The theory of concept lattices is approached from the point of view of fuzzy logic. The notions of partial order, lattice order, and formal concept are generalized for fuzzy setting. Presented is a theorem characterizing the hierarchical structure of formal fuzzy concepts arising in a given formal fuzzy context. Also, as an application of the present approach, Dedekind–MacNeille completion of a partial fuzzy order is described. The approach and results provide foundations for formal concept analysis of (...) vague data—the propositions “object x has attribute y”, which form the input data to formal concept analysis, are now allowed to have also intermediate truth values, meeting reality better. (shrink)

In the paper, we study connections between rough concept lattices and domains. The main result is representation theorems of complete lattices and algebraic lattices by concepts based on Rough Set Theory. It is shown that there is a deep relationship between Rough Set Theory and Domain Theory.

In the last decade, formal concept analysis in a fuzzy setting has received more attention for knowledge processing tasks in various fields. The hierarchical order visualisation of generated formal concepts is a major concern for the practical application of FCA. In this process, a major issue is the huge number of formal concepts generated from ‘a large context’, and another problem is their ‘storage’ complexity. To deal with these issues a method is proposed in this paper based on Shannon (...) entropy and Huffman coding. The proposed method is illustrated using crisply generated concepts such that the changes between obtained concepts can be measured using Levenshtein distance. The analysis derived from the proposed method is illustrated with an example for FCA in a fuzzy setting. (shrink)

Uncertainty natural language processing has always been a research focus in the artificial intelligence field. In this paper, we continue to study the linguistic truth-valued concept lattice and apply it to the disease intelligent diagnosis by building an intelligent model to directly handle natural language. The theoretical bases of this model are the classical concept lattice and the lattice implication algebra with natural language. The model includes the case library formed by patients, attributes matching, and (...) the matching degree calculation about the new patient. According to the characteristics of the patients, the disease attributes are firstly divided into intrinsic invariant attributes and extrinsic variable attributes. The calculation algorithm of the linguistic truth-valued formal concepts and the constructing algorithm of the linguistic truth-valued concept lattice based on the extrinsic attributes are proposed. And the disease bases of the different treatments for different patients with the same disease are established. Secondly, the matching algorithms of intrinsic attributes and extrinsic attributes are given, and all the linguistic truth-valued formal concepts that match the new patient’s extrinsic attributes are found. Lastly, by comparing the similarity between the new patients and the matching formal concepts, we calculate the best treatment options to realize the intelligent diagnosis of the disease. (shrink)

We give a characterization of the fixed points and of the lattices of fixed points of fuzzy Galois connections. It is shown that fixed points are naturally interpreted as concepts in the sense of traditional logic.

This new edition of Introduction to Lattices and Order presents a radical reorganization and updating, though its primary aim is unchanged. The explosive development of theoretical computer science in recent years has, in particular, influenced the book's evolution: a fresh treatment of fixpoints testifies to this and Galois connections now feature prominently. An early presentation of concept analysis gives both a concrete foundation for the subsequent theory of complete lattices and a glimpse of a methodology for data analysis that (...) is of commercial value in social science. Classroom experience has led to numerous pedagogical improvements and many new exercises have been added. As before, exposure to elementary abstract algebra and the notation of set theory are the only prerequisites, making the book suitable for advanced undergraduates and beginning graduate students. It will also be a valuable resource for anyone who meets ordered structures. (shrink)

In 1982, Wolniewicz proposed a formal ontology of situations based on the lattice of elementary situations (cf. [7, 8]). In [3], I constructed some types of formal structure Porphyrian Tree Structures (PTS), Concepts Structures (CS) and the Structures of Individuals (U) that formally represent ontologically fundamental categories: species and genera (PTS), concepts (CS) and individual beings (U) (cf. [3, 4]). From an ontological perspective, situations and concepts belong to different categories. But, unexpectedly, as I shall show, some variants of (...) CS and Wolniewicz’s lattice are similar. The main theorem states that a subset of a modified concepts structure (called CS+) based on CS fulfils the axioms of Wolniewicz’ lattice. Finally, I shall draw some philosophical conclusions and state some formal facts. (shrink)

The lattices of the title generalize the concept of a De Morgan lattice. A representation in terms of ordered topological spaces is described. This topological duality is applied to describe homomorphisms, congruences, and subdirectly irreducible and free lattices in the category. In addition, certain equational subclasses are described in detail.

In this paper authors for the first time define a new notion called neutrosophic lattices. We define few properties related with them. Three types of neutrosophic lattices are defined and the special properties about these new class of lattices are discussed and developed. This paper is organised into three sections. First section introduces the concept of partially ordered neutrosophic set and neutrosophic lattices. Section two introduces different types of neutrosophic lattices and the final section studies neutrosophic Boolean algebras. Conclusions (...) and results are provided in section three. (shrink)

The goal of the paper is to develop a universal semantic approach to derivable rules of propositional multiple-conclusion sequent calculi with structural rules, which explicitly involve not only atomic formulas, treated as metavariables for formulas, but also formula set variables, upon the basis of the conception of model introduced in :27–37, 2001). One of the main results of the paper is that any regular sequent calculus with structural rules has such class of sequent models that a rule is derivable in (...) the calculus iff it is sound with respect to each model of the semantics. We then show how semantics of admissible rules of such calculi can be found with using a method of free models. Next, our universal approach is applied to sequent calculi for many-valued logics with equality determinant. Finally, we exemplify this application by studying sequent calculi for some of such logics. (shrink)

