We present an algebraic theory of structured objects based on and generalizing Aczel's theory of form systems. Notions of identity of structured objects and of transformations of systems of such objects are discussed. A generalization of Aczel's representation theorem is proven.

The contribution of this paper lies with providing a systematically specified and intuitive interpretation pattern and delineating a class of relational structures and models providing a natural interpretation of logical operators on an underlying propositional calculus of Positive Lattice Logic and subsequently proving a generic completeness theorem for the related class of logics, sometimes collectively referred to as Generalized Galois Logics.

This article addresses and resolves some issues of relational, Kripke-style, semantics for the logics of bounded lattice expansions with operators of well-defined distribution types, focusing on the case where the underlying lattice is not assumed to be distributive. It therefore falls within the scope of the theory of Generalized Galois Logics, introduced by Dunn, and it contributes to its extension. We introduce order-dual relational semantics and present a semantic analysis and completeness theorems for non-distributive lattice logic with n -ary additive (...) or multiplicative operators, with negation operators modally interpreted as impossibility and unnecessity, as well as with implication connectives. Order-dual relational semantics shares with the generalized Kripke frames, or the bi-approximation semantics approach, the use of both a satisfaction and a co-satisfaction relation, but it also responds to the recently voiced concerns of Craig, Haviar and Conradie about the relative non-intuitiveness of the 2-sorted semantics of the aforementioned approaches. In this article, we provide a standard interpretation of modalities and natural interpretations of both negation and implication, despite the absence of distribution. Thereby, our results contribute in creating the necessary background for research in non-distributive logics with modalities variously interpreted as dynamic, temporal etc, by analogy to the classical case. (shrink)

Part I of this paper is developed in the tradition of Stone-type dualities, where we present a new topological representation for general lattices (influenced by and abstracting over both Goldblatt's [17] and Urquhart's [46]), identifying them as the lattices of stable compact-opens of their dual Stone spaces (stability refering to a closure operator on subsets). The representation is functorial and is extended to a full duality.In part II, we consider lattice-ordered algebras (lattices with additional operators), extending the Jónsson and Tarski (...) representation results [30] for Boolean algebras with Operators. Our work can be seen as developing, and indeed completing, Dunn's project of gaggle theory [13, 14]. We consider general lattices (rather than Boolean algebras), with a broad class of operators, which we dubb normal, and which includes the Jónsson-Tarski additive operators. Representation of l-algebras is extended to full duality. (shrink)

This article presents an approach to the semantics of non-distributive propositional logics that is based on a lattice representation theorem that delivers a canonical extension of the lattice. Our approach supports both a plain Kripke-style semantics and, by restriction, a general frame semantics. Unlike the framework of generalized Kripke frames, the semantic approach presented in this article is suitable for modeling applied logics, as it respects the intended interpretation of the logical operators. This is made possible by restricting admissible interpretations.

One of the main problems of the orthoframe approach to quantum logic was that orthomodularity could not be captured by any first-order condition. This paper studies an elementary and natural class of orthomodular frames that can work around this limitation. Set-theoretically, the frames we propose form a natural subclass of the orthoframes, where is an irreflexive and symmetric relation on X. More specifically, they are partially-ordered orthoframes with a designated subset. Our frame class contains the canonical orthomodular frame of the (...) logic and therefore characterises orthomodular quantum logic. To prove soundness, a further restriction is placed on admissible valuations which requires that they be point-generated in addition to the requirement that they return stable sets. (shrink)

ABSTRACTCorrespondence and Shalqvist theories for Modal Logics rely on the simple observation that a relational structure is at the same time the basis for a model of modal logic and for a model of first-order logic with a binary predicate for the accessibility relation. If the underlying set of the frame is split into two components,, and, then frames are at the same time the basis for models of non-distributive lattice logic and of two-sorted, residuated modal logic. This suggests that (...) a reduction of the first to the latter may be possible, encoding Positive Lattice Logic as a fragment of Two-Sorted, Residuated Modal Logic. The reduction is analogous to the well-known Gödel-McKinsey-Tarski translation of Intuitionistic Logic into the S4 system of normal modal logic. In this article, we carry out this reduction in detail and we derive some properties of PLL from corresponding properties of First-Order Logic. The reduction we present is extendible to the case of lattices with operators, making use of recent results by this author on the relational representation of normal lattice expansions. (shrink)

