Categorical Logic Research Articles

Syntactic categories for Nori motives

We give a new construction, based on categorical logic, of Nori’s $$\mathbb Q$$ -linear abelian category of mixed motives associated to a cohomology or homology functor with values in finite-dimensional vector spaces over $$\mathbb Q$$ . This new construction makes sense for infinite-dimensional vector spaces as well, so that it associates a $$\mathbb Q$$ -linear abelian category of mixed motives to any (co)homology functor, not only Betti homology (as Nori had done) but also, for instance, $$\ell $$ -adic, p-adic or motivic cohomology. We prove that the $$\mathbb Q$$ -linear abelian categories of mixed motives associated to different (co)homology functors are equivalent if and only a family (of logical nature) of explicit properties is shared by these different functors. The problem of the existence of a universal cohomology theory and of the equivalence of the information encoded by the different classical cohomology functors thus reduces to that of checking these explicit conditions.

A Stone-type Duality Theorem for Separation Logic Via its Underlying Bunched Logics

Stone-type duality theorems, which relate algebraic and relational/topological models, are important tools in logic because — in addition to elegant abstraction — they strengthen soundness and completeness to a categorical equivalence, yielding a framework through which both algebraic and topological methods can be brought to bear on a logic. We give a systematic treatment of Stone-type duality theorems for the structures that interpret bunched logics, starting with the weakest systems, recovering the familiar Boolean BI, and concluding with Separation Logic. Our results encompass all the known existing algebraic approaches to Separation Logic and prove them sound with respect to the standard store-heap semantics. We additionally recover soundness and completeness theorems of the specific truth-functional models of these logics as presented in the literature. This approach synthesises a variety of techniques from modal, substructural and categorical logic and contextualises the ‘resource semantics’ interpretation underpinning Separation Logic amongst them. As a consequence, theory from those fields — as well as algebraic and topological methods — can be applied to both Separation Logic and the systems of bunched logics it is built upon. Conversely, the notion of indexed resource frame (generalizing the standard model of Separation Logic) and its associated completeness proof can easily be adapted to other non-classical predicate logics.

On Models of Higher-Order Separation Logic

We show how tools from categorical logic can be used to give a general account of models of higher-order separation logic with a sublogic of so-called persistent predicates satisfying the usual rules of higher-order logic. The models of separation logic are based on a notion of resource, a partial commutative monoid, and the persistent predicates can be defined using a modality. We classify well-behaved sublogics of persistent predicates in terms of interior operators on the partial commutative monoid of resources. We further show how the general constructions can be used to recover the model of Iris, a state-of-the-art higher-order separation logic with guarded recursive predicates.

THE UNIVERSAL LAW OF INVARIANT TRANSFORMATIONS IN SPACE OF CONTRARY AND CONTRADICTORY LOGICAL FORMS

The article describes the features of the human mind in constructing logical categories and his ability to construct reality, which constitutes itself in the form of a network of interrelated logical categories belonging to contrary or contradictory type, based on the concept of constructivism. Consideration of the relationship between grammatical and logical forms leads to the understanding that the latter can manifest itself as combinatorial - topological structures that are well modeled by the trigrams and hexagrams of the ancient Chinese symbolism of the Book of Changes (I Ching). The use of such symbolism allows to identify linkages between structures, hidden in the analytical record through the expressive means of hexagrams. Subsequent application of operations of contrary and contradicting type allowed to identify four types of invariant transformations that are combinations of contrary and contradictory relations in a model space with a basis of two opposite elements, indicating new, previously hidden ontology of reality. In this ontology, invariants are not represented by objects but with transitions between different states, connecting contrary characteristics with contradictory ones. Examples from various fields of knowledge illustrate the fundamental role of a quad-state in a variety of real world processes.