This book presents the foundations of a general theory of algebras. Often called “universal algebra”, this theory provides a common framework for all algebraic systems, including groups, rings, modules, fields, and lattices. Each chapter is replete with useful illustrations and exercises that solidify the reader's understanding. The book begins by developing the main concepts and working tools of algebras and lattices, and continues with examples of classical algebraic systems like groups, semigroups, monoids, and categories. The essence of the book lies (...) in Chapter 4, which provides not only basic concepts and results of general algebra, but also the perspectives and intuitions shared by practitioners of the field. The book finishes with a study of possible uniqueness of factorizations of an algebra into a direct product of directly indecomposable algebras. There is enough material in this text for a two semester course sequence, but a one semester course could also focus primarily on Chapter 4, with additional topics selected from throughout the text. (shrink)

The paper introduces the concept of B-Almost distributive fuzzy lattice in terms of its principal ideal fuzzy lattice. Necessary and sufficient conditions for an ADFL to become a B-ADFL are investigated. We also prove the equivalency of B-algebra and B-fuzzy algebra. In addition, we extend PSADL to PSADFL and prove that B-ADFL implies PSADFL.

The literature on quantum logic emphasizes that the algebraic structures involved with orthodox quantum mechanics are non distributive. In this paper we develop a particular algebraic structure, the quasi-lattice ( $${\mathfrak{I}}$$ -lattice), which can be modeled by an algebraic structure built in quasi-set theory $${\mathfrak{Q}}$$. This structure is non distributive and involve indiscernible elements. Thus we show that in taking into account indiscernibility as a primitive concept, the quasi-lattice that ‘naturally’ arises is non distributive.

In this paper, we prove that the lattice of all closure operators of a complete Almost Distributive Lattice L with fixed maximal element m is dual atomistic. We define the concept of a completely meet-irreducible element in a complete ADL and derive a necessary and sufficient condition for a dual atom of Φ (L) to be complemented.

Brain, Mind and Consciousness are the research concerns of psychiatrists, psychologists, neurologists, cognitive neuroscientists and philosophers. All of them are working in different and important ways to understand the workings of the brain, the mysteries of the mind and to grasp that elusive concept called consciousness. Although they are all justified in forwarding their respective researches, it is also necessary to integrate these diverse appearing understandings and try and get a comprehensive perspective that is, hopefully, more than the sum (...) of their parts. There is also the need to understand what each one is doing, and by the other, to understand each other's basic and fundamental ideological and foundational underpinnings. This must be followed by a comprehensive and critical dialogue between the respective disciplines. Moreover, the concept of mind and consciousness in Indian thought needs careful delineation and critical/evidential enquiry to make it internationally relevant. The brain-mind dyad must be understood, with brain as the structural correlate of the mind, and mind as the functional correlate of the brain. To understand human experience, we need a triad of external environment, internal environment and a consciousness that makes sense of both. We need to evolve a consensus on the definition of consciousness, for which a working definition in the form of a Consciousness Tetrad of Default, Aware, Operational and Evolved Consciousness is presented. It is equally necessary to understand the connection between physical changes in the brain and mental operations, and thereby untangle and comprehend the lattice of mental operations. Interdisciplinary work and knowledge sharing, in an atmosphere of healthy give and take of ideas, and with a view to understand the significance of each other's work, and also to critically evaluate the present corpus of knowledge from these diverse appearing fields, and then carry forward from there in a spirit of cooperative but evidential and critical enquiry - this is the goal for this monograph, and the work to follow. (shrink)

The Management Centre for Human Values along with the participants of the Post-Graduate Program for Executives and the Oil and Natural Gas Corporation Limited on the occasion of the Golden Jubilee of the Indian Institute of Management Calcutta arranged a seminar on Socially Conscious Leadership, or the Lattice 2010, on 19 December 2010. The seminar debate on the role of Corporate Social Responsibility in contemporary business makes for an interesting note that would befit the Journal of Human Values. This (...) is so because the invitees to the seminar, through their singular and idiosyncratic narration of experiences, have inevitably problematized the basic concepts of CSR itself. Concepts like ‘human values’, ‘ethics’, ‘social consciousness’ no longer exist as water-tight compartments of traditional or even autonomous sanctuaries of humane goodness. These concepts have been transformed from ‘value-adding’ propositions to ‘value-appropriating’ propositions, or in other words, social issues have become business issues. A wide spectrum of views coming from Gen. Bajwa, representing the army; Prof. Chaudhuri and Prof. Bhatta, representing administrative academia; Sri Salvi and Sri Tyagi, endorsing real-life CSR; Mr Ahir and Mr Rayaprolu, representing the corporate; Ms Vatsa and Ms Swami, representing NGOs; and Prof. Sarkar, Prof. Mohanty and Prof. Chatterjee, representing the academicians; along with the presence of Ms M. Bhattacharya from ONGC—all of them have aided and abetted in establishing the contemporary position of CSR as a successful and competitive business proposition. The note addresses the issues developed from their individual experiences and opinions, and attempts to establish, but on an individual level, an intellectual point of departure for an academic discussion of CSR that would initiate further potent research and theorization in this regard. (shrink)

