We investigate a hitherto under-considered avenue of response for the logical pluralist to collapse worries. In particular, we note that standard forms of the collapse arguments seem to require significant order-theoretic assumptions, namely that the collection of admissible logics for the pluralist should be closed under meets and joins. We consider some reasons for rejecting this assumption, noting some prima facie plausible constraints on the class of admissible logics which would lead a pluralist admitting those logics to resist such closure (...) conditions. (shrink)

In this paper the propositional logic LTop is introduced, as an extension of classical propositional logic by adding a paraconsistent negation. This logic has a very natural interpretation in terms of topological models. The logic LTop is nothing more than an alternative presentation of modal logic S4, but in the language of a paraconsistent logic. Moreover, LTop is a logic of formal inconsistency in which the consistency and inconsistency operators have a nice topological interpretation. This constitutes a new proof of (...) S4 as being "the logic of topological spaces", but now under the perspective of paraconsistency. (shrink)

In the study of modal and nonclassical logics, translations have frequently been employed as a way of measuring the inferential capabilities of a logic. It is sometimes claimed that two logics are “notational variants” if they are translationally equivalent. However, we will show that this cannot be quite right, since first-order logic and propositional logic are translationally equivalent. Others have claimed that for two logics to be notational variants, they must at least be compositionally intertranslatable. The definition of compositionality these (...) accounts use, however, is too strong, as the standard translation from modal logic to first-order logic is not compositional in this sense. In light of this, we will explore a weaker version of this notion that we will call schematicity and show that there is no schematic translation either from first-order logic to propositional logic or from intuitionistic logic to classical logic. (shrink)

The book presents a thoroughly elaborated logical theory of generalized truth-values understood as subsets of some established set of truth values. After elucidating the importance of the very notion of a truth value in logic and philosophy, we examine some possible ways of generalizing this notion. The useful four-valued logic of first-degree entailment by Nuel Belnap and the notion of a bilattice constitute the basis for further generalizations. By doing so we elaborate the idea of a multilattice, and most notably, (...) a trilattice of truth values – a specific algebraic structure with information ordering and two distinct logical orderings, one for truth and another for falsity. Each logical order not only induces its own logical vocabulary, but determines also its own entailment relation. We consider both semantic and syntactic ways of formalizing these relations and construct various logical calculi. (shrink)

The book provides a historical and systematic exposition of the semantic theory of truth formulated by Alfred Tarski in the 1930s. This theory became famous very soon and inspired logicians and philosophers. It has two different, but interconnected aspects: formal-logical and philosophical. The book deals with both, but it is intended mostly as a philosophical monograph. It explains Tarski’s motivation and presents discussions about his ideas as well as points out various applications of the semantic theory of truth to philosophical (...) problems. (shrink)

This book celebrates the work of Don Pigozzi on the occasion of his 80th birthday. In addition to articles written by leading specialists and his disciples, it presents Pigozzi’s scientific output and discusses his impact on the development of science. The book both catalogues his works and offers an extensive profile of Pigozzi as a person, sketching the most important events, not only related to his scientific activity, but also from his personal life. It reflects Pigozzi's contribution to the rise (...) and development of areas such as abstract algebraic logic, universal algebra and computer science, and introduces new scientific results. Some of the papers also present chronologically ordered facts relating to the development of the disciplines he contributed to, especially abstract algebraic logic. The book offers valuable source material for historians of science, especially those interested in history of mathematics and logic. (shrink)

Angell's logic of analytic containment AC has been shown to be characterized by a 9-valued matrix NC by Ferguson, and by a 16-valued matrix by Fine. We show that the former is the image of a surjective homomorphism from the latter, i.e., an epimorphic image. The epimorphism was found with the help of MUltlog, which also provides a tableau calculus for NC extended by quantifiers that generalize conjunction and disjunction.

I investigate syntactic notions of theoretical equivalence between logical theories and a recent objection thereto. I show that this recent criticism of syntactic accounts, as extensionally inadequate, is unwarranted by developing an account which is plausibly extensionally adequate and more philosophically motivated. This is important for recent anti-exceptionalist treatments of logic since syntactic accounts require less theoretical baggage than semantic accounts.