The Actualization of the Concept «Free Finding Law» in the Works of Eugen Ehrlich

In this article the author questions the role of Eugen Ehrlich’s sociological jurisprudence in contemporary debates about legal pluralism. This dualist understanding of law was not typical for Ehrlich who considered that law always manifests itself as a unified social integrity, though it consists of many social orders. Law can be conceptually divided into two logical categories: official law (the law of a state and lawyers) and living (social) law. Neither was Ehrlich inclined to advocate for mechanical transformation of facticity into normativity – this requires additional creative work of lawyers. From this point of view, it is not practicable to use Ehrlich’s socio-legal theory for justification of the project of anthropological jurisprudence to include all normative systems of social regulation in the category of law under the slogan of legal pluralism. Citing Ehrlich as one of the founding fathers of legal pluralism is not undisputable because legal pluralism itself is not a unified scientific doctrine, and many assertions of legal pluralists contradict Ehrlich’s scientific position. Ehrlich was not inclined to undermine the role of law and judiciary in legal reality, which sometimes is typical for some legal pluralists.   Keywords: sociology of law, living law, legal pluralism, normativity, state, social unions, society, law-bound state. https://doi.org/ 10.31861/ehrlichsjournal2017.01.036

An Embarrassment of Riches and a Surplus of Shame: Amartya Sen on Poverty and Deprivation

This theoretical contribution to poverty studies investigates Amartya Sen's work as a basis for examining poverty. Sen discusses two social capabilities, each essential to the avoidance of poverty; one is the ability to appear in public without feeling shame, the other the ability to participate in the life of the community. This essay analyzes the intricacies of using the concept of community as a reference group for judging a person's poverty. It develops a notion of the “affluent poor,” which is a logical category of the capability perspective that Sen has developed. Although the affluent poor might appear to be oxymoronic, those who embrace the capability perspective should acknowledge it as a necessary implication of that perspective.

Auditing the Assignments of Top-Level Semantic Types in the UMLS Semantic Network to UMLS Concepts.

The Unified Medical Language System (UMLS) is an important terminological system. By the policy of its curators, each concept of the UMLS should be assigned the most specific Semantic Types (STs) in the UMLS Semantic Network (SN). Hence, the Semantic Types of most UMLS concepts are assigned at or near the bottom (leaves) of the UMLS Semantic Network. While most ST assignments are correct, some errors do occur. Therefore, Quality Assurance efforts of UMLS curators for ST assignments should concentrate on automatically detected sets of UMLS concepts with higher error rates than random sets. In this paper, we investigate the assignments of top-level semantic types in the UMLS semantic network to concepts, identify potential erroneous assignments, define four categories of errors, and thus provide assistance to curators of the UMLS to avoid these assignments errors. Human experts analyzed samples of concepts assigned 10 of the top-level semantic types and categorized the erroneous ST assignments into these four logical categories. Two thirds of the concepts assigned these 10 top-level semantic types are erroneous. Our results demonstrate that reviewing top-level semantic type assignments to concepts provides an effective way for UMLS quality assurance, comparing to reviewing a random selection of semantic type assignments.

Frege on Syntax, Ontology, and Truth's Pride of Place

AbstractFrege's strict alignment between his syntactic and ontological categories is not, as is commonly assumed, some kind of a philosophical thesis. There is no thesis that proper names refer only to objects, say, or that what refers to an object is a proper name. Rather, the alignment of categories is internal to Frege's conception of what syntax and ontology are. To understand this, we need to recognise the pride of place Frege assigns within his theorising to the notion of truth. For both language and the world, the Fregean categories are logical categories, categories, that is, of truth. The elaboration of this point makes clear the incoherence of supposing that they might not align.

Algebraizable logics and a functorial encoding of its morphisms

The present work presents some results about the categorial relation between logics and its categories of structures. A (propositional, finitary) logic is a pair given by a signature and Tarskian consequence relation on its formula algebra. The logics are the objects in our categories of logics; the morphisms are certain signature morphisms that are translations between logics (\cite{AFLM1},\cite{AFLM2},\cite{AFLM3} \cite{FC}). Morphisms between algebraizable logics (\cite{BP}) are translations that preserves algebraizing pairs (\cite{MaMe}): they can be completely encoded by certain functors defined on the quasi-variety canonically associated to the algebraizable logics. This kind of results will be useful in the development of a categorial approach to the representation theory of general logics (\cite{MaPi1}, \cite{MaPi2}, \cite{AJMP}).