Brain, Mind and Consciousness are the research concerns of psychiatrists, psychologists, neurologists, cognitive neuroscientists and philosophers. All of them are working in different and important ways to understand the workings of the brain, the mysteries of the mind and to grasp that elusive concept called consciousness. Although they are all justified in forwarding their respective researches, it is also necessary to integrate these diverse appearing understandings and try and get a comprehensive perspective that is, hopefully, more than the sum (...) of their parts. There is also the need to understand what each one is doing, and by the other, to understand each other's basic and fundamental ideological and foundational underpinnings. This must be followed by a comprehensive and critical dialogue between the respective disciplines. Moreover, the concept of mind and consciousness in Indian thought needs careful delineation and critical/evidential enquiry to make it internationally relevant. The brain-mind dyad must be understood, with brain as the structural correlate of the mind, and mind as the functional correlate of the brain. To understand human experience, we need a triad of external environment, internal environment and a consciousness that makes sense of both. We need to evolve a consensus on the definition of consciousness, for which a working definition in the form of a Consciousness Tetrad of Default, Aware, Operational and Evolved Consciousness is presented. It is equally necessary to understand the connection between physical changes in the brain and mental operations, and thereby untangle and comprehend the lattice of mental operations. Interdisciplinary work and knowledge sharing, in an atmosphere of healthy give and take of ideas, and with a view to understand the significance of each other's work, and also to critically evaluate the present corpus of knowledge from these diverse appearing fields, and then carry forward from there in a spirit of cooperative but evidential and critical enquiry - this is the goal for this monograph, and the work to follow. (shrink)

We give natural examples of factors of the Muchnik lattice which capture intuitionistic propositional logic , arising from the concepts of lowness, 1-genericity, hyperimmune-freeness and computable traceability. This provides a purely computational semantics for IPC.

A Kleene lattice is a distributive lattice equipped with an antitone involution and satisfying the so-called normality condition. These lattices were introduced by J. A. Kalman. We extended this concept also for posets with an antitone involution. In our recent paper (Chajda, Länger and Paseka, in: Proceeding of 2022 IEEE 52th International Symposium on Multiple-Valued Logic, Springer, 2022), we showed how to construct such Kleene lattices or Kleene posets from a given distributive lattice or poset and (...) a fixed element of this lattice or poset by using the so-called twist product construction, respectively. We extend this construction of Kleene lattices and Kleene posets by considering a fixed subset instead of a fixed element. Moreover, we show that in some cases, this generating poset can be embedded into the resulting Kleene poset. We investigate the question when a Kleene poset can be represented by a Kleene poset obtained by the mentioned construction. We show that a direct product of representable Kleene posets is again representable and hence a direct product of finite chains is representable. This does not hold in general for subdirect products, but we show some examples where it holds. We present large classes of representable and non-representable Kleene posets. Finally, we investigate two kinds of extensions of a distributive poset $${\textbf{A}}$$ A, namely its Dedekind-MacNeille completion $${{\,\mathrm{\textbf{DM}}\,}}({\textbf{A}})$$ DM ( A ) and a completion $$G({\textbf{A}})$$ G ( A ) which coincides with $${{\,\mathrm{\textbf{DM}}\,}}({\textbf{A}})$$ DM ( A ) provided $${\textbf{A}}$$ A is finite. In particular we prove that if $${\textbf{A}}$$ A is a Kleene poset then its extension $$G({\textbf{A}})$$ G ( A ) is also a Kleene lattice. If the subset X of principal order ideals of $${\textbf{A}}$$ A is involution-closed and doubly dense in $$G({\textbf{A}})$$ G ( A ) then it generates $$G({\textbf{A}})$$ G ( A ) and it is isomorphic to $${\textbf{A}}$$ A itself. (shrink)

It is shown that the Boolean center of complemented elements in a bounded integral residuated lattice characterizes direct decompositions. Generalizing both Boolean products and poset sums of residuated lattices, the concepts of poset product, Priestley product and Esakia product of algebras are defined and used to prove decomposition theorems for various ordered algebras. In particular, we show that FLw-algebras decompose as a poset product over any finite set of join irreducible strongly central elements, and that bounded n-potent GBL-algebras are (...) represented as Esakia products of simple n-potent MV-algebras. (shrink)

Based on Alasdair Urquhart’s representation of not necessarily distributive bounded lattices we exhibit several discrete dualities in the spirit of the “duality via truth” concept by Orłowska and Rewitzky. We also exhibit a discrete duality for Urquhart’s relevant algebras and their frames.

To ascertain that a formalization of the intuitive notion of a ‘concept’ is linguistically interesting, one has to check whether it allows to get a grip on distinctions and notions from lexical semantics. Prime candidates are notions like ‘prototype’, ‘stereotypical attribute’, ‘essential attribute versus accidental attribute’, ‘intension versus extension’. We will argue that although the current paradigm of formal concept analysis as an application of lattice theory is not rich enough for an analysis of these notions, a (...) lattice theoretical approach to concepts is a suitable starting point for formalizing them. (shrink)

To cope with problems arising in the description of (1) contextual interactions, and (2) the generation of new states with new properties when quantum entities become entangled, the mathematics of quantum mechanics was developed. Similar problems arise with concepts. We use a generalization of standard quantum mechanics, the mathematical lattice theoretic formalism, to develop a formal description of the contextual manner in which concepts are evoked, used, and combined to generate meaning.

