1 Introduction
The discovery of Kripke incompleteness, the existence of normal modal logics that are not sound and complete with respect to any class of Kripke frames, has been called one of the two forces that gave rise to the “modern era” of modal logic (Blackburn et al., 2001, p. 44). In the Lemmon Notes of 1966, it was conjectured that all normal modal logics are Kripke complete (Lemmon and Scott, 1977, p. 74).^{3}^{3}3In fact, Wolter and Zakharyaschev (2006, p. 428) claim that much of early research in the area was motivated by the still more optimistic Big Programme or globalist’s dream. See § 6.3 and especially Footnote 25 for a further discussion. But this was not to be. Kripke incompleteness was first demonstrated with a bimodal logic (Thomason, 1972), shortly thereafter with complicated unimodal logics (Fine, 1974; Thomason, 1974a), and later with simple unimodal logics (van Benthem, 1978, 1979; Boolos and Sambin, 1985). The significance of these discoveries can be viewed from several angles.
From one angle, they show that Kripke frames are too blunt an instrument to characterize normal modal logics in general. More finegrained semantic structures are needed. From another angle, they show that the notion of derivability in a normal modal logic with a set of axioms is too weak to capture the notion of Kripke frame consequence, in the sense where iff every Kripke frame that validates every also validates . A deep result of Thomason (1975a) showed such weakness to be inevitable: the standard consequence relation for monadic secondorder formulas with a single binary relation is reducible to the Kripke frame consequence relation for modal formulas.^{4}^{4}4To be more specific, Thomason (1975a) showed that such secondorder formulas can be translated into unimodal formulas in such a way that a secondorder formula is a consequence of a set of secondorder formulas over standard secondorder structures iff the translation of is valid in every Kripke frame that validates for a fixed modal formula . Since the former consequence relation is not recursively axiomatizable, neither is the latter.
Both of these angles on Kripke incompleteness lead to closely related questions and lines of investigation, with slightly differing focus. As we shall see, both of these lines of investigation will come together nicely in this paper.
1.1 The Semantic Angle
The first angle on Kripke incompleteness—the realization that Kripke frames are not finegrained enough for the study of normal modal logics in general—renewed interest in the algebraic semantics for normal modal logics based on Boolean algebras with operators (baos) (Jónsson and Tarski, 1951, 1952). A bao is a Boolean algebra together with one or more unary^{5}^{5}5Jónsson and Tarski considered operators of higher arity, but we will consider only unary operators. operators, i.e., unary operations such that for all elements of the algebra, , and for the bottom element of the algebra, . Every normal modal logic is sound and complete with respect to a bao, namely, the LindenbaumTarski algebra of the logic, according to a straightforward definition of when a modal formula is valid over a bao. Kripke incompleteness can be better understood in light of the fact that Kripke frames correspond to baos that are complete (), atomic (), and completely additive (), or baos for short (see § 3). A bao is complete or atomic, respectively, according to whether its Boolean reduct is complete or atomic in the standard sense, while it is completely additive iff the following holds:

for any set of elements, if exists, then exists and
which in the case of a complete bao reduces to the distributivity of over arbitrary joins. (The letter ‘’ is intended to suggest the big join .)
Given the correspondence between Kripke frames and baos, the fact that a normal modal logic is not the logic of any class of Kripke frames means that it is not the logic of any class of baos. To put these points in more algebraic terms: normal modal logics correspond to varieties of baos, and Kripke incompleteness is the phenomenon that not every variety of baos can be generated as the smallest variety containing some class of baos. The last point is underscored by the dramatic Blok Dichotomy (Blok, 1978): a variety of baos is either uniquely generated by the baos it contains (the associated modal logic is strictly Kripke complete) or else there are continuummany other varieties of baos that contain exactly the same baos (other modal logics that are valid over exactly the same Kripke frames).
From this algebraic perspective, a series of natural questions arises: what happens if we drop or weaken one or more of the properties , , and ? Do we thereby obtain distinct notions of completeness for normal modal logics? Can we represent the resulting baos with some kind of frames with richer structure than Kripke frames? Does incompleteness persist even if we retain only one of the properties , , or ? Does the analogue of the Blok Dichotomy hold if we drop one or more of , , or ?
These questions and the general landscape of subKripkean notions of completeness from an algebraic perspective were the subject of the PhD research of Litak (2004; 2005a; 2005b; 2008). As it turns out, none of the following notions of completeness are equivalent to any of the others: completeness with respect to atomic baos (baos), complete baos (baos), baos that admit residuals (baos), atomic and completely additive baos (baos), complete and completely additive baos (baos), and baos. A rich hierarchy of different notions of completeness thereby comes into view. As for the representation question: Došen (1989) showed that baos are the duals of normal neighborhood frames; ten Cate and Litak (2007) showed that baos are the duals of discrete general frames; and Holliday (2015) has given a dual representation of baos in the framework of possibility semantics. As for the question of whether incompleteness persists if we retain only one of the properties , , or : Litak (2004) showed that there are incomplete logics, and Venema (2003) showed that there are incomplete logics (by proving the much stronger result that there is a variety of baos all of whose members are atomless). Finally, as for extending the Blok Dichotomy: Zakharyaschev et al. (2001) noted that it extends to baos, Chagrova (1998) showed that it extends to baos, and Litak (2008) showed that it extends to all baos (indeed, even to complete baos, only requiring countable joins) as well as baos.
One piece of the puzzle remained missing for over a decade: are there incomplete logics (Litak 2004, Litak 2005b, Ch. 9, Litak 2008, § 7)? Venema (2007, § 6.1) also asked whether there are inconsistent logics, i.e., normal modal logics that are not sound over any bao. In this paper, we answer these questions affirmatively. Our solution involves a firstorder reformulation of the ostensibly secondorder condition of complete additivity, which arose in the context of possibility semantics mentioned above (Holliday, 2015).^{6}^{6}6Around the same time that we proved that complete additivity of an operator in a bao is a firstorder condition, in January 2015, Hajnal Andréka, Zalán Gyenis, and István Németi independently proved that complete additivity of an operator on a poset is preserved under ultraproducts. After they learned of our result on the firstorderness of complete additivity in baos from Steven Givant, Andréka et al. (2016) extended it from baos to arbitrary posets. Using this firstorder reformulation, we revisit an intriguing Kripkeincomplete logic of van Benthem (1979) and show that van Benthem’s logic, previously known to be  and incomplete, is the missing example of a incomplete logic. Furthermore, it can be easily modified to answer Venema’s question on inconsistency. Building on this example, we extend the Blok Dichotomy to incompleteness.
Given these results, the question arises of whether there are “naturally occurring” incomplete logics. In the case of bimodal logic, we answer this question affirmatively. We show that the bimodal provability logic (Japaridze, 1988; Boolos, 1993), wellknown to be Kripke incomplete and hence incomplete, is also incomplete.
1.2 The Syntactic Angle
Van Benthem’s logic was designed to illuminate the second angle on Kripke incompleteness mentioned above—the weakness of the notion of derivability in a normal modal logic with axioms—so our story will bring these two angles together. Let mean that belongs to the smallest normal modal logic that contains the formulas in as axioms; so thinking in terms of derivations, not only modus ponens but also necessitation and uniform substitution may be applied to formulas from . Van Benthem observed that Kripke incompleteness results can be viewed as nonconservativity results with respect to . These results show that (i) for some modal formulas and , , yet (ii) every Kripke frame that validates also validates . As is well known, every modal formula can be translated into a sentence of monadic secondorder logic such that is valid over a Kripke frame in the sense of Kripke semantics iff is true in the frame as a standard secondorder structure. Thus, (ii) can be rephrased as the fact that every Kripke frame that makes true also makes true. Van Benthem observed that the proof of (ii) typically shows that is derivable from using some weak system of monadic secondorder logic plus an axiom of choice. In this sense, the secondorder system is not conservative with respect to , in light of (i). To better gauge the weakness of , van Benthem asked whether there exist such a and for which the derivation of from can be carried out using only what he considered the weakest reasonable secondorder system, dubbed weak secondorder logic. This would be a striking example of the weakness of compared to secondorder logic. Van Benthem indeed found such a and . We will call the smallest normal modal logic containing that , which does not contain by (i), the logic .