We consider the notion of structural completeness with respect to arbitrary (finitary and/or infinitary) inferential rules. Our main task is to characterize structurally complete intermediate logics. We prove that the structurally complete extension of any pure implicational in termediate logic C can be given as an extension of C with a certain family of schematically denned infinitary rules; the same rules are used for each C. The cardinality of the family is continuum and, in the case of (the pure implicational (...) fragment of) intuitionistic logic, the family cannot be reduced to a countable one. It means that the structurally complete extension of the intuitionistic logic is not countably axiomatizable by schematic rules. (shrink)

The title of this paper is a theorem, which I am going to state and prove. The theorem extends from prepositional to predicate languages the result I presented in [5].

The concepts of question evocation and erotetic implication play central role in Inferential Erotetic Logic. In this paper, deduction theorems for question evocation and erotetic implication are proven. Moreover, it is shown how question evocation by a finite non-empty set of declaratives can be reduced to question evocation by the empty set, and how erotetic implication based on a finite non-empty set of declaratives can be reduced to a relation between questions only.

I examine notions of equivalence between logics (understood as languages interpreted model-theoretically) and develop two new ones that invoke not only the algebraic but also the string-theoretic structure of the underlying language. As an application, I show how to construe modal operator languages as what might be called typographical notational variants of _bona fide_ first-order languages.

According to Suszko’s Thesis, there are but two logical values, true and false. In this paper, R. Suszko’s, G. Malinowski’s, and M. Tsuji’s analyses of logical twovaluedness are critically discussed. Another analysis is presented, which favors a notion of a logical system as encompassing possibly more than one consequence relation. [A] fundamental problem concerning many-valuedness is to know what it really is. [13, p. 281].

A new combined temporal logic called synchronized linear-time temporal logic (SLTL) is introduced as a Gentzen-type sequent calculus. SLTL can represent the n -Cartesian product of the set of natural numbers. The cut-elimination and completeness theorems for SLTL are proved. Moreover, a display sequent calculus δ SLTL is defined.

In this paper we develop an abstract theory of adequacy. In the same way as the theory of consequence operations is a general theory of logic, this theory of adequacy is a general theory of the interactions and connections between consequence operations and its sound and complete semantics. Addition of axioms for the connectives of propositional logic to the basic axioms of consequence operations yields a unifying framework for different systems of classical propositional logic. We present an abstract model-theoretical semantics (...) based on model mappings and theory mappings. Between the classes of models and theories, i.e., the set of sentences verified by a model, it obtains a connection that is well-known within algebra as Galois correspondence. Many basic semantical properties can be derived from this observation. A sentence A is a semantical consequence of T if every model of T is also a model of A. A model mapping is adequate for a consequence operation if its semantical inference operation is identical with the consequence operation. We study how properties of an adequate model mapping reflect the properties of the consequence operation and vice versa. In particular, we show how every concept of the theory of consequence operations can be formulated semantically. (shrink)

The paper deals with reconstruction of the unique reductivecounterpart of the deductive logic. The procedure results in the deductivereductive form of logic. This extension is illustrated on the base of intuitionistic logics: Heyting’s, Brouwerian and Heyting-Brouwer’s ones.

The properties of monotonic inference systems and the properties of their theories are strongly linked. These links, however, are much weaker in nonmonotonic inference systems. In this paper we introduce the notion of anaxiomatic variety for a theory and show how this notion, instead of the notion of a theory, can be used for the syntactic and semantic analysis of nonmonotonic inferences.

This paper discusses the resolution principle in the context of non-classical logics.

This paper develops a new framework for combining propositional logics, called "juxtaposition". Several general metalogical theorems are proved concerning the combination of logics by juxtaposition. In particular, it is shown that under reasonable conditions, juxtaposition preserves strong soundness. Under reasonable conditions, the juxtaposition of two consequence relations is a conservative extension of each of them. A general strong completeness result is proved. The paper then examines the philosophically important case of the combination of classical and intuitionist logics. Particular attention is (...) paid to the phenomenon of collapse. It is shown that there are logics with two stocks of classical or intuitionist connectives that do not collapse. Finally, the paper briefy investigates the question of which rules, when added to these logics, lead to collapse. (shrink)