Priestley duality can be used to study subalgebras of Heyting algebras and related structures. The dual concept is that of congruence on the dual space and the congruence lattice of a Heyting space is dually isomorphic to the subalgebra lattice of the dual algebra. In this paper we continue our investigation of the congruence lattice of a Heyting space that was undertaken in [10], [8] and [12]. Our main result is a characterization of the modularity of (...) this lattice (Theorem 2.12). Partial results about its complementedness are also given, and among other things a characterization of those finite Heyting algebras with a complemented subalgebra lattice (Theorem 3.5). (shrink)

Compared to existing classical approaches to semiotics which are dyadic (signifier/signified, F. de Saussure) and triadic (symbol/concept/object, Ch. S. Peirce), this theory can be characterized as tetradic ([sign/semion]//[object/noema]) and is the result of either doubling the dyadic approach along the semiotic/ordinary dimension or splitting the ‘concept’ of the triadic one into two (semiotic/ordinary). Other important features of this approach are (a) the distinction made between concepts (only functional pairs of extent and intent) and categories (as representations of expressions) (...) and (b) the indication of the need for providing the mathematical passage from the duality between two sets (where one is a singleton) within systems of sets to category-theoretical monoids within systems of categories while waiting for the solution of this problem in the field of logic. Last but not least, human language expressions are the most representative physical instances of semiotic objects. Moreover, as computational experiments which are possible with linguistic objects present a high degree of systematicity (of oppositions), in general, it is relatively easy to elucidate their dependence on the concepts underlying signs. This new semiotic theory or rather this new research program emerged as the fruit of experimentation and reflection on the application of data science tools elaborated within the frameworks of Rough Set Theory (RST), Formal Context Analysis (FCA) and, though only theoretically, Distributed Information Logic (DIL). The semiotic objects (s-objects) of this theory can be described in tabular datasets. Nevertheless, at this stage of formalisation of the theory, lattices (not trees) can be used as working representation structures for characterizing the components of concept systems and graphs for categories of each layer. (shrink)

We propose a theory for modeling concepts that uses the state-context-property theory (SCOP), a generalization of the quantum formalism, whose basic notions are states, contexts and properties. This theory enables us to incorporate context into the mathematical structure used to describe a concept, and thereby model how context influences the typicality of a single exemplar and the applicability of a single property of a concept. We introduce the notion `state of a concept' to account for this contextual (...) influence, and show that the structure of the set of contexts and of the set of properties of a concept is a complete orthocomplemented lattice. The structural study in this article is a preparation for a numerical mathematical theory of concepts in the Hilbert space of quantum mechanics that allows the description of the combination of concepts. (shrink)

A proposition is associated in classical mechanics with a subset of phase space, in quantum logic with a projection in Hilbert space, and in both cases with a 2-valued observable or test. A theoretical statement typically assigns a probability to such a pure test. However, since a pure test is an idealization not realizable experimentally, it is necessary — to give such a statement a practical meaning — to describe how it can be approximated by feasible tests. This gives rise (...) to a search for a formal representation of feasible tests, which leads via mixed tests (weighted means of pure tests) to vague tests (convex sets of mixed tests). A model is described in which the latter form a continuous lattice; the pure and mixed tests are the maximal elements and the feasible tests form a basis. Each type of test has its own logic; this is illustrated by the passage from mixed tests to pure tests, which corresponds to the transition from L to classical logic. (shrink)

This paper introduces current acoustic theories relating to the phenomenology of sound as a framework for interrogating concepts relating to the ecologies of acoustic and landscape phenomena in a Japanese stroll garden. By applying the technique of Formal Concept Analysis, a partially ordered lattice of garden objects and attributes is visualized as a means to investigate the relationship between elements of the taxonomy.

There are well-known isomorphisms between the complete lattice of all partitions of a given set A and the lattice of all equivalence relations on A. In the paper the notion of partition is generalized in order to work correctly for wider classes of binary relations than equivalence ones such as quasiorders or tolerance relations. Some others classes of binary relations and corresponding counterparts of partitions are considered.

The aim of this paper is to provide a complete Natural Kind Semantics for an Essentialist Theory of Kinds. The theory is formulated in two-sorted first order monadic modal logic with identity. The natural kind semantics is based on Rudolf Willes Theory of Concept Lattices. The semantics is then used to explain several consequences of the theory, including results about the specificity (species–genus) relations between kinds, the definitions of kinds in terms of genera and specific differences and the existence (...) of negative kinds. First, I show under which conditions the Hierarchy principle, which has been subjected to counterexamples in the literature, holds. I also show that a different principle about the species–genus relations between kinds, namely Kant’s Law, follows from the essentialist theory. Second, I introduce two new operations for kinds and show that they can be used to provide traditional definitions of kinds in terms of genera and specific differences. Finally, I show that these operations of specific difference induce, for each kind, a uniquely specified contrary kind and a uniquely specified subcontrary kind, which can be used as semantic values for non-classical predicate negations of kind terms. (shrink)

We prove completeness for some language-theoretic models of the full Lambek calculus and its various fragments. First we consider syntactic concepts and syntactic concepts over regular languages, which provide a complete semantics for the full Lambek calculus \. We present a new semantics we call automata-theoretic, which combines languages and relations via closure operators which are based on automaton transitions. We establish the completeness of this semantics for the full Lambek calculus via an isomorphism theorem for the syntactic concepts (...) class='Hi'>lattice of a language and a construction for the universal automaton recognizing the same language. Finally, we use automata-theoretic semantics to prove completeness of relation models of binary relations and finite relation models for the Lambek calculus without and with empty antecedents and \), thus solving a problem left open by Pentus. (shrink)