We will show that is a incomplete logic by showing that every bao that validates also validates . At the end of the paper, we will follow a path in the spirit of van Benthem 1979: how can we strengthen our base logic to derive the formula from , and what does this show about the weakness of the base logic? We begin by reviewing van Benthem’s approach of translation into weak secondorder logic. As it turns out, even weak secondorder logic is much more powerful than what one needs to derive van Benthem’s from his and thereby demonstrate nonconservativity with respect to basic modal logic. We consider two ways of extending the basic modal syntax for this purpose. One way leads to the nominal^{7}^{7}7 The more recent use of the corresponding adjective in theoretical computer science (Pitts, 2013, 2016) has nothing to do with the term as used in § 8.2 and references quoted therein. calculus that characterizes consequence over baos. Another way leads to the tense calculus that characterizes consequence over baos. Finally, we find a common core for these weakenings: we show how the firstorder reformulation of complete additivity inspires additional modal inference rules admissible over baos that allow us to derive van Benthem’s from his . Of course, characterizing a consequence relation by means of a (set of) rule(s), in a possibly extended syntax, is much more than just admissibility: in addition to soundness, one requires a generic completeness result. We will leave as an open question whether the rules we are proposing yield a syntactic characterization of consequence.
1.3 Organization
The paper is organized as follows. In § 2, we review the proof that van Benthem’s logic is Kripke incomplete, which we find to be a simple, vivid, and hence pedagogically useful example of Kripke incompleteness. In § 3, we review the algebraic approach to modal (in)completeness as in Litak 2005b. With this background, we proceed to the main part of the paper: in § 4, we present the firstorder reformulation of complete additivity, and in § 5, we use this reformulation to prove that the unimodal logic and the bimodal logic are incomplete as above (and that the quasinormal logic is even inconsistent in a suitably adjusted sense). In § 6, we discuss issues of decidability and complexity: we show that even inconsistent logics do not have to be more complex than the classical propositional calculus, that the property of completeness is in general undecidable, but that the associated notion of consequence (unlike Kripke frame consequence) is recursively axiomatizable. In § 7, we build on the example of to generalize the Blok Dichotomy to incompleteness. In the remaining sections we discuss the second, syntactic angle presented above: in § 8, we give syntactic proofs in existing derivation systems of the formula witnessing the incompleteness of ; and in § 9, we discuss extending our base logic with new sound rules of inference. We conclude the paper in § 10 with open problems for future research.
2 Kripke Incompleteness
In order to put Kripke incompleteness in context, let us review some basic definitions.
Let be the set of formulas of propositional modal logic generated from a set of propositional variables. We use the usual notation for connectives: , , , , , . A normal modal logic is a set such that: (a) contains all tautologies of classical propositional logic; (b) is closed under modus ponens, i.e., if and , then ; (c) is closed under uniform substitution, i.e., if and is the result of uniformly substituting formulas for propositional variables in , then ; (d) is closed under necessitation, i.e., if , then ; and (e) . Let be the smallest normal modal logic.
The definition of a normal modal logic extends to polymodal languages with multiple modalities , , etc., by requiring (d) and (e) for each . In this section, we focus on the unimodal language, but polymodal languages will become important in § 5.
We assume familiarity with Kripke frames , Kripke models , and the Kripke semantic definition of when a formula is true at a : . We abuse notation and write ‘’ or ‘’ to mean . For convenient additional notation, given , let , and given , let . A formula is globally true in a Kripke model , written ‘’, iff for every ; and is valid over a Kripke frame , written ‘’, iff for every model based on . A formula is valid over a class of Kripke frames iff it is valid over every frame in the class. Let be the set of formulas valid over , which is always a normal modal logic.
A logic is Kripke complete if there is a class of Kripke frames for which . Otherwise it is Kripke incomplete. For any logic , we can consider the class of frames that validate it: . For a Kripke complete logic, , whereas for a Kripke incomplete logic, .
Everything said above was in terms of validity, but we could also put our discussion in terms of consequence. To avoid confusion, it is important to distinguish between the following consequence relations, following van Benthem (1983, p. 37):