This paper is an historical study of Tarski's methodology of deductive sciences (in which a logic S is identified with an operator Cn S, called the consequence operator, on a given set of expressions), from its appearance in 1930 to the end of the 1970s, focusing on the work done in the field by Roberto Magari, Piero Mangani and by some of their pupils between 1965 and 1974, and comparing it with the results achieved by Tarski and the Polish school (...) (Łoś, Suszko, Słupecki, Pogorzelski, Wójcicki). In the last section of the paper we will then compare these works with some recent developments in algebraic logic: this will lead to a better understanding of the results of the methodology of deductive science, but at the same time will show some intrinsic limits to such an approach to logic. Even if Magari's work on diagonizable algebras and universal algebra and Mangani's axiomatization of MV-algebras and results in model theory are rather famous, the articles on closure operators, published in the 1960s, are almost totally unknown outside Italy (mainly because of a linguistic limitation, the papers we analyse having been written and published in Italian). This paper aims to fill the gap in the literature and to enable the international community to get acquainted with this part of Italian logic. The same applies to some works published in Barcelona (in Catalan) at the end of the 1970s, analysed in the last section. (shrink)

Peirce’s triadic logic has been under discussion since its discovery in the 1960s by Fisch and Turquette. The experiments with matrices of three-valued logic are recorded in a few pages of unpublished manuscripts dated 1909, a decade before similar systems have been developed by logicians. The purposes of Peirce’s work on such logic, as well as semantical aspects of his system, are disputable. In the most extensive work about it, Turquette suggested that the matrices are related in dual pairs of (...) axiomatic Hilbert-style systems. In this paper, we present a simple tableau proof for a fragment of Peirce three-valued logic, called P3, based on similar approaches in many-valued literature. We demonstrated that this proof is sound and complete. Besides that, taking the false as the only undesignated value and adding non-classical negations to the calculus, we can explore paraconsistent and paracompleteness theories into P3. Keywords: Charles S. Peirce, many-valued logics, theory of proof, tableau method. (shrink)

We consider the logics determined by the set of all natural implicative expansions of Kleene’s strong 3-valued matrix and select the class of all logics functionally equivalent to Łukasiewicz’s 3-valued logic Ł3. The concept of a “natural implicative matrix” is based upon the notion of a “natural conditional” defined in Tomova.

The present paper is a sequel to Robles et al. :349–374, 2020. https://doi.org/10.1007/s10849-019-09306-2). A class of implicative expansions of Kleene’s 3-valued logic functionally including Łukasiewicz’s logic Ł3 is defined. Several properties of this class and/or some of its subclasses are investigated. Properties contemplated include functional completeness for the 3-element set of truth-values, presence of natural conditionals, variable-sharing property and vsp-related properties.

A recent paper by Jakl, Jung and Pultr succeeded for the first time in establishing a very natural link between bilattice logic and the duality theory of d-frames and bitopological spaces. In this paper we further exploit, extend and investigate this link from an algebraic and a logical point of view. In particular, we introduce classes of algebras that extend bilattices, d-frames and N4-lattices to a setting in which the negation is not necessarily involutive, and we study corresponding logics. We (...) provide product representation theorems for these algebras, as well as completeness, algebraizability results for the corresponding logics. (shrink)

We look at extensions (i.e., stronger logics in the same language) of the Belnap–Dunn four-valued logic. We prove the existence of a countable chain of logics that extend the Belnap–Dunn and do not coincide with any of the known extensions (Kleene’s logics, Priest’s logic of paradox). We characterise the reduced algebraic models of these new logics and prove a completeness result for the first and last element of the chain stating that both logics are determined by a single finite logical (...) matrix. We show that the last logic of the chain is not finitely axiomatisable. (shrink)

Let V be a s.f.b. (strongly finitely based, see below) variety of algebras. The central result is Theorem 2 saying that the logic defined by all matrices (A, d) with d A V is finitely based iff the A V have 1st order definable cosets for their congruences. Theorem 3 states a similar axiomatization criterion for the logic determined by all matrices (A, A), A V, a term which is constant in V. Applications are given in a series of examples.

In this work we define, in a general way, an algebraic semantics for the logics of the hierarchy InPk. This semantics is defined by means of an alternative construction, with respect to the usual algebraic semantics, and it is known in the literature as Twist-structures semantics. Besides that, we modify such construction, defining the so-called ω-Twist-structures here. This adaptation allows us to prove adequacy theorems for every logic of the hierarchy InPk.