In [This Zeitschrift 25 , 45-52, 119-134, 447-464], Pavelka systematically discussed propositional calculi with values in enriched residuated lattices and developed a general framework for approximate reasoning. In the first part of this paper we introduce the concept of generalized quantifiers into Pavelka's logic and establish the fundamental theorem of ultraproduct in first order Pavelka's logic with generalized quantifiers. In the second part of this paper we show that the fundamental theorem of ultraproduct in first order Pavelka's logic is (...) preserved under some direct product of lattices of truth values. (shrink)

The concept of Galois connection between power sets is generalized from the point of view of fuzzy logic. Studied is the case where the structure of truth values forms a complete residuated lattice. It is proved that fuzzy Galois connections are in one-to-one correspondence with binary fuzzy relations. A representation of fuzzy Galois connections by Galois connections is provided.

In this dissertation, a logical analysis of points of view is presented. It is based on the concept of determinable presented by Johnson in his book Logic. A point of view is a set of Determinables. Determinables generate a many-dimensional conceptual space. Concepts are subsets of this space, and their relations form a lattice. A logical system to present points of view is introduced and proved to be complete. Some applications of this logic are demonstrated (relative identity, scientific (...) change, inductive logic etc.). (shrink)

This paper is an experiment in Leibnizian analysis. The reader will recall that Leibniz considered all true sentences to be analytically so. The difference, on his account, between necessary and contingent truths is that sentences reporting the former are finitely analytic; those reporting the latter require infinite analysis of which God alone is capable. On such a view at least two competing conceptions of entailment emerge. According to one, a sentence entails another when the set of atomic requirements for the (...) first is included in the corresponding set for the other; according to the other conception, every atomic requirement of the entailed sentence is underwritten by an atomic constituent of the entailing one. The former conception is classical on the twentieth century understanding of the term; the latter is the one we explore here. Now if we restrict ourselves to the formal language of the propositional calculus, every sentence has a finite analysis into its conjunctive normal form. Semantically, then, every sentence of that language can be represented as a simple hypergraph, H, on the powerset of a universe of states. Entailment of the sort we wish to study can be represented as a known relation, subsumption between hypergraphs. Since the lattice of hypergraphs thus ordered is a DeMorgan lattice, the logic of entailment thus understood is the familiar system, FDE of first-degree entailment. We observe that, extensionalized, the relation of subsumption is itself a DeMorgan Lattice ordered by higher-order subsumption. Thus the semantic idiom that hypergraph-theory affords reveals a hierarchy of lattices capable of representing entailments of every finite degree. (shrink)

This paper looks at how the idea of pointless topology itself evolved during its pre-localic phase by analyzing the definitions of the concept of topological space of Menger and Nöbeling. Menger put forward a topology of lumps in order to generalize the definition of the real line. As to Nöbeling, he developed an abstract theory of posets so that a topological space becomes a particular case of topological poset. The analysis emphasizes two points. First, Menger's geometrical perspective was superseded (...) by an algebraic one, a lattice-theoretical one to be precise. Second, Menger's bottom–up approach was replaced by a top–down one. (shrink)

We examine the concept of almost everywhere domination from the viewpoint of mass problems. Let AED and MLR be the sets of reals which are almost everywhere dominating and Martin-Löf random, respectively. Let b1, b2, and b3 be the degrees of unsolvability of the mass problems associated with AED, MLR × AED, and MLR ∩ AED, respectively. Let [MATHEMATICAL SCRIPT CAPITAL P]w be the lattice of degrees of unsolvability of mass problems associated with nonempty Π01 subsets of 2ω. (...) Let 1 and 0 be the top and bottom elements of [MATHEMATICAL SCRIPT CAPITAL P]w. We show that inf, inf, and inf belong to [MATHEMATICAL SCRIPT CAPITAL P]w and 0 < inf < inf < inf < 1. Under the natural embedding of the recursively enumerable Turing degrees into [MATHEMATICAL SCRIPT CAPITAL P]w, we show that inf and inf but not inf are comparable with some recursively enumerable Turing degrees other than 0 and 0′. In order to make this paper more self-contained, we exposit the proofs of some recent theorems due to Hirschfeldt, Miller, Nies, and Stephan. (shrink)

We study the lattice of local operators in Hylandʼs Effective Topos. We show that this lattice is a free completion under internal sups indexed by the natural numbers object, generated by what we call basic local operators.We produce many new local operators and we employ a new concept, sight, in order to analyze these.We show that a local operator identified by A.M. Pitts in his thesis, gives a subtopos with classical arithmetic.

Based on the concepts of pseudocomplement of L -subsets and the implication operator where L is a completely distributive lattice with order-reversing involution, the definition of countable RL -fuzzy compactness degree and the Lindelöf property degree of an L -subset in RL -fuzzy topology are introduced and characterized. Since L -fuzzy topology in the sense of Kubiak and Šostak is a special case of RL -fuzzy topology, the degrees of RL -fuzzy compactness and the Lindelöf property are generalizations of (...) the corresponding degrees in L -fuzzy topology. (shrink)

We define the concepts of minimal p-morphic image and basic p-morphism for transitive Kripke frames. These concepts are used to determine effectively the least number of variables necessary to axiomatize a tabular extension of K4, and to describe the covers and co-covers of such a logic in the lattice of the extensions of K4.

