Almost structural completeness; an algebraic approach

A deductive system is structurally complete if its admissible inference rules are derivable. For several important systems, like modal logic S5, failure of structural completeness is caused only by the underivability of passive rules, i.e. rules that can not be applied to theorems of the system. Neglecting passive rules leads to the notion of almost structural completeness, that means, derivablity of admissible non-passive rules. Almost structural completeness for quasivarieties and varieties of general algebras is investigated here by purely algebraic means. The results apply to all algebraizable deductive systems. Firstly, various characterizations of almost structurally complete quasivarieties are presented. Two of them are general: expressed with finitely presented algebras, and with subdirectly irreducible algebras. One is restricted to quasivarieties with finite model property and equationally definable principal relative congruences, where the condition is verifiable on finite subdirectly irreducible algebras. Secondly, examples of almost structurally complete varieties are provided Particular emphasis is put on varieties of closure algebras, that are known to constitute adequate semantics for normal extensions of S4 modal logic. A certain infinite family of such almost structurally complete, but not structurally complete, varieties is constructed. Every variety from this family has a finitely presented unifiable algebra which does not embed into any free algebra for this variety. Hence unification in it is not unitary. This shows that almost structural completeness is strictly weaker than projective unification for varieties of closure algebras

Complete Additivity and Modal Incompleteness

In this paper, we tell a story about incompleteness in modal logic. The story weaves together a paper of van Benthem, `Syntactic aspects of modal incompleteness theorems,' and a longstanding open question: whether every normal modal logic can be characterized by a class of completely additive modal algebras, or as we call them, V-BAOs. Using a first-order reformulation of the property of complete additivity, we prove that the modal logic that starred in van Benthem's paper resolves the open question in the negative. In addition, for the case of bimodal logic, we show that there is a naturally occurring logic that is incomplete with respect to V-BAOs, namely the provability logic GLB. We also show that even logics that are unsound with respect to such algebras do not have to be more complex than the classical propositional calculus. On the other hand, we observe that it is undecidable whether a syntactically defined logic is V-complete. After these results, we generalize the Blok Dichotomy to degrees of V-incompleteness. In the end, we return to van Benthem's theme of syntactic aspects of modal incompleteness

Algebraic proof theory for LE-logics

In this paper we extend the research programme in algebraic proof theory from axiomatic extensions of the full Lambek calculus to logics algebraically captured by certain varieties of normal lattice expansions (normal LE-logics). Specifically, we generalise the residuated frames in [16] to arbitrary signatures of normal lattice expansions (LE). Such a generalization provides a valuable tool for proving important properties of LE-logics in full uniformity. We prove semantic cut elimination for the display calculi D.LE associated with the basic normal LE-logics and their axiomatic extensions with analytic inductive axioms. We also prove the finite model property (FMP) for each such calculus D.LE, as well as for its extensions with analytic structural rules satisfying certain additional properties

Logics of variable inclusion and the lattice of consequence relations

In this paper, firstly, we determine the number of sublogics of variable inclusion of an arbitrary finitary logic L with partition function. Then, we investigate their position into the lattice of consequence relations over the language of L.Comment: arXiv admin note: text overlap with arXiv:1804.08897, arXiv:1809.0676

Expanding FLew with a Boolean connective

We expand FLew with a unary connective whose algebraic counterpart is the operation that gives the greatest complemented element below a given argument. We prove that the expanded logic is conservative and has the Finite Model Property. We also prove that the corresponding expansion of the class of residuated lattices is an equational class.Comment: 15 pages, 4 figures in Soft Computing, published online 23 July 201

Polyatomic Logics and Generalised Blok-Esakia Theory

This paper presents a novel concept of a Polyatomic Logic and initiates its systematic study. This approach, inspired by Inquisitive semantics, is obtained by taking a variant of a given logic, obtained by looking at the fragment covered by a selector term. We introduce an algebraic semantics for these logics and prove algebraic completeness. These logics are then related to translations, through the introduction of a number of classes of translations involving selector terms, which are noted to be ubiquitous in algebraic logic. In this setting, we also introduce a generalised Blok-Esakia theory which can be developed for special classes of translations. We conclude by showing some systematic connections between the theory of Polyatomic Logics and the general Blok-Esakia theory for a wide class of interesting translations.Comment: 48 pages, 2 figure

Degrees of the finite model property: the antidichotomy theorem

A classic result in modal logic, known as the Blok Dichotomy Theorem, states that the degree of incompleteness of a normal extension of the basic modal logic $\sf K$ is $1$ or $2^{\aleph_0}$. It is a long-standing open problem whether Blok Dichotomy holds for normal extensions of other prominent modal logics (such as $\sf S4$ or $\sf K4$) or for extensions of the intuitionistic propositional calculus $\mathsf{IPC}$. In this paper, we introduce the notion of the degree of finite model property (fmp), which is a natural variation of the degree of incompleteness. It is a consequence of Blok Dichotomy Theorem that the degree of fmp of a normal extension of $\sf K$ remains $1$ or $2^{\aleph_0}$. In contrast, our main result establishes the following Antidichotomy Theorem for the degree of fmp for extensions of $\mathsf{IPC}$: each nonzero cardinal $\kappa$ such that $\kappa \leq \aleph_0$ or $\kappa = 2^{\aleph_0}$ is realized as the degree of fmp of some extension of $\mathsf{IPC}$. We then use the Blok-Esakia theorem to establish the same Antidichotomy Theorem for normal extensions of $\sf S4$ and $\sf K4$