Logics that do not have a deduction-detachment theorem (briefly, a DDT) may still possess a contextual DDT —a syntactic notion introduced here for arbitrary deductive systems, along with a local variant. Substructural logics without sentential constants are natural witnesses to these phenomena. In the presence of a contextual DDT, we can still upgrade many weak completeness results to strong ones, e.g., the finite model property implies the strong finite model property. It turns out that a finitary system has a contextual (...) DDT iff it is protoalgebraic and gives rise to a dually Brouwerian semilattice of compact deductive filters in every finitely generated algebra of the corresponding type. Any such system is filter distributive, although it may lack the filter extension property. More generally, filter distributivity and modularity are characterized for all finitary systems with a local contextual DDT, and several examples are discussed. For algebraizable logics, the well-known correspondence between the DDT and the equational definability of principal congruences is adapted to the contextual case. (shrink)

The article concerns two axiom systems of Słupecki for the functionally complete three-valued propositional logic: W1–W6 and A1–A9. The article proves that both of them are inadequate—W1–W6 is semantically incomplete, on the other hand, A1–A9 governs a functionally incomplete calculus, and thus, it cannot be a semantically complete axiom system for the functionally complete three-valued logic.

The aim of this paper is to study the paraconsistent deductive systemP 1 within the context of Algebraic Logic. It is well known due to Lewin, Mikenberg and Schwarse thatP 1 is algebraizable in the sense of Blok and Pigozzi, the quasivariety generated by Sette's three-element algebraS being the unique quasivariety semantics forP 1. In the present paper we prove that the mentioned quasivariety is not a variety by showing that the variety generated byS is not equivalent to any algebraizable (...) deductive system. We also show thatP 1 has no algebraic semantics in the sense of Czelakowski. Among other results, we study the variety generated by the algebraS. This enables us to prove in a purely algebraic way that the only proper non-trivial axiomatic extension ofP 1 is the classical deductive systemPC. Throughout the paper we also study those abstract logics which are in a way similar toP 1, and are called hereabstract Sette logics. We obtain for them results similar to those obtained for distributive abstract logics by Font, Verdú and the author. (shrink)

The aim of this paper is technically to study Belnap's four-valued sentential logic . First, we obtain a Gentzen-style axiomatization of this logic that contains no structural rules while all they are still admissible in the Gentzen system what is proved with using some algebraic tools. Further, the mentioned logic is proved to be the least closure operator on the set of {Λ, V, ⌝}-formulas satisfying Tarski's conditions for classical conjunction and disjunction together with De Morgan's laws for negation. It (...) is also proved that Belnap's logic is the only sentential logic satisfying the above-mentioned conditions together with Anderson-Belnap's Variable-Sharing Property. Finally, we obtain a finite Hilbert-style axiomatization of this logic. As a consequence, we obtain a finite Hilbert-style axiomatization of Priest's logic of paradox. (shrink)

The present paper concerns a technical study of PRIEST'S logic of paradox [Pri 79], We prove that this logic has no proper paraconsistent strengthening. It is also proved that the mentioned logic is the largest paraconsistent one satisfaying TARSKI'S conditions for the classical conjunction and disjunction together with DE MORGAN'S laws for negation. Finally, we obtain for the logic of paradox an algebraic completeness result related to Kleene lattices.

In this paper, first, we determine the number of sublogics of variable inclusion of an arbitrary finitary logic ⊢ with a composition term. Then, we investigate their position into the lattice of co...

A 4-valued framework is proposed to distinguish intuitionistically valid formulas within classical validities.

This paper discusses the general problem of translation functions between logics, given in axiomatic form, and in particular, the problem of determining when two such logics are "synonymous" or "translationally equivalent." We discuss a proposed formal definition of translational equivalence, show why it is reasonable, and also discuss its relation to earlier definitions in the literature. We also give a simple criterion for showing that two modal logics are not translationally equivalent, and apply this to well-known examples. Some philosophical morals (...) are drawn concerning the possibility of having two logical systems that are "empirically distinct" but are both translationally equivalent to a common logic. (shrink)

http://dx.doi.org/10.5007/1808-1711.2009v13n3p357 Neste trabalho concorda-se com a tese de I. Hacking segundo a qual o significado das constantes lógicas é dado pelas Regras de Introdução e Eliminação do cálculo de sequentes de Gentzen que caracterizam a concepção da noção de consequência lógica abstrata. Perguntamos quais são as regras mínimas que um conectivo deve satisfazer para que seja considerado uma negação genuína. Tomaremos como referência para tratar dessa questão os C-sistemas de Newton da Costa e o sistema LP de Graham Priest. Finalmente, (...) analisaremos esses sistemas na lógica de sequentes a fim de mostrar que a negação paraconsistente ou bem carece das regras puras de eliminação e negação da negação ou ela envolve outros conectivos, o que torna difícil atribuir um significado unívoco à negação paraconsistente. (shrink)