We show how Peirce's architectonics folds on itself and finds local consequences that correspond to the major global hypotheses of the system. In particular, we study how the pragmaticist maxim can be technically represented in Peirce's existential graphs, well-suited to reveal an underlying continuity in logical operations, and can provide suggestive philosophical analogies. Further, using the existential graphs, we formalize — and prove one direction of — a “local proof of pragmaticism,” trying thus to explain the prominent place that existential (...) graphs can play in the architectonics of pragmaticism, as Peirce persistently advocated. Finally, we present a web of “continuous iterations” of some key Peircean concepts that supports a “lattice of partial proofs” of pragmaticism. (shrink)

In Philosophical Logic, the Liar Paradox has been used to motivate the introduction of both truth value gaps and truth value gluts. Moreover, in the light of “revenge Liar” arguments, also higher-order combinations of generalized truth values have been suggested to account for so-called hyper-contradictions. In the present paper, Graham Priest's treatment of generalized truth values is scrutinized and compared with another strategy of generalizing the set of classical truth values and defining an entailment relation on the resulting sets of (...) higher-order values. This method is based on the concept of a multilattice. If the method is applied to the set of truth values of Belnap's “useful four-valued logic”, one obtains a trilattice, and, more generally, structures here called Belnap-trilattices. As in Priest's case, it is shown that the generalized truth values motivated by hyper-contradictions have no effect on the logic. Whereas Priest's construction in terms of designated truth values always results in his Logic of Paradox, the present construction in terms of truth and falsity orderings always results in First Degree Entailment. However, it is observed that applying the multilattice-approach to Priest's initial set of truth values leads to an interesting algebraic structure of a “bi-and-a-half” lattice which determines seven-valued logics different from Priest's Logic of Paradox. (shrink)