iff for every Kripke model and , if for all , then (local consequence over models).

iff for every Kripke frame , if for all , then (global consequence over frames).
These two notions of consequence are related to two different notions of when is derivable from a set of formulas. Let iff belongs to the closure of under modus ponens. Let iff belongs to the smallest normal modal logic , which is the closure of under modus ponens, necessitation, and uniform substitution. Our is what van Benthem (1979) denotes by ‘’ and calls ‘the minimal modal logic ’. It would be reasonable to call derivability from axioms and to call derivability from premises, since intuitively necessitation and uniform substitution should be applicable to logical axioms but not to arbitrary premises.
The relation is axiomatized by : iff , which is equivalent to there being such that . By contrast, by the result of Thomason (1975a) mentioned in § 1, is not recursively axiomatizable. We have that implies , but the converse is not guaranteed. The “weakness” of referred to in § 1.2 is the fact that does not guarantee .
While seems to capture an intuitive notion of modal consequence, the relation implicitly prefixes all premises by arbitrary sequences of boxes and universal quantifiers over propositional variables, so it yields striking consequences like and . Thus, it does not enjoy a deduction theorem and one should not think about sets of formulas closed under as “local theories”. There is, however, a better way to think about . In modal logic, we are often not interested in the class of all frames, but rather in some restricted classes of frames, perhaps defined as for some set . We then want to know what formulas are valid over this class, i.e., whether . This is equivalent to asking whether . Also note that is Kripke complete as above iff for every , implies .
Let be the smallest normal modal logic containing the axiom
which we will call the axiom. Van Benthem (1979) proved that the logic is Kripke incomplete. While may at first seem an entirely ad hoc example of a Kripkeincomplete logic, we will observe a striking connection between the incompleteness of and the incompleteness of an important provability logic in § 5.2. In this connection, it is noteworthy that the axiom is a theorem of the provability logic , the smallest normal modal logic containing the Löb axiom, . Substituting for in the Löb axiom yields , which in the context of provability logic is a modal version of Gödel’s Second Incompleteness Theorem.^{8}^{8}8When describing the origin of the axiom, van Benthem (1979) hints at a quasinormal logic that one recognizes as (a subsystem of) Solovay’s system (see, e.g., Boolos 1993, p. 65), another central formalism in the area of provability logic. The formula under in the consequent of , i.e., , is a theorem of , and as shown by a syntactic derivation in van Benthem 1979, this formula alone makes it impossible to characterize in terms of Kripke semantics with distinguished worlds, which is the standard relational semantics for quasinormal systems. We will discuss such provabilityrelated quasinormal systems in § 5.3. Clearly the axiom is derivable from . In the other direction, van Benthem showed that is a Kripkeframe consequence of the axiom. However, he also showed that is not a theorem of . Together these facts imply the Kripkeincompleteness of .
So that our presentation is somewhat selfcontained, we will include van Benthem’s proof of the Kripkeincompleteness of . It should be emphasized that this is one of the simplest proofs of Kripke incompleteness.^{9}^{9}9It is not, however, the most natural example of a Kripke incomplete unimodal logic. That honor goes to the logic of the Henkin sentence (see Boolos and Sambin 1985), which is also a simplest possible Kripke incomplete unimodal logic in the following sense: it is axiomatized by a formula with only one propositional variable and modal depth 2. Lewis (1974) showed that all normal unimodal logics axiomatizable by formulas of modal depth are Kripke complete. Despite its charms, the Henkin logic is irrelevant for our puposes in this paper, for a reason that can be explained using notions introduced in § 3: the proof of its Kripke incompleteness does not attack complete additivity, but rather closure under countable joins/meets, i.e., completeness. This property is exploited by most incompleteness proofs involving and its relatives, like the failure of strong completeness of itself or the Kripkeinconsistency of various tense logics containing (although counterexamples related to the original one by Thomason 1972 clash with full rather than its restriction to for any fixed cardinality ) (Litak, 2005b). It should also be emphasized that in § 5 we will prove a much more general result than the following lemma; but we include a proof of Lemma 2.1 for later reference in § 8.
Lemma 2.1.
Any Kripke frame that validates also validates .
Proof.
Let be a Kripke frame that validates . We need to show that if is such that , then there is a such that (think of the contraposition: ). Hence, consider an such that and a . Let be a model based on with . For reductio, suppose there is no with , so . Then since validates , we have , which with implies , which with our valuation for implies . Thus, there is a such that but . From , we have . Then from and , we have and , so , a contradiction. Hence there is a with , as desired. ∎
All that remains to show is that . We can do so by exhibiting a Kripke model and showing that every is globally true in , while is not. Since every is globally true in every Kripke model, and the set of formulas that are globally true in a given Kripke model is closed under modus ponens and necessitation, to show that every is globally true in , it suffices to show that every substitution instance of the axiom is globally true in (in the terminology of Fine 1974, the axiom is strongly verified in ).
In the literature on Kripke incompleteness, rather than directly exhibiting a model as above, authors typically exhibit an appropriate general frame^{10}^{10}10Readers familiar with the algebraic approach to modal logic will of course note that one can directly define a modal algebra (bao) instead. We will introduce modal algebras soon in § 3. where is a Kripke frame and is a family of subsets of that is closed under union, complement relative to , and the operation . An admissible model based on a general frame is a model such that for every propositional variable . An easy induction then shows that for every , . It follows that if a formula is globally true in every admissible model based on —in which case is valid over , written ‘’—then for any particular admissible model based on , all substitution instances of are globally true in . Thus, to obtain a model as in the previous paragraph, it suffices to exhibit a general frame over which the axiom is valid, while is not.
Another way to motivate going to a general frame here is by the following observation. Define a consequence relation by iff for every general frame , if for all , then . Then it can be shown that there is an exact match between and the derivability relation above: iff belongs to the smallest normal modal logic containing . So to show that does not belong to , we simply show that , which is again to show that there is a general frame over which the axiom is valid, while is not.
Definition 2.2 (van Benthem frame).
Observe that is closed under union, relative complement, and .
We now add the final piece of the argument.
Lemma 2.3.
is valid over , while is not. Thus, .
Proof.
Consider any admissible model based on . First observe that , and for all , , so ; but , so we have . Thus, we need only show that , which is equivalent to . If , then for every , , whence an obvious induction shows that . Hence is cofinite, so . This shows that , which completes the proof that is valid over .
Finally, observe that .∎
Theorem 2.4 (van Benthem 1979).
The logic is Kripke incomplete.
Van Benthem’s main point was not that is Kripke incomplete^{12}^{12}12Indeed, the very existence of Kripke incomplete logics was not much of a revelation anymore at the time. We have already mentioned in the opening paragraph of this paper that Kripke incompleteness was first demonstrated with a bimodal logic (Thomason, 1972) and shortly thereafter with complicated unimodal logics (Fine, 1974; Thomason, 1974a) located in increasingly specific areas of the lattice of extensions of . Van Benthem himself devoted an earlier paper (1978) to simple examples of incomplete logics. Moreover, at that time two crucial results (which are going to be our main concern in § 6.3 and §§ 78) that make explicit the ubiquity and unavoidability of Kripke incompleteness were already known: Thomason 1975a and Blok 1978. but that it is a special example of such incompleteness: it can be used to show that the derivability relation falls short of capturing not only the consequence relation itself, but also syntactically inspired weakenings of . We will return to this in § 8.
The main point we wish to make about is that it is special in another way: it provides the long missing example of a incomplete logic. To explain what this means and its context, let us now review the algebraic perspective on modal logic.
3 The Algebraic Approach to Modal (In)completeness
As noted in § 1, the discovery of Kripke incompleteness renewed interest in the algebraic semantics for normal modal logics based on Boolean algebras with operators (baos) (Jónsson and Tarski, 1951, 1952). An ary operator on a Boolean algebra with universe is a function that preserves finite joins in each coordinate (including the join of the empty set, ); a dual operator preserves finite meets in each coordinate (including the meet of the empty set, ). A bao is a Boolean algebra equipped with a collection of operators. In this paper, we consider only baos with unary operators. If the collection of these operators in a bao has cardinality , we call a bao. Per tradition, we call a bao a modal algebra (ma).
The language of basic unimodal logic can be interpreted in an ma in the obvious way: any mapping of propositional variables to elements of extends to a mapping of arbitrary formulas to elements of , taking , , , and , where , , and are the complement, join, and operator in , respectively. The ma validates a modal formula (notation: ) iff every such mapping sends to the top element of . For a given class of mas (see below for important examples of such classes), we define a consequence relation , analogous to the global Kripke frame consequence relation from § 2:

iff for every , if for all , then .
All of the notions above extend to interpreting a polymodal language with modal operators in baos in the obvious way.
Definition 3.1.
Let be a class of baos and a normal modal logic in a language with modal operators. We say that is complete if for all formulas , we have iff . Otherwise is incomplete.
Equivalently, is complete if is the logic of some class , i.e., is exactly the set of formulas validated by all baos in .
Each normal modal logic is the logic of a bao: the LindenbaumTarski algebra of , whose elements are the equivalence classes of modal formulas under the relation defined by iff , and whose operations are defined in the obvious way: , , and . This general algebraic completeness theorem via LindenbaumTarski algebras can be seen as a special case of an even more general approach: since the derivability relation^{13}^{13}13The relation is defined by: iff belongs to the closure of under modus ponens and necessitation. For , is the derivability relation that matches global consequence over Kripke models: iff for every Kripke model , if for all , then . associated with a given normal modal logic is (Rasiowa) implicative and hence strongly finitely algebraizable, one obtains a strong completeness theorem for with respect to algebraic semantics using the standard machinery of abstract algebraic logic (AAL) (Rasiowa, 1974; Blok and Pigozzi, 1989; Czelakowski, 2001; Andréka et al., 2001; Font, 2006; Font et al., 2003, 2009); see § 5.3 and especially Footnote 18 for historical origins of this approach.
Before proceeding further, let us fix notation for dealing with algebras. We use gothic letters for names of algebras and for elements of algebras. Whenever it is not confusing, we blur the distinction between an algebra and its carrier, writing statements like ‘’. We also blur the distinction between modal formulas and baoterms, and henceforth we will simply use , , and for the complement, join, and meet, respectively, in our algebras, trusting that no confusion will arise. In an ma, we take to be the operator and to be a dual operator, defined by . In baos, we may add indices to distinguish between multiple operators, e.g., taking and to be operators and and to be their duals.
The generic completeness result described above made the algebraic semantics historically the first to be studied, prior to the invention of Kripke frames (see Goldblatt 2003, § 3; Blackburn et al. 2001, § 1.7). However, one can also obtain a generic completeness result with respect to the general frames of § 2. This result is implicit already in the work of Jónsson and Tarski (1951; 1952), who proposed an extension of Stone’s Representation Theorem from Boolean algebras to baos. The general frames obtained via this representation are known as descriptive frames; thus, every normal modal logic is sound and complete with respect to a class of descriptive frames. Furthermore, for a large class of modal axioms/equations, especially socalled Sahlqvist axioms and their various generalizations (see, e.g., Conradie et al. 2006 or Conradie et al. 2014 and references therein), one can in addition observe their persistence in passing from a descriptive frame to its underlying Kripke frame. This phenomenon is known in the contemporary literature as canonicity or dpersistence (Chagrov and Zakharyaschev 1997, Ch. 10, Blackburn et al. 2001, Ch. 5), but in hindsight the JónssonTarski work can be seen as its earliest study. In short, algebra combined with duality theory provides a viable route towards Kripke completeness results for suitably wellbehaved logics. On the other hand, as far as weak completeness (which is the main subject of the present paper) with respect to finite models is concerned, it is not necessary to phrase such completeness results in algebraic terms or to involve the StoneJónssonTarski duality in the proof. Think, e.g., of tableauxstyle extraction of countermodels from failed proof search in suitable Gentzenstyle calculi (in fact, close to Kripke’s original work) or the technique of normal forms. Such finitary approaches are not restricted to dpersistent logics. See Fine 1975, Moss 2007, and Bezhanishvili and Ghilardi 2014; the relationship with duality theory and construction of free algebras is discussed in Ghilardi 1995, Bezhanishvili and Kurz 2007, and Coumans and van Gool 2013.
Remark 3.2.
Henkinstyle generalframe strong completeness of logics in countably many variables is equivalent to the weak König Lemma even over , the weak base theory for reverse mathematics (Simpson, 2009, IV.3.3). Recall that the weak König Lemma holds in or even in the Zermelo set theory (i.e., without replacement). On the other hand, for uncountably many propositional variables, one needs representation theorems in the style of Jónsson and Tarski. They rely on the Ultrafilter Theorem, or equivalently, the Boolean Prime Ideal Theorem (). It would thus seem that completeness based on canonicity is rather nonconstructive, but with some care it is possible to prove more finegrained results along these lines—see Ghilardi and Meloni 1997 and Suzuki 2010 or strong completeness with respect to the general possibility frames of Holliday 2015. It is also worth mentioning that strong Kripke completeness does not imply dpersistence, as in the case of the tense logic of the reals (Wolter, 1996b); if one is willing to extend the notions of strong completeness and canonicity to neighborhood frames, another counterexample is provided by the McKinsey logic (Surendonk, 2001).
In the reverse direction, we of course do not need Kripke frames (or any other semantics) as an intermediate step in proving algebraic completeness; the LindenbaumTarski construction provides a direct route. Nevertheless, Kripke completeness results can be reformulated and understood from an algebraic point of view: they establish that the equational class (variety) of baos corresponding to a given modal logic is determined by its elements with special additional properties. In other words, they show that when looking for algebraic models refuting a given formula/equation, one can restrict attention to a wellbehaved subclass of algebras.
Let us discuss this in more detail. Recall the standard construction associating with a given Kripke frame its dual ma , whose universe is , whose Boolean operations are interpreted using the settheoretic ones, and whose operator is defined by . This is just a special case of taking the dual of a general frame , where the ma in question is provided by ; in the case of Kripke frames, . As observed already by Jónsson and Tarski (1951; 1952), such an ma always has the following three special (and mutually independent) properties.