This paper is an investigation into the logic of information flow. The basic perspective is that logic flows in virtue of constraints and that constraints classify channels connecting particulars In this paper we explore some logics intended to model reasoning in the case of idealized information flow, that is, where the constraints involved are exceptionless. We look at this as a step toward the far more challenging task of understanding the logic of imperfect information flow, that is where the constraints (...) admit of exceptional connections. This paper continues and amplifies work presented by the same authors in [10]. (shrink)

In an article dating back in 1992, Kosta Došen initiated a project of modal translations in substructural logics, aiming at generalising the well-known Gödel–McKinsey–Tarski translation of intuitionistic logic into S4. Došen's translation worked well for (variants of) BCI and stronger systems (BCW, BCK), but not for systems below BCI. Dropping structural rules results in logic systems without distribution. In this article, we show, via translation, that every substructural (indeed, every non-distributive) logic is a fragment of a corresponding sorted, residuated (multi) (...) modal logic. At the conceptual and philosophical level, the translation provides a classical interpretation of the meaning of the logical operators of various non-distributive propositional calculi. Technically, it allows for an effortless transfer of results, such as compactness and the Löwenheim-Skolem property and it opens up new directions for deducing properties of substructural logic systems by establishing appropriate transfer theorems. (shrink)

We present dualities for implicative and residuated lattices. In combination with our recent article on a discrete duality for lattices with unary modal operators, the present article contributes in filling in a gap in the development of Orłowska and Rewitzky’s research program of discrete dualities, which seemed to have stumbled on the case of non-distributive lattices with operators. We discuss dualities via truth, which are essential in relating the non-distributive logic of two-sorted frames with their sorted, residuated modal logic, as (...) well as full Stone duality for residuated lattices. Our results have immediate applications to the semantics of related substructural logical calculi. (shrink)

We present a Hilbert style axiomatization and an equational theory for reasoning about actions and capabilities. We introduce two novel features in the language of propositional dynamic logic, converse as backwards modality and abstract processes specified by preconditions and effects, written as \({\varphi \Rightarrow \psi}\) and first explored in our recent paper (Hartonas, Log J IGPL Oxf Univ Press, 2012), where a Gentzen-style sequent calculus was introduced. The system has two very natural interpretations, one based on the familiar relational (...) semantics and the other based on type semantics, where action terms are interpreted as types of actions (sets of binary relations). We show that the proof systems do not distinguish between the two kinds of semantics, by completeness arguments. Converse as backwards modality together with action types allow us to produce a new purely equational axiomatization of Dynamic Algebras, where iteration is axiomatized independently of box and where the fixpoint and Segerberg induction axioms are derivable. The system also includes capabilities operators and our results provide then a finitary Hilbert-style axiomatization and a decidable system for reasoning about agent capabilities, missing in the KARO framework. (shrink)

A systematic approach to the algebraic and Kripke semantics for logics with restricted structural rules, notably for logics on an underlying non-distributive lattice, is developed. We provide a new topological representation theorem for general lattices, using the filter space X. Our representation involves a galois connection on subsets of X, hence a closure operator $\Gamma$, and the image of the representation map is characterized as the collection of $\Gamma$-stable, compact-open subsets of the filter space . The original lattice ${\cal L}$ (...) is thus imbedded in the complete lattice of stable sets. The representation is shown to be functorial and is extended to a Stone-type duality. ;In the presence of additional operators, the extension problem is addressed in terms of what we call adjoint completions of normal operators. An operator f on a lattice L is normal if f: $L\sp{} \times \cdots \times L\sp{} \to L\sp{}$ is a monotone map, where each $L\sp{}$, $i \in n$ + 1, is either L or $L\sp{op}$ and f sends finite joins of $L\sp{}$ to joins in $L\sp{}$. Every familiar operator, such as $\circ$ , + , $\gets, \to, \neg, \square$ O is normal. By a uniform argument we show that normal operators $f\sb{i}$ on L extend to operators $F\sb{i}$ on $\Gamma X$ that distribute over arbitrary joins or meets and that \sb{i\in I})$ is a universal such extension. ;We also raise and resolve the question of representability of operators on stable sets by relations. We give an extensive number of examples of representation of specific operators and provide explicit constructions of canonical Kripke frames, based on our extension arguments, for a number of logical systems, notably for Intuitionistic and Classical Linear Logics, or Non-commutative Linear Logic and of fragments thereof. (shrink)