THE ART OF CAUSAL CONJECTURE Glenn Shafer Table of Contents Chapter 1. Introduction........................................................................................ ...........1 1.1. Probability Trees..........................................................................................3 1.2. Many Observers, Many Stances, Many Natures..........................................8 1.3. Causal Relations as Relations in Nature’s Tree...........................................9 1.4. Evidence............................................................................................ ...........13 1.5. Measuring the Average Effect of a Cause....................................................17 1.6. Causal Diagrams..........................................................................................20 1.7. Humean Events............................................................................................23 1.8. Three Levels of Causal Language................................................................27 1.9. An Outline of the Book................................................................................27 Chapter 2. Event Trees............................................................................................... .....31 2.1. Situations and Events...................................................................................32 2.2. The Ordering of Situations and Moivrean Events.......................................35 2.3. Cuts................................................................................................ ..............39 2.4. Humean Events............................................................................................43 2.5. (...) Moivrean Variables......................................................................................49 2.6. Humean Variables........................................................................................53 2.7. Event Trees for Stochastic Processes...........................................................54 2.8. Timing in Event Trees.................................................................................56 2.9. Intersecting Event Trees...............................................................................60 2.10. Notes on the Literature...............................................................................61 Chapter 3. Probability Trees...........................................................................................63 3.1. Some Types of Probability Trees.................................................................64 ii 3.2. Axioms for the Probabilities of Moivrean Events.......................................68 3.3. Zero Probabilities....................................................................................... ..70 3.4. A Sample-Space Analysis of the Event-Tree Axioms.................................72 3.5. Probabilities and Expected Values for Variables.........................................74 3.6. Martingales......................................................................................... .........79 3.7. The Expectation of a Variable in a Cut........................................................83 3.8. Conditional Expected Value and Conditional Expectation.........................87 Chapter 4. The Meaning of Probability...........................................................................91 4.1. The Interpretation of Expected Value..........................................................92 4.2. The Interpretation of Expectation................................................................95 4.3. The Long Run..............................................................................................98 4.4. Changes in Belief.........................................................................................101 4.5. The Empirical Validation of Probability......................................................106 4.6. The Diversity of Uses of Probability...........................................................108 4.7. Notes on the Literature.................................................................................110 Chapter 5. Independent Events.......................................................................................113 5.1. Independence........................................................................................ .......114 5.2. Weak Independence.....................................................................................118 5.3. The Principle of the Common Cause...........................................................121 5.4. Conditional Independence............................................................................128 5.5. Notes on the Literature.................................................................................133 Chapter 6. Events Tracking Events.................................................................................135 6.1. Tracking............................................................................................ ...........137 6.2. Tracking and Conditional Independence.....................................................142 6.3. Stochastic Subsequence...............................................................................143 6.4. Singular Diagrams for Stochastic Subsequence...........................................147 6.5. Conjunctive and Interactive Forks...............................................................149 Chapter 7. Events as Signs of Events..............................................................................153 iii 7.1. Sign................................................................................................ ..............154 7.2. Weak Sign................................................................................................ ....159 7.3. The Ethics of Causal Talk............................................................................160 7.4. Screening Off...............................................................................................16 2 Chapter 8. Independent Variables...................................................................................167 8.1. Unconditional Independence........................................................................170 8.2. Conditional Independence............................................................................175 8.3. Independence for Partitions.........................................................................177 8.4. Independence for Families of Variables......................................................182 8.5. Individual Properties of the Independence Relations...................................186 Chapter 9. Variables Tracking Variables........................................................................189 9.1. Tracking and Conditional Independence: A Summary...............................190 9.2. Strong Tracking............................................................................................ 192 9.3. Strong Tracking and Conditional Independence..........................................198 9.4. Stochastic Subsequence...............................................................................201 9.5. Functional Dependence................................................................................203 9.6. Tracking in Mean.........................................................................................204 9.7. Linear Tracking............................................................................................ 207 9.8. Tracking by Partitions..................................................................................210 9.9. Tracking by Families of Variables...............................................................212 Chapter 10. Variables as Signs of Variables...................................................................215 10.1. Sign................................................................................................ ............219 10.2. Linear Sign................................................................................................ .222 10.3. Scored Sign................................................................................................ 225 10.4. Families of Variables.................................................................................227 Chapter 11. AnTheory of Event Trees.............................................................229 11.1. Event Trees as Sets of Sets........................................................................230 11.2. Event Trees as Partially Ordered Sets........................................................232 iv 11.3. Regular Event Trees...................................................................................240 11.4. The Resolution of Moivrean Variables......................................................244 11.5. Humean Events and Variables...................................................................246 Chapter 12. Martingale Trees..........................................................................................247 12.1. Examples of Decision Trees......................................................................249 12.2. The Meaning of Probability in a Decision Tree.........................................253 12.3. Martingales......................................................................................... .......257 12.4. The Structure of Martingale Trees.............................................................261 12.5. Probability and Causality...........................................................................265 12.6. Lower and Upper Probability.....................................................................269 12.7. The Law of Large Numbers.......................................................................272 12.8. Notes on the Literature...............................................................................274 Chapter 13. Refining............................................................................................ ...........275 13.1. Examples of Refinement............................................................................277 13.2. A Constructive Definition of Finite Refinement.......................................281 13.3. Axioms for Refinement..............................................................................282 13.4. Lifting Moivrean Events and Variables.....................................................288 13.5. Refining Martingale Trees.........................................................................288 13.6. Grounding........................................................................................... .......294 Chapter 14. Principles of Causal Conjecture..................................................................299 14.1. The Diversity of Causal Explanation.........................................................302 14.2. The Mean Effect of the Happening of a Moivrean Event..........................305 14.3. The Effect of a Humean Variable..............................................................311 14.4. Attribution and Generality.........................................................................316 14.5. The Statistical Measurement of the Effect of a Cause...............................319 14.6. Measurement by Experiment.....................................................................320 14.7. Using Our Knowledge of How Things Work............................................322 v 14.8. Sampling Error...........................................................................................329 14.9. The Sampling Frame..................................................................................329 14.10. Notes on the Literature.............................................................................330 Chapter 15. Causal Models.............................................................................................3 31 15.1. The Causal Interpretation of Statistical Prediction....................................333 15.2. Generalizing to a Family of Exogenous Variables....................................337 15.3 Some Joint Causal Diagrams......................................................................339 15.4. Causal Path Diagrams................................................................................342 15.5. Causal Relevance Diagrams.......................................................................346 15.6. The Meaning of Latent Variables..............................................................352 15.7. Notes on the Literature...............................................................................357 Chapter 16. Representing Probability Trees...................................................................359 16.1. Three Graphical Representations...............................................................361 16.2. Skeletal Simplifications.............................................................................368 16.3. Martingale Trees in Type Theory...............................................................371 Appendix A. Huygens’s Probability Trees.....................................................................379 Huygens’s Manuscript in Translation..................................................................380 Appendix B. Some Elements of Graph Theory..............................................................385 B1. Undirected Graphs........................................................................................385 B2. Directed Graphs............................................................................................38 6 Appendix C. Some Elements of Order Theory...............................................................393 C1. Partial and Quasi Orderings.........................................................................393 C2. Singular and Joint Diagrams for Binary Relations.......................................394 C3. Lattices............................................................................................ .............395 C4. The Lattice of Partitions of a Set..................................................................396 Appendix D. The Sample-Space Framework for Probability.........................................399 D1. Probability Measures....................................................................................399 D2. Variables........................................................................................... ...........400 vi D3. Families of Variables...................................................................................401 D4. Expected Value............................................................................................402 D5. The Law of Large Numbers.........................................................................405 D6. Conditional Probability................................................................................406 D7. Conditional Expected Value........................................................................407 Appendix E. Prediction in Probability Spaces................................................................409 E1. Conditional Distribution...............................................................................411 E2. Regression on a Single Variable...................................................................412 E3. Regression on a Partition or a Family of Variables......................................415 E4. Linear Regression on a Single Variable.......................................................418 E5. Linear Regression on a Family of Variables................................................422 Appendix F. Sample-Space Concepts of Independence.................................................425 F1. Overview............................................................................................ ...........426 F2. Independence Proper.....................................................................................432 F3. Unpredictability in Mean..............................................................................434 F4. Simple Uncorrelatedness..............................................................................437 F5. Mixed Uncorrelatedness...............................................................................438 F6. Partial Uncorrelatedness...............................................................................440 F7. Independence for Partitions..........................................................................442 F8. Independence for Families of Variables.......................................................445 F9. The Basic Role of Uncorrelatedness.............................................................448 F10. Dawid’s Axioms.........................................................................................449 Appendix G. Prediction Diagrams..................................................................................453 G1. Path Diagrams............................................................................................ ..454 G2. Generalized Path Diagrams..........................................................................462 G3. Relevance Diagrams.....................................................................................466 G4. Bubbled Relevance Diagrams......................................................................475 Appendix H. Abstract Stochastic Processes...................................................................479 vii H1. Probability Conditionals and Probability Distributions...............................477 H2. Abstract Stochastic Processes......................................................................479 H3. Embedding Variables and Processes in a Sample Space.............................480 References.......................................................................................... ..............................491. (shrink)