Minimal Categorical System and Predication Theory In Porphyry

The article considers some problematic aspects of Porphyry’s typology of Aristotle’s categories and the theory of predication. Minimal (_________) class of categories in Porphyry is revealed. The work has shed some light on the opposition between $\textit{explanation}$ and $\textit{description}$ (__________ / ___________) within the framework of ancient categorical logic. A fourfold pattern of predication theory in Porphyry is described. The study aims to illuminate the development of Porphyry’s predication theory towards the archaic doctrine of quantifiers. Particular attention is paid to Porphyry’s account of semantic relation between sets. The paper represents Porphyry’s nine kinds of class / item relationships. The article focuses on the awakening of academic interest to the logical heritage of Porphyry.

Topological Representation of Intuitionistic and Distributive Abstract Logics

We continue work of our earlier paper (Lewitzka and Brunner: Minimally generated abstract logics, Logica Universalis 3(2), 2009), where abstract logics and particularly intuitionistic abstract logics are studied. Abstract logics can be topologized in a direct and natural way. This facilitates a topological study of classes of concrete logics whenever they are given in abstract form. Moreover, such a direct topological approach avoids the often complex algebraic and lattice-theoretic machinery usually applied to represent logics. Motivated by that point of view, we define in this paper the category of intuitionistic abstract logics with stable logic maps as morphisms, and the category of implicative spectral spaces with spectral maps as morphisms. We show the equivalence of these categories and conclude that the larger categories of distributive abstract logics and distributive sober spaces are equivalent, too.

Entre fenomenologia e lógica: o conceito hegeliano do absoluto

ABSTRACT: The aim of this paper is to investigate how the concept of the Absolute (das Absolute) was designed and exposed by Hegel in two of his major works. Our interpretative hypothesis is that the way in which Hegel uses this concept in a phenomenological framework – namely, the way in which this concept is set out in the Introduction of the Phenomenology of Spirit (1807) and developed in the figures of consciousness of Spirit, from the sensible consciousness to self-consciousness as the ‘absolute Knowledge’ –, partly anticipates various other uses which will be discussed within the first part of the Encyclopedia of the Philosophical Sciences (1830), the Science of Logic. In this book, the exposition of the logical categories within pure thought made it necessary to anticipate that all logical determinations are ‘definitions of the absolute’ or ‘metaphysical definitions of God.’

L’assujettissement des textes: quelques réflexions sur le classement par catégories dans les « encyclopédies » (leishu 類書) en Chine ancienne

Abstract This paper focuses on the logic of categorization of three Chinese « encyclopedias » (leishu 類書), and the influence the categories have on the texts they contain. The choices made by the compilers are in fact not always easy to understand, and are indeed often quite subjective, which leads to think that leishu may not always prove reliable when used for research on specific topics. In order to exemplify this idea, I present here the case of a short narrative which can be found in three different leishu, namely the Yiwen leiju, the Taiping guangji and the Taiping yulan. Despite the text’s different versions being roughly the same, I argue that its meaning is modified by the way it is categorized in each of these books. By their choices, the compilers therefore influence our perception of the writings found in leishu, which are one of our most important sources concerning ancient Chinese literature.

The Category of Number in Dıfferent Language Systems

Philosophical or logical category of number is based on the existence of quantitative characteristics of things and events of the reality. A human being first comprehended the number of things and objects and then tried to express them by means of languages. Gradually there appeared the existence of more than one things and there emerged the necessity of expressing them. In Turkic languages, including Azerbaijani, the abundance of things began to be expressed by the numeral three, then it reached forty, which was the maximal limit. It found its expression in fairy-tales and eposes, as well as in proverbs. For instance: Atalar üçəcən deyiblər (Our forefathers counted up to theree or allowed do something only thrice); Üç gün, üç gecə yol getdilər (They walked three days and nights); Qırx gün möhlət vermək(They granted him a delay for forty days); Qırx gün, qırx gecə döyüşmək (They fought for forty days and nights); qırx qapını açmaq (to open forty doors); divin canını qırxıncı otaqda tapmaq (to find the vigour (life) of the giant in the fortieth room), etc. All this shows that the expression of the category of number in language has undergone a long historical process.

Tensor product of no-signaling boxes in the framework of quantum logics

In the quantum logic framework we show that the no-signaling box model is a particular type of tensor product with single box logics. Such notion of a tensor product is too strong to apply in the category of logics of quantum mechanical systems. In the light of the obtained results, the statement that no-signaling box models are generalizations of quantum models is questionable.