latticecompleteness: given any set of elements of , its join exists in . This also implies the existence of arbitrary meets.

atomicity: any nonbottom element is above an atom, i.e., a minimal nonbottom element.

complete additivity: for any set of elements, if exists, then exists and
For complete mas, complete additivity reduces to distributivity of over arbitrary joins. Of course, can be equivalently stated with replacing and replacing .
Remark 3.3.
In the case of (duals of) Kripke frames, where every subset is admissible, is simply . Nevertheless—and this is an important point!—it does not have to be the case with general frames whose is latticecomplete. In particular, in any descriptive frame associated with an infinite bao, there are instances of joins and meets not coinciding with unions and intersections. The atoms of (duals of) Kripke frames are obviously singleton sets . Again, for arbitrary general frames this does not have to be the case, but this is less crucial: in differentiated general frames, which are in an important sense the only relevant ones, admissible atoms have to be singleton sets.
Remark 3.4.
As we shift from unimodal to polymodal contexts, will be the class of complete baos with the appropriate number of operators in the context, and similarly for , , etc. In principle, when we say a polymodal logic is complete, we should mean that it is complete with respect to baos in which every operator is completely additive. But most of the time, complete additivity of all operators occurring is not needed, and we may wish to be more finegrained: given a logic with modalities indexed by natural numbers, we can say that is complete if it is complete, as in Definition 3.1, where is the class of baos in which the th operator is completely additive.
As it turns out, the combination of the three properties above is a defining feature of duals of Kripke frames. One can say even more: the category of Kripke frames with bounded morphisms is dually equivalent to that of baos with complete morphisms (Thomason, 1975b). In particular, taking any Kripke frame/bao, converting it into its dual bao/Kripke frame, and then going back produces an output isomorphic to the original input. Therefore, Kripke completeness is just completeness.
In this way, we are led to the first of the two angles on Kripke incompleteness discussed in the introduction: the semantic angle. Given that the properties , , and are independent of each other, will arbitrary combinations of these three lead to distinct notions of completeness, each more general than Kripke completeness but less general than algebraic completeness? Or is the propositional modal language too coarse to care about differences between all or at least some of these semantics? And how about other notions contained somewhere in between? For example, can be weakened to completeness (), i.e., closure under countable meets and joins. One can then ask if there are logics that are complete but not complete. For another important example, consider the property of admissibility of residuals/conjugates. Recall that admits conjugates if there is a function such that for every , iff . Alternatively, we can say that an algebra admits residuals if there is a function such that for every , iff . These two definitions are equivalent, taking .^{14}^{14}14Observe that we do not require that residuals are termdefinable. This is the difference between baos admitting residuals and Jipsen’s (1993) residuated baos. Some wellknown facts include (i.e., admissibility of residuals implies complete additivity) and = (i.e., in the presence of latticecompleteness, the converse implication also holds). Once again, one may ask: how does completeness relate to all the other completeness notions?
As mentioned in § 1, a systematic investigation into these questions has been undertaken by Litak (2004; 2005a; 2005b; 2008), unifying, expanding, and building on earlier results by Thomason, Fine, Gerson, van Benthem, Chagrova, Chagrov, Wolter, Zakharyaschev, Venema, and other researchers. There is no place here to discuss most of the results in detail, but an executive summary of those most relevant for the present paper is as follows:

almost any conceivable combination of the above properties of baos leads to a distinct notion of completeness. In other words, for almost any pair of such combinations, there is a logic complete in one sense, but not in the other;

the Blok Dichotomy (§ 7), the result of Wim Blok showing that Kripke incompleteness is in a certain mathematical sense the norm rather than an exception among normal modal logics, generalizes to most of these weaker notions of completeness;

many of these notions admit syntactic characterizations in terms of conservativity of certain types of extensions, e.g., completeness in terms of conservativity of minimal nominal extensions (§ 8.2), completeness in terms of conservativity of minimal tense extension (§ 8.3), or completeness in terms of conservativity of minimal infinitary extensions with countable conjunctions and disjunctions.
The possibility of incompleteness, however, was left completely open, and in fact it was the sole reason for the “almost any” hedge above (Litak 2004, Litak 2005b, Ch. 9, Litak 2008, § 7). Discussing this line of research in the Handbook of Modal Logic, Venema (2007, § 6.1) singled out a slightly stronger version of the same question: whether there are inconsistent logics, i.e., normal modal logics that are not sound over any bao.
Why was this question so puzzling? First of all, note that while a free algebra on infinitely many generators in any variety of baos can never be latticecomplete or atomic, it can be completely additive.^{15}^{15}15For a characterization of when the LindenbaumTarski algebra of a normal modal logic is completely additive, see Holliday 2015, § 7.2. Ghilardi (1995) showed that free algebras in the variety of all baos are baos (and hence baos), while Bezhanishvili and Kurz (2007) extended this to all varieties of baos axiomatized by rank1 formulas. Holliday (2014) added an analogous result for , , , and (and reproved it for , , and all extensions of ). One can imagine that if is not inconsistent with freeness, then there might be a general way of turning any bao into a completely additive one without changing the set of valid equations. But there are other ways in which complete additivity seemed somewhat intangible. Unlike its closest relatives and , for which van Benthem’s logic can be shown to be incomplete, did not seem definable in a language with a usable model theory.
Let us make this more precise. There is an obvious firstorder correspondence language for baos, whose connectives can be written as (to avoid notational clashes with baoterms and modal formulas). For classes of algebras definable in this language, one can even blur the distinction between the class itself and its defining formula. , , and are firstorder properties, in fact even properties. But how could one define without using infinitary formulas (of unrestricted cardinality!) or a powerful secondorder language (with full rather than Henkin semantics)?
4 as an Elementary Class
Surprisingly, complete additivity is in fact a firstorder property. To prove this, it will be helpful to use some abbreviations for describing relations between elements of a bao . Let lowercase letters range over elements of and define:
stands for  

stands for  
stands for 
Consider a property of baos formulated in our correspondence language as follows:
.
The origin of this condition is in the duality theory for classes of baos and possibility frames in Holliday 2015. There the condition is viewed as follows. Given a bao , define a binary relation on the universe of by: iff for all , we have . Then satisfies iff whenever , there is a such that . Any such bao can be turned into a possibility frame with the accessibility relation provided by and the validity relation coinciding with that of the original algebra.
Remark 4.1.
We will now prove that is equivalent to complete additivity.
Theorem 4.2.
implies .
Proof.
We prove this by contraposition. Assume is not a bao. This means there is a such that exists in , but there is an such that:

for all ,

.
By (ii), , so in order to refute , it is enough to show that
Pick a , so . By the joininfinite distributive law
holding in any Boolean algebra, it follows that there is a such that is not , so . But then
by (i), so . ∎
Theorem 4.3.
implies .
Proof.
Again we reason by contraposition. Assume for some that
(1) 
Consider . To refute it is enough to show that
for then , yet for all , , which implies
By definition of , is an upper bound of , so we need only show that it is the least. Suppose there is an upper bound of such that . Hence is not , so . Then (1) implies there is a such that . Therefore , and since is an upper bound of , . But then , contradicting . ∎
Remark 4.4.
implies in a more direct way: where is the conjugate of , take , so implies and hence ; then if were such that , we would have and hence , contradicting . We shall see a derivation of a similar form at the end of § 9.2.
Corollary 4.5.
.
Not only does this show that is a firstorder property but furthermore that it is of a rather convenient syntactic shape: . Such conditions are particularly convenient for reformulation as nonstandard inference rules, which we will discuss in § 9.
Remark 4.6.
In response to our proof that is equivalent to the firstorder property , Johan van Benthem (p. c.) devised a proof of the firstorderness of in the style of correspondence theory (van Benthem, 2001). First note that in the equality for ,
the direction is immediate from the motonicity of . Thus, using the equivalence of and , we can rewrite as the following sentence in the secondorder language of baos:
(2) 
where ‘’ abbreviates the firstorder sentence expressing that is the least upper bound of . Since does not occur in the consequent of the outer conditional in (2), we can rewrite (2) as
(3) 
Now we observe that the antecedent of the outer conditional in (3) is equivalent to a firstorder sentence. For there exists an as in the antecedent iff is the least upper bound of the following firstorder definable set:
One direction of the ‘iff’ is immediate. For the other, if a set is such that and , then and hence . Then since and , we have , and by definition of , we have . Thus, if there is any witness for the , then is a witness. Hence (3) can be equivalently rewritten as the following firstorder sentence:
(4) 
where ‘’ abbreviates the firstorder sentence expressing that is the least upper bound of . This completes the proof that is firstorder. Further manipulations are required to show that (4) is equivalent to in particular.
5 Incompleteness
Using the equivalence of and , we can now prove a result from which all of our incompleteness theorems will derive.
Theorem 5.1.
Let be a bao, , and and operators on (not necessarily in the signature) such that

is completely additive, and

for any , .
Then .
Proof.
Suppose that . Since is completely additive, by Theorem 4.3 we can apply condition to with to obtain:
(5) 
Pick such a and observe that if , then (using this very inequality to substitute for the first occurrence of on the righthand side) we get
a contradiction. Hence, we have that
Then by taking in (5) to be , we have
which contradicts condition 2 in the statement of the theorem. ∎
The fact that we do not insist on these operators to be primitive allows a lot of flexibility in instantiating this theorem, as we will witness below in Theorems 5.2, 5.4, 5.6, 7.3, 7.4, and 7.5. Note that working with this more general notion of operator would in fact allow us to replace with in the above statement without loss of generality: the present statement would follow after replacing with (as this transformation preserves complete additivity).
5.1 The van Benthem Logic
We are now ready to prove that the logic of § 2, the smallest normal modal logic containing , is incomplete. To state this result in a more general form, let us borrow notation from Cresswell (1984): let be the logic of the van Benthem frame from Definition 2.2 (see § 6.1 for an explanation of this name).
Theorem 5.2.
Any logic between and is incomplete.
Proof.
We have now come a long way from the initial Kripkeincompleteness results, i.e., incompleteness results, of the 1970s. It turns out that each of the properties (Litak, 2004), (Venema, 2003), and finally gives rise to incompleteness by itself.
Next, we will show that there are syntactically consistent bimodal logics that are inconsistent in the sense that they are not even sound with respect to any bao. This is not possible in the unimodal case, since by Makinson’s Theorem (Makinson, 1971), every normal unimodal logic is sound with respect to a Kripke frame—either the single reflexive point or the single irreflexive point—and hence with respect to a bao.
For the inconsistency result, consider a bimodal language with modalities and and let be the smallest normal logic in this language containing the axiom for and the axiom . In addition, let be the general frame for the same language that extends such that the accessibility relation for in is the universal relation on the frame. Finally, let be the logic of .
Theorem 5.3.

Any logic extending (in particular ) is inconsistent, yet is consistent.

Any logic extended by (in particular ) is consistent, and is complete.
Proof.
For the inconsistency of extensions of , by the proof of Theorem 5.2, every bao that validates the axiom for is such that , so . For the consistency of , observe that validates .
For part 2, observe that the bao underlying is atomic. ∎
The logic was introduced by van Benthem to prove a point about modal incompleteness, which we have pushed all the way to incompleteness. Theorem 5.2 raises the question: are there also “naturally occurring” examples of incomplete logics?
As soon as we have at least two modal operators at our disposal, the answer turns out to be an emphatic “yes”.
5.2 The Provability Logic
To motivate the main logic of this section, we recall that formulas of propositional modal logic can be translated into sentences of Peano Arithmetic () as follows: map each atomic to a sentence of arithmetic, send the modal to the arithmetized provability predicate of , and make the translation commute with the Boolean connectives in the obvious way. Solovay (1976) showed that the modal logic , the smallest normal modal logic containing the Löb axiom is arithmetically sound and complete: a modal formula is a theorem of iff for every mapping of atomic sentences to sentences of arithmetic, the induced arithmetic translation of is a theorem of . Thus, captures the logic of the provability predicate of .
Japaridze (1988) introduced a polymodal extension of , the bimodal version of which we will treat here. Let us interpret a bimodal language with operators and in the language of by sending to the provability predicate of as before and sending to a predicate encoding provability from with one application of the rule.^{16}^{16}16Equivalently, encodes provability from together with all arithmetical truths. A sentence of arithmetic is provable in with one application of the rule if for some formula , proves and proves for every numeral . The bimodal system that captures the combined logic of provability and provability in is the smallest normal bimodal logic containing the following axioms:

for ;