This paper discusses how facet-like structures occur as a commonplace feature in a variety of computer science disciplines as a means for structuring class hierarchies. The paper then focuses on a mathematical model for facets (and class hierarchies in general), called formal concept analysis, and discusses graphical representations of faceted systems based on this model.

We systematically study the completion of choice problems in the Weihrauch lattice. Choice problems play a pivotal rôle in Weihrauch complexity. For one, they can be used as landmarks that characterize important equivalences classes in the Weihrauch lattice. On the other hand, choice problems also characterize several natural classes of computable problems, such as finite mind change computable problems, non-deterministically computable problems, Las Vegas computable problems and effectively Borel measurable functions. The closure operator of completion generates the (...) class='Hi'>concept of total Weihrauch reducibility, which is a variant of Weihrauch reducibility with total realizers. Logically speaking, the completion of a problem is a version of the problem that is independent of its premise. Hence, studying the completion of choice problems allows us to study simultaneously choice problems in the total Weihrauch lattice, as well as the question which choice problems can be made independent of their premises in the usual Weihrauch lattice. The outcome shows that many important choice problems that are related to compact spaces are complete, whereas choice problems for unbounded spaces or closed sets of positive measure are typically not complete. (shrink)

Elementary set theory accustoms the students to mathematical abstraction, includes the standard constructions of relations, functions, and orderings, and leads to a discussion of the various orders of infinity. The material on logic covers not only the standard statement logic and first-order predicate logic but includes an introduction to formal systems, axiomatization, and model theory. The section on algebra is presented with an emphasis on lattices as well as Boolean and Heyting algebras. Background for recent research in natural language semantics (...) includes sections on lambda-abstraction and generalized quantifiers. Chapters on automata theory and formal languages contain a discussion of languages between context-free and context-sensitive and form the background for much current work in syntactic theory and computational linguistics. The many exercises not only reinforce basic skills but offer an entry to linguistic applications of mathematical concepts. For upper-level undergraduate students and graduate students in theoretical linguistics, computer-science students with interests in computational linguistics, logic programming and artificial intelligence, mathematicians and logicians with interests in linguistics and the semantics of natural language. (shrink)

Many results in fuzzy logic depend on the mathematical structure the truth value set obeys. In this textbook the algebraic foundations of many-valued and fuzzy reasoning are introduced. The book is self-contained, thus no previous knowledge in algebra or in logic is required. It contains 134 exercises with complete answers, and can therefore be used as teaching material at universities for both undergraduated and post-graduated courses. Chapter 1 starts from such basic concepts as order, lattice, equivalence and residuated (...) class='Hi'>lattice. It contains a full section on BL-algebras. Chapter 2 concerns MV-algebra and its basic properties. Chapter 3 applies these mathematical results on Lukasiewicz-Pavelka style fuzzy logic, which is studied in details; besides semantics, syntax and completeness of this logic, a lot of examples are given. Chapter 4 shows the connection between fuzzy relations, approximate reasoning and fuzzy IF-THEN rules to residuated lattices. (shrink)

This paper utilizes Scott domains (continuous lattices) to provide a mathematical model for the use of idealized and approximately true data in the testing of scientific theories. Key episodes from the history of science can be understood in terms of this model as attempts to demonstrate that theories are monotonic, that is, yield better predictions when fed better or more realistic data. However, as we show, monotonicity and truth of theories are independent notions. A formal description is given of the (...) pragmatic virtues of theories which are monotonic. We also introduce the stronger concept of continuity and show how it relates to the finite nature of scientific computations. Finally, we show that the space of theories also has the structure of a Scott domain. This result provides an analysis of how one theory can be said to approximate another. (shrink)

Prompted by the thesis that an organism’s umwelt possesses not just a descriptive dimension, but a normative one as well, some have sought to annex semiotics with ethics. Yet the pronouncements made in this vein have consisted mainly in rehearsing accepted moral intuitions, and have failed to concretely further our knowledge of why or how a creature comes to order objects in its environment in accordance with axiological charges of value or disvalue. For want of a more explicit account, theorists (...) writing on the topic have relied almost exclusively on semiotic insights about perception originally designed as part of a sophisticated refutation of idealism. The end result, which has been a form of direct givenness, has thus been far from convincing. In an effort to bring substance to the right-headed suggestion that values are rooted in the biological and conform to species-specific requirements, we present a novel conception that strives to make explicit the elemental structure underlying umwelt normativity. Building and expanding on the seminal work of Ayn Rand in metaethics, we describe values as an intertwined lattice which takes a creature’s own embodied life as its ultimate standard; and endeavour to show how, from this, all subsequent valuations can in principle be determined. (shrink)