An Abstract Approach to Consequence Relations

We generalise the Blok-J\'onsson account of structural consequence relations, later developed by Galatos, Tsinakis and other authors, in such a way as to naturally accommodate multiset consequence. While Blok and J\'onsson admit, in place of sheer formulas, a wider range of syntactic units to be manipulated in deductions (including sequents or equations), these objects are invariably aggregated via set-theoretical union. Our approach is more general in that non-idempotent forms of premiss and conclusion aggregation, including multiset sum and fuzzy set union, are considered. In their abstract form, thus, deductive relations are defined as additional compatible preorderings over certain partially ordered monoids. We investigate these relations using categorical methods, and provide analogues of the main results obtained in the general theory of consequence relations. Then we focus on the driving example of multiset deductive relations, providing variations of the methods of matrix semantics and Hilbert systems in Abstract Algebraic Logic

Computer-Aided Discovery and Categorisation of Personality Axioms

We propose a computer-algebraic, order-theoretic framework based on intuitionistic logic for the computer-aided discovery of personality axioms from personality-test data and their mathematical categorisation into formal personality theories in the spirit of F.~Klein's Erlanger Programm for geometrical theories. As a result, formal personality theories can be automatically generated, diagrammatically visualised, and mathematically characterised in terms of categories of invariant-preserving transformations in the sense of Klein and category theory. Our personality theories and categories are induced by implicational invariants that are ground instances of intuitionistic implication, which we postulate as axioms. In our mindset, the essence of personality, and thus mental health and illness, is its invariance. The truth of these axioms is algorithmically extracted from histories of partially-ordered, symbolic data of observed behaviour. The personality-test data and the personality theories are related by a Galois-connection in our framework. As data format, we adopt the format of the symbolic values generated by the Szondi-test, a personality test based on L.~Szondi's unifying, depth-psychological theory of fate analysis.Comment: related to arXiv:1403.200

Fuzzy algebras of concepts

Preconcepts are basic units of knowledge that form the basis of formal concepts in formal concept analysis (FCA). This paper investigates the relations among different kinds of preconcepts, such as protoconcepts, meet and join-semiconcepts and formal concepts. The first contribution of this paper, is to present a fuzzy powerset lattice gradation, that coincides with the preconcept lattice at its 1-cut. The second and more significant contribution, is to introduce a preconcept algebra gradation that yields different algebras for protoconcepts, semiconcepts, and concepts at different cuts. This result reveals new insights into the structure and properties of the different categories of preconcepts.Partial funding for open access charge: Universidad de Málag

Conference Program

Document provides a list of the sessions, speakers, workshops, and committees of the 32nd Summer Conference on Topology and Its Applications

Tameness in generalized metric structures

We broaden the framework of metric abstract elementary classes (mAECs) in several essential ways, chiefly by allowing the metric to take values in a well-behaved quantale. As a proof of concept we show that the result of Boney and Zambrano on (metric) tameness under a large cardinal assumption holds in this more general context. We briefly consider a further generalization to partial metric spaces, and hint at connections to classes of fuzzy structures, and structures on sheaves

An algebraic study of exactness in partial contexts

DMF@?s are the natural algebraic tool for modelling reasoning with Korner@?s partial predicates. We provide two representation theorems for DMF@?s which give rise to two adjunctions, the first between DMF and the category of sets and the second between DMF and the category of distributive lattices with minimum. Then we propose a logic L"{"1"} for dealing with exactness in partial contexts, which belongs neither to the Leibniz, nor to the Frege hierarchies, and carry on its study with techniques of abstract algebraic logic. Finally a fully adequate and algebraizable Gentzen system for L"{"1"} is given

Defining rough sets as core-support pairs of three-valued functions

We answer the question what properties a collection $\mathcal{F}$ of three-valued functions on a set $U$ must fulfill so that there exists a quasiorder $\leq$ on $U$ such that the rough sets determined by $\leq$ coincide with the core--support pairs of the functions in $\mathcal{F}$. Applying this characterization, we give a new representation of rough sets determined by equivalences in terms of three-valued {\L}ukasiewicz algebras of three-valued functions.Comment: This version is accepted for publication in Approximate Reasoning (May 2021