We characterize, in syntactic terms, the ranges of epimorphisms in an arbitrary class of similar first-order structures. This allows us to strengthen a result of Bacsich, as follows: in any prevariety having at most \ non-logical symbols and an axiomatization requiring at most \ variables, if the epimorphisms into structures with at most \ elements are surjective, then so are all of the epimorphisms. Using these facts, we formulate and prove manageable ‘bridge theorems’, matching the surjectivity of all epimorphisms in (...) the algebraic counterpart of a logic \ with suitable infinitary definability properties of \, while not making the standard but awkward assumption that \ comes furnished with a proper class of variables. (shrink)

The logic TK was introduced as a propositional logic extending the classical propositional calculus with a new unary operator which interprets some conceptions of Tarski’s consequence operator. TK-algebras were introduced as models to TK . Thus, by using algebraic tools, the adequacy (soundness and completeness) of TK relatively to the TK-algebras was proved. This work presents a neighbourhood semantics for TK , which turns out to be deductively equivalent to the non-normal modal logic EMT4 . DOI:10.5007/1808-1711.2011v15n2p287.

What is the fundamental insight behind truth-functionality ? When is a logic interpretable by way of a truth-functional semantics? To address such questions in a satisfactory way, a formal definition of truth-functionality from the point of view of abstract logics is clearly called for. As a matter of fact, such a definition has been available at least since the 70s, though to this day it still remains not very widely well-known. A clear distinction can be drawn between logics characterizable through: (...) (1) genuinely finite-valued truth-tabular semantics; (2) no finite-valued but only an infinite-valued truthtabular semantics; (3) no truth-tabular semantics at all. Any of those logics, however, can in principle be characterized through non-truth-functional valuation semantics, at least as soon as their associated consequence relations respect the usual tarskian postulates. So, paradoxical as that might seem at first, it turns out that truth-functional logics may be adequately characterized by non-truth-functional semantics . Now, what feature of a given logic would guarantee it to dwell in class (1) or in class (2), irrespective of its circumstantial semantic characterization? (shrink)

We show that for an arbitrary logic being locally tabular is a strictly weaker property than being locally finite. We describe our hunt for a logic that allows us to separate the two properties, revealing weaker and weaker conditions under which they must coincide, and showing how they are intertwined. We single out several classes of logics where the two notions coincide, including logics that are determined by a finite set of finite matrices, selfextensional logics, algebraizable and equivalential logics. Furthermore, (...) we identify a closure property on models of a logic that, in the presence of local tabularity, is equivalent to local finiteness. (shrink)

In Computer Science stepwise refinement of algebraic specifications is a well-known formal methodology for rigorous program development. This paper illustrates how techniques from Algebraic Logic, in particular that of interpretation, understood as a multifunction that preserves and reflects logical consequence, capture a number of relevant transformations in the context of software design, reuse, and adaptation, difficult to deal with in classical approaches. Examples include data encapsulation and the decomposition of operations into atomic transactions. But if interpretations open such a new (...) research avenue in program refinement, (conceptual) tools are needed to reason about them. In this line, the paper’s main contribution is a study of the correspondence between logical interpretations and morphisms of a particular kind of coalgebras. This opens way to the use of coalgebraic constructions, such as simulation and bisimulation, in the study of interpretations between (abstract) logics. (shrink)

Referential semantics importantly subscribes to the programme of theory of logical calculi. Defined by Wójcicki in [8], it has been subsequently studied in a series of papers of the author, till the full exposition of the framework in [9] and its intuitive characterisation in [10]. The aim of the article is to present several generalizations of referential semantics as compared and related to the matrix semantics for propositional logics. We show, in a uniform way, some own generalizations of referentiality: the (...) first, directed to unrestricted cluster referential semantics, [4], its "discrete" version, a counterpart of algebraic semantics and a many-valued referentiality based on matrices, whose elements are functions from the set of indices to a finite n-element set of values, n≥2, [3]. Next to this we outline pragmatic matrices introduced by Tokarz in [6] as an alternative for cluster referential approach and discuss together all presented versions of referential semantics. (shrink)

For decades Ryszard Wójcicki has been a highly influential scholar in the community of logicians and philosophers. Our aim is to outline and comment on some essential issues on logic, methodology of science and semantics as seen from the perspective of distinguished contributions of Wójcicki to these areas of philosophical investigations.