We introduce game-theoretic semantics for systems without the conveniences of either a De Morgan negation, or of distribution of conjunction over disjunction and conversely. Much of game playing rests on challenges issued by one player to the other to satisfy, or refute, a sentence, while forcing him/her to move to some other place in the game’s chessboard-like configuration. Correctness of the game-theoretic semantics is proven for both a training game, corresponding to Positive Lattice Logic and for more advanced games for (...) the logics of lattices with weak negation and modal operators. (shrink)

In this article we will focus our attention on the variety of distributive bisemilattices and some linguistic expansions thereof: bounded, De Morgan, and involutive bisemilattices. After extending Balbes’ representation theorem to bounded, De Morgan, and involutive bisemilattices, we make use of Hartonas–Dunn duality and introduce the categories of 2spaces and 2spaces\. The categories of 2spaces and 2spaces\ will play with respect to the categories of distributive bisemilattices and De Morgan bisemilattices, respectively, a role analogous to the category of Stone (...) spaces with respect to the category of Boolean algebras. Actually, the aim of this work is to show that these categories are, in fact, dually equivalent. (shrink)

A class of ordered relational topological spaces is described, which we call orthomodular spaces. Our construction of these spaces involves adding a topology to the class of orthomodular frames introduced by Hartonas, along the lines of Bimbó's topologization of the class of orthoframes employed by Goldblatt in his representation of ortholattices. We then prove that the category of orthomodular lattices and homomorphisms is dually equivalent to the category of orthomodular spaces and certain continuous frame morphisms, which we call continuous (...) weak p‐morphisms. It is well‐known that orthomodular lattices provide an algebraic semantics for the quantum logic. Hence, as an application of our duality, we develop a topological semantics for using orthomodular spaces and prove soundness and completeness. (shrink)

The dissertation deals with problems in "logic", more precisely, it deals with particular formal systems aiming at capturing patterns of valid reasoning. Sequent calculi were proposed to characterize logical connectives via introduction rules. These systems customarily also have structural rules which allow one to rearrange the set of premises and conclusions. In the "structurally free logic" of Dunn and Meyer the structural rules are replaced by combinatory rules which allow the same reshuffling of formulae, and additionally introduce an explicit marker (...) for the application of the rule---the combinator. The dissertation gives syntactic extensions of such systems introducing conjunction and disjunction . Cut elimination is proved for these systems. ;Syntactic systems as a rule are accompanied by a semantics which interprets them. The extended systems are supplied with semantics using algebraic methods. First, the Lindenbaum algebras are formed, and then these algebras are represented in an algebra of sets. This gives a so called Kripke-style semantics for the systems. The representation for the distributive extension utilizes well-known techniques; it uses sets of prime filters and set theoretical operations on them to deal with conjunction and disjunction, plus a ternary accessibility relation to accommodate the intensional operators as fusion and implication. The representation theorem is proven generally, i.e., for arbitrary sets of combinators using the "squeeze lemma" of Routley and Meyer. ;The non-distributive calculi require a different representation since union and intersection are distributive. Two previous representations for lattices are overviewed, and a modification of the Hartonas-Dunn representation is successfully augmented with the intensional operations. Instead of sets of filters, sets of pairs of minimally inconsistent filters and ideals are used and the representation theorem is proven. The representation results provide soundness and completeness for the systems. The dissertation also investigates properties of dual combinators , proving a series of lemmas about dual combinatory systems not being Church-Rosser and being inconsistent. Combinatorially definable classes of lambda-terms are considered as well. (shrink)

We investigate an information based generalization of the incompatibility-frame treatment of logics with non-classical negation connectives. Our framework can be viewed as an alternative to the neighbourhood semantics for extensions of lattice logic by various negation connectives, investigated by Hartonas. We set out the basic semantic framework, along with some correspondence results for extensions. We describe three kinds of constructions of canonical models and show that double negation law is not canonical with respect to any of these constructions. We (...) also compare our semantics to Hartonas’. (shrink)