Proofs as Cognitive or Computational: Ibn Sı̄nā’s Innovations

We record the advances made by the eleventh century Persian logician Ibn Sina—known in the West as Avicenna—away from a purely cognitive view of proofs and towards a more computational view, and the kinds of consideration that led him to these advances. Some of Ibn Sina’s new logics, which stand somewhere between Aristotle’s categorical syllogisms and modern first-order logic, can serve as a kind of laboratory for testing what are the differences between Aristotelian and modern logic, and where these differences come from.

A meta-theoretical approach to the history and theory of semiotics

Abstract The object of this paper is the domain of semiotic theories, from “traditional” semiotics to poststructuralism and postmodernism, excluding “semiotizing” approaches such as phenomenology or cultural studies. Thus, it is metatheoretical. It is based on two matrices. The first maps semiotic theories on the basis of the continuity or discontinuity between them. The second displays the logical categories of the relationship between semiotics and Marxism, which has historically been an important influence on the field. The paper presents the views of the main authors of the domain in terms of these two matrices. Some of the conclusions are: (a) the irreconcilability between Saussurean and Peircean semiotics; (b) the greater historical development of the former in comparison to the latter; (c) the different orientation between Central and Eastern European semiotics on the one hand and French semiotics on the other; (d) the strong influence of Saussure and Levi-Straussian structuralism on poststructuralism; (e) the increase of the influence of Marxism from structuralism to poststructuralism; and (f) the transformation from poststructuralism to North American postmodernism. The paper closes with some thoughts about the present status of the main semiotic currents and a proposal for a fertile future orientation for semiotics.

Canonical Syllogistic Moods in Traditional Aristotelian Logic

A novel theoretical formulation of Categorical Logic based on two properties of categorical propositions and three simple axioms has been introduced recently. This formulation allowed for the suppression of the distinction between immediate and mediate inferences, and also provided a theoretical framework to study opposition relations, thus restoring the theoretical unity of traditional Aristotelian logic. By using this approach, it has been reported that a total of 3072 conclusive syllogistic moods can be found when including indefinite terms in classical syllogistic, but this result has yet to be proven. This paper presents an overview of the recently proposed theoretical formulation of Categorical Logic, along with the derivation of the 48 canonical syllogistic moods that are capable of generating the 3072 conclusive moods previously reported.

Dooyeweerd’s Idea of Modalities: The Pivotal 1922 Article

Dooyeweerd says that the first rudimental conception of his philosophy had ripened even before he started work at the Kuyper Foundation in October 1922. He had not even studied Kuyper’s works, although he would later find some similarities in Kuyper. A detailed analysis of an article written earlier in 1922 shows us how Dooyeweerd developed his philosophy. This article is ‘Normatieve rechtsleer. Een kritisch-methodologische onderzoeking naar Kelsen’s normatieve rechtsbeschouwing.’ It includes these ideas: the rejection of the autonomy of thought, the idea of intuitive beholding [schouwen], and the idea of modalities or modes of consciousness. Previous historians of reformational philosophy have not adequately researched Dooyeweerd’s sources for these ideas. None of these sources are Calvinistic. Dooyeweerd used these ideas to critique neo-Kantianism. He dismantles Kant’s logical categories and instead puts forward the idea of intuited modalities. And Dooyeweerd uses the scholastic idea of ‘meaning-moments’ to individuate these modalities from totality.

A comparison between monoidal and substructural logics

Monoidal logics were introduced as a foundational framework to analyse the proof theory of deontic logic. Building on Lambek’s work in categorical logic, logical systems are defined as deductive systems, that is, as collections of equivalence classes of proofs satisfying specific rules and axiom schemata. This approach enables the classification of deductive systems with respect to their categorical structure. When looking at their proof theory, however, one can see that there are similarities between monoidal and substructural logics. The purpose of the present paper is to address this issue and highlight the differences between these two approaches. We argue that monoidal logics provide a more flexible foundational framework that enables a finer analysis of the relationship between negation(s) and other logical connectives. We show that the elimination of double negation(s) is independent from the de Morgan dualities, that monoidal deductive systems are not necessarily weakly distributive and that deductive systems satisfying the elimination of double negation(s) and the law of excluded middle are not necessarily classical.