The actual introduction of a non-reflexive and non-idempotent q -consequence gave birth to the concept of logical three-valuedness based on the idea of noncomplementary categories of rejection and acceptance. A q -consequence may not have bivalent description, the property claimed by Suszko’s Thesis on logical two-valuedness, ( ST ), of structural logics, i.e. structural consequence operations. Recall that ( ST ) shifts logical values over the set of matrix values and it refers to the division of matrix universe into two (...) subsets of designated and undesignated elements using their characteristic functions as logical valuations, cf. [4] The extension of the idea operates with three-valued function, with the third value ascribed to those elements of the matrix which are neither rejected nor accepted. Accordingly, the logical three-valuedness departs naturally from the division of the matrix universe into three subsets and the ( ST ) counterpart says that any inference based on a structural q -consequence may have a bivalent or a three-valued description. After a short presentation of the three-valued inferential framework, we discuss a solution for further exploration of the idea leading to logical n -valuedness for n > 3. Apparently, the first step in that direction is easy and it consists of a division of the matrix universe into more than three subsets. The next move, i.e. a definition of a matrix consequence-like relation being neither a consequence nor a q -consequence, seems extremely difficult. Therefore, here we consider only finite linear matrices with one-argument functions “labelling” respective matrix subsets. By means of these functions it is possible to represent a q-consequence as a “partial” Tarski’s consequence and, ultimately, to define a logically more-valued consequence-like relation. We believe, that the present partial proposal deserves an attention by itself but also that it may lead to a general approach to logically many-valued inference. (shrink)

In a previous paper [ 21 ] all extensions of Johansson’s minimal logic J with the weak interpolation property WIP were described. It was proved that WIP is decidable over J. It turned out that the weak interpolation problem in extensions of J is reducible to the same problem over a logic Gl, which arises from J by adding tertium non datur. In this paper we consider extensions of the logic Gl. We prove that only finitely many logics over Gl (...) have the Craig interpolation property CIP, the restricted interpolation property IPR or the projective Beth property PBP. The full list of Gl-logics with the mentioned properties is found, and their description is given. We note that IPR and PBP are equivalent over Gl. It is proved that CIP, IPR and PBP are decidable over the logic Gl. (shrink)

ABSTRACT The two main directions pursued in the present paper are the following. The first direction was started by Pigozzi in 1969. In [Mak 91] and [Mak 79] Maksimova proved that a normal modal logic has the Craig interpolation property iff the corresponding class of algebras has the superamalgamation property. In this paper we extend Maksimova's theorem to normal multi-modal logics with arbitrarily many, not necessarily unary modalities, and to not necessarily normal multi-modal logics with modalities of ranks smaller than (...) 2. To extend the characterization beyond multi-modal logics, we look at arbitrary algebraizable logics. We will introduce an algebraic property equivalent with the Craig interpolation property in algebraizable logics, and prove that the superamalgamation property implies the Craig interpolation property. The problem of extending the characterization result to non-normal non-unary modal logics also will be discussed. In the second direction pursued herein: for non-normal modal logic with one unary modality Lemmon [Lem 66] gave a possible worlds semantics. Here we give a more general possible worlds semantics for not necessarily normal multi-modal logics with arbitrarily many not necessarily unary modalities. Strongly related to the above is the theorem, proved, e.g., in Jóns son-Tarski [JT 52] and Henkin-Monk-Tarski [HMT 71], that every normal Boolean algebra with operators can be represented as a subalgebra of the complex algebra of some relational structure. We extend this result to not necessarily normal BAO's as follows. We define partial relational structures and show that every not necessarily normal BAO is embeddable into the complex algebra of a partial relational structure. This gives a possible worlds semantics for not necessarily normal multi-modal logics. (shrink)