). ∀f and ∃f satisfy the free quantifier rules and make the Barcan formula invalid. Therefore everything which can be expressed in free modal logics is also expressible in wQMLs.

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