;

.
Japaridze (1988) proved that this logic, now known as (Boolos, 1993), is arithmetically sound and complete in the sense analogous to that of above.
While is Kripke complete (Segerberg, 1971), is Kripke incomplete (Japaridze 1988, see also Boolos 1993, p. 194). To see this, recall that over Kripke frames, the Löb axiom (i) for corresponds to the associated accessibility relation being transitive and Noetherian (conversely wellfounded); axiom (ii), which we can equivalently take in the diamond form , corresponds to the property that implies ; and axiom (iii) corresponds to the property that if and , then .
Such a combination of axioms, however, makes an empty relation. Suppose there are for which in some Kripke frame for . Hence by the property corresponding to axiom (ii). Then using , , and the property corresponding to axiom (iii), we obtain , which contradicts the Noetherianity of given by the Löb axiom for (note that the argument does not use the Löb axiom for ). But Japaridze’s arithmetical soundness theorem for shows that is not a theorem of (a semantic argument can be extracted from Theorem 5.5 below). Thus, is a nontheorem of that is valid in all Kripke frames for .
Viewed algebraically, Japaridze’s Kripkeincompleteness result shows that is incomplete. Beklemishev et al. (2010) show that is complete with respect to a class of topological spaces, which implies that it is complete. In light of this result, it is a natural question whether the incompleteness of is due to the interaction of with the other properties or whether it is due to the property by itself. Using the equivalence of and from § 4, we are able to answer this question and show that by itself is to blame.
Theorem 5.4.
The logic is incomplete.
Proof.
First, for any bao validating and any , we have
Note that this proof uses the complete additivity only of the operator, so it shows that is in fact incomplete in the sense of Remark 3.4. Similar remarks apply to later results involving variants of , though we will not mention this again.
We can get still more mileage out of this result by following the pattern of Theorem 5.3 to obtain a somewhat more “natural” logic answering the inconsistency question of Venema (2007, § 6.1). As in the case of , the problematic formula derivable over baos for is variable free. Therefore, let us define as the smallest normal modal logic in the language with three modalities and containing the axioms for and as above as well as the axiom .
Theorem 5.5.
Proof sketch.
Given Theorem 5.4, it is enough to prove the parenthetical claim. Recall that the topological semantics for is defined in terms of spaces with two suitably related scattered topologies; diamonds are interpreted by the derived set operator rather than Kuratowski’s closure operator.^{17}^{17}17For all the notions undefined in this proof, see Beklemishev and Gabelaia 2014. Ordinals provide particularly important instances of spaces, with interpreted by the order topology and by the club topology. In order to extend this semantics to , it is enough to interpret , e.g., with the interior operator of the trivial topology on any . ∎
One may hope to find not only a good topological interpretation but also a good arithmetical interpretation of validating . This would make a natural example of inconsistency among normal logics. As things stand now, we have at least an extraordinarily natural example of incompleteness in Theorem 5.4. And for inconsistency, we can do better than in the quasinormal realm of § 5.3.
5.3 Inconsistency of
If one is willing to broaden somewhat the setup of the present paper, a very natural example of inconsistency can be found among modal logics without the necessitation rule. In analogy with § 2, given a normal modal logic , we can define to mean that belongs to the closure of under modus ponens and uniform substitution. A set of formulas closed under is called a quasinormal logic over or simply a quasinormal logic. If , we simply use the name quasinormal logic. These notions transfer without any changes to the polymodal setting.
Algebraic semantics for these quasinormal logics has been well investigated. One can find a standard presentation in, e.g., Chagrov and Zakharyaschev 1997, Ch. 7; an early exhaustive discussion is provided by Blok and Köhler (1983), who indicate that the basic notion of a filtered modal algebra is a special case of the notion of a matrix that dates back to the prewar work of the Warsaw school.^{18}^{18}18“A wellknown result, going back to the twenties, states that, under some reasonable assumptions, any logic can be characterized as the set of formulas satisfied by a matrix , where is an algebra of the appropriate type, and a subset of the domain …” (Blok and Köhler, 1983, p. 941). In this opening quote, Blok and Köhler were presumably referring to Łukasiewicz and Tarski (1930). The exact references and more history can be found, e.g., in Font et al. 2003, § 1.2. As discussed by Jansana (2006), nonnormal modal logics were an important inspiration for Blok’s later work on abstract algebraic logic, and more generally, such logics have been a major source of examples and applications in the area (see Blok and Pigozzi 1989; Andréka et al. 2001; Czelakowski 2001; Font 2006; Font et al. 2003, 2009).
Instead of reproducing the whole apparatus here, let us just recall what is most relevant for our purposes. In this section, we are only interested in quasinormal logics where each box operator obeys the Löb axiom. baos in which each dual operator validates the Löb axiom are called diagonalizable baos (also known as Magari algebras).
Let be a bao. Recall that a nonempty subset is called a filter if

for any , iff .
is proper if . A maximal proper filter is called an ultrafliter.
Let be a quasinormal Löb logic, i.e., a quasinormal, polymodal logic such that for any in the signature, the unimodal restriction of to is an extension of . We say that a pair is a matrix for if

is a diagonalizable bao and is a filter on it, and

for any (i.e., any theorem of ) and any valuation on , .
It is a standard fact (see the references above) that every quasinormal Löb logic is sound and complete with respect to its class of matrices.
We can apply our terminology for baos to matrices as well: is a
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