Modal Incompleteness Revisited

In this paper, we are going to analyze the phenomenon of modal incompleteness from an algebraic point of view. The usual method of showing that a given logic L is incomplete is to show that for some L and some cannot be separated from by a suitably wide class of complete algebras — usually Kripke algebras. We are going to show that classical examples of incomplete logics, e.g., Fine logic, are not complete with respect to any class of complete BAOs. Even above Grz it is possible to find a continuum of such logics, which immediately implies the existence of a continuum of neighbourhood-incomplete Grz logics. Similar results can be proved for Löb logics. In addition, completely incomplete logics above Grz may be found uniformly as a result of failures of some admissible rule of a special kind.     

On Neat Reducts of Algebras of Logic

SC , CA , QA and QEA stand for the classes of Pinter's substitution algebras, Tarski's cylindric algebras, Halmos' quasipolyadic algebras, and quasipolyadic equality algebras of dimension , respectively. Generalizing a result of Németi on cylindric algebras, we show that for K {SC, CA, QA, QEA} and ordinals < , the class Nr K of -dimensional neat reducts of -dimensional K algebras, though closed under taking homomorphic images and products, is not closed under forming subalgebras (i.e. is not a variety) if and only if > 1.From this it easily follows that for 1 < < , the operation of forming -neat reducts of algebras in K does not commute with forming subalgebras, a notion to be made precise.We give a contrasting result concerning Halmos' polyadic algebras (with and without equality). For such algebras, we show that the class of infinite dimensional neat reducts forms a variety.We comment on the status of the property of neat reducts commuting with forming subalgebras for various reducts of polyadic algebras that are also expansions of cylindric-like algebras. We try to draw a borderline between reducts that have this property and reducts that do not.Following research initiated by Pigozzi, we also emphasize the strong tie that links the (apparently non-related) property of neat reducts commuting with forming subalgebras with proving amalgamation results in cylindric-like algebras of relations. We show that, like amalgamation, neat reducts commuting with forming subalgebras is another algebraic expression of definability and, accordingly, is also strongly related to the well-known metalogical properties of Craig, Beth and Robinson in the corresponding logics.     

Categorical Ontology of Levels and Emergent Complexity: An Introduction

An overview of the following three related papers in this issue presents the Emergence of Highly Complex Systems such as living organisms, man, society and the human mind from the viewpoint of the current Ontological Theory of Levels. The ontology of spacetime structures in the Universe is discussed beginning with the quantum level; then, the striking emergence of the higher levels of reality is examined from a categorical—relational and logical viewpoint. The ontological problems and methodology aspects discussed in the first two papers are followed by a rigorous paper based on Category Theory, Algebraic Topology and Logic that provides a conceptual and mathematical basis for a Categorical Ontology Theory of Levels. The essential links and relationships between the following three papers of this issue are pointed out, and further possible developments are being considered.     

An Interpolation Theorem for First Order Logic with Infinitary Predicates

An interpolation Theorem is proved for first order logic withinfinitary predicates. Our proof is algebraic via cylindricalgebras.¹     

Squares in Fork Arrow Logic

In this paper we show that the class of fork squares has a complete orthodox axiomatization in fork arrow logic (FAL). This result may be seen as an orthodox counterpart of Venema's non-orthodox axiomatization for the class of squares in arrow logic. FAL is the modal logic of fork algebras (FAs) just as arrow logic is the modal logic of relation algebras (RAs). FAs extend RAs by a binary fork operator and are axiomatized by adding three equations to RAs equational axiomatization. A proper FA is an algebra of relations where the fork is induced by an injective operation coding pair formation. In contrast to RAs, FAs are representable by proper ones and their equational theory has the expressive power of full first-order logic. A square semantics (the set of arrows is U×U for some set U) for arrow logic was defined by Y. Venema. Due to the negative results about the finite axiomatizability of representable RAs, Venema provided a non-orthodox finite axiomatization for arrow logic by adding a new rule governing the applications of a difference operator. We address here the question of extending the type of relational structures to define orthodox axiomatizations for the class of squares. Given the connections between this problem and the finitization problem addressed by I. Németi, we suspect that this cannot be done by using only logical operations. The modal version of the FA equations provides an orthodox axiomatization for FAL which is complete in view of the representability of FAs. Here we review this result and carry it further to prove that this orthodox axiomatization for FAL also axiomatizes the class of fork squares.     

Computing with Cylindric Modal Logics and Arrow Logics,Lower Bounds

The complexity of the satisfiability problems of various arrow logics and cylindric modal logics is determined. As is well known, relativising these logics makes them decidable. There are several parameters that can be set in such a relativisation. We focus on the following three: the number of variables involved, the similarity type and the kind of relativised models considered. The complexity analysis shows the importance and relevance of these parameters.     

On Amalgamation in Algebras of Logic

We show that not all epimorphisms are surjective in certain classes of infinite dimensional cylindric algebras, Pinter's substitution algebras and Halmos' quasipolyadic algebras with and without equality. It follows that these classes fail to have the strong amalgamation property. This answers a question in [3] and a question of Pigozzi in his landmark paper on amalgamation [9]. The cylindric case was first proved by Judit Madarasz [7]. The proof presented herein is substantially different. By a result of Németi, our result implies that the Beth-definability Theorem fails for certain expansions of first order logic     

Free Q-distributive lattices

The dual spaces of the free distributive lattices with a quantifier are constructed, generalizing Halmos' construction of the dual spaces of free monadic Boolean algebras.