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

An Algebraic Approach to Valued Constraint Satisfaction

[EN]We study the complexity of the valued CSP (VCSP, for short) over arbitrary templates, taking the general framework of integral bounded linearly order monoids as valuation structures. The class of problems considered here subsumes and generalizes the most common one in VCSP literature, since both monoidal and lattice conjunction operations are allowed in the formulation of constraints. Restricting to locally finite monoids, we introduce a notion of polymorphism that captures the pp-definability in the style of Geiger’s result. As a consequence, sufficient conditions for tractability of the classical CSP, related to the existence of certain polymorphisms, are shown to serve also for the valued case. Finally, we establish the dichotomy conjecture for the VCSP, modulo the dichotomy for classical CSP.The work was partly supported by the grant No. GA17-04630S of the Czech Science Foundation and partly by the long-term strategic development financing of the Institute of Computer Science (RVO:67985807).Peer reviewe

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

Strutture, logiche e pensiero

L'articolo si pone l'obiettivo di studiare un metodo per categorizzare la realtà che ci circonda e analizzarne le relazioni con il linguaggio in cui intendiamo parlarne. Vengono poi discussi i problemi filosofici che la formalizzazione comporta e le loro relazioni con il nostro processo di chiarificazione e conoscenza del mondo. Dopo aver mostrato come tutto ci si risolve nello studio dei rapporti che intercorrono tra la nozione di realtà, quella di linguaggio e quella di logica vengono presentate alcune possibili interpretazioni filosofiche, alcune confutate altre accettate, di questi rapporti. Infine si cerca di caratterizzare brevemente la relazione che il nostro pensiero deve instaurare con il linguaggio formale nel corso dei suoi processi deduttivi cercando così di legare questa nozione a quelle precedentemente introdotte

Degrees of the finite model property: the antidichotomy theorem

A classic result in modal logic, known as the Blok Dichotomy Theorem, states that the degree of incompleteness of a normal extension of the basic modal logic $\sf K$ is $1$ or $2^{\aleph_0}$. It is a long-standing open problem whether Blok Dichotomy holds for normal extensions of other prominent modal logics (such as $\sf S4$ or $\sf K4$) or for extensions of the intuitionistic propositional calculus $\mathsf{IPC}$. In this paper, we introduce the notion of the degree of finite model property (fmp), which is a natural variation of the degree of incompleteness. It is a consequence of Blok Dichotomy Theorem that the degree of fmp of a normal extension of $\sf K$ remains $1$ or $2^{\aleph_0}$. In contrast, our main result establishes the following Antidichotomy Theorem for the degree of fmp for extensions of $\mathsf{IPC}$: each nonzero cardinal $\kappa$ such that $\kappa \leq \aleph_0$ or $\kappa = 2^{\aleph_0}$ is realized as the degree of fmp of some extension of $\mathsf{IPC}$. We then use the Blok-Esakia theorem to establish the same Antidichotomy Theorem for normal extensions of $\sf S4$ and $\sf K4$

Positive Modal Logic Beyond Distributivity

We develop a duality for (modal) lattices that need not be distributive, and use it to study positive (modal) logic beyond distributivity, which we call weak positive (modal) logic. This duality builds on the Hofmann, Mislove and Stralka duality for meet-semilattices. We introduce the notion of $\Pi_1$-persistence and show that every weak positive modal logic is $\Pi_1$-persistent. This approach leads to a new relational semantics for weak positive modal logic, for which we prove an analogue of Sahlqvist correspondence result

The algebraic significance of weak excluded middle laws

Please read abstract in the article.National Research Foundation of South Africa; Ministry of Science and Innovation of Spain; Agència de Gestió d'Ajuts Universitaris i de Recerca.https://onlinelibrary.wiley.com/journal/15213870hj2023Mathematics and Applied Mathematic

Varieties of De Morgan monoids : covers of atoms

The variety DMM of De Morgan monoids has just four minimal subvarieties. The join-irreducible covers of these atoms in the subvariety lattice of DMM are investigated. One of the two atoms consisting of idempotent algebras has no such cover; the other has just one. The remaining two atoms lack nontrivial idempotent members. They are generated, respectively, by 4{element De Morgan monoids C4 and D4, where C4 is the only nontrivial 0{generated algebra onto which nitely subdirectly irreducible De Morgan monoids may be mapped by non-injective homomorphisms. The homomorphic pre-images of C4 within DMM (together with the trivial De Morgan monoids) constitute a proper quasivariety, which is shown to have a largest subvariety U. The covers of the variety V(C4) within U are revealed here. There are just ten of them (all nitely generated). In exactly six of these ten varieties, all nontrivial members have C4 as a retract. In the varietal join of those six classes, every subquasivariety is a variety|in fact, every nite subdirectly irreducible algebra is projective. Beyond U, all covers of V(C4) [or of V(D4)] within DMM are discriminator varieties. Of these, we identify in nitely many that are nitely generated, and some that are not. We also prove that there are just 68 minimal quasivarieties of De Morgan monoids.The European Union's Horizon 2020 research and innovation programme under the Marie Sklodowska-Curie grant, RVO 67985807 and by the CAS-ICS postdoctoral fellowship, the National Research Foundation of South Africa and DST-NRF Centre of Excellence in Mathematical and Statistical Sciences (CoE-MaSS), South Africa.https://www.cambridge.org/core/journals/review-of-symbolic-logic2021-06-01am2021Mathematics and Applied Mathematic

Singly generated quasivarieties and residuated structures

Please read abstract in the article.H2020 Marie Skłodowska-Curie Actions; DST-NRF Centre of Excellence in Mathematical and Statistical Sciences (CoE-MaSS), South Africa and National Research Foundation of South Africa.https://onlinelibrary.wiley.com/journal/15213870hj2021Mathematics and Applied Mathematic

Epimorphisms, definability and cardinalities

We characterize, in syntactic terms, the ranges of epimorphisms in an arbitrary class of similar first-order structures (as opposed to an elementary class). This allows us to strengthen a result of Bacsich, as follows: in any prevariety having at most s non-logical symbols and an axiomatization requiring at most m variables, if the epimorphisms into structures with at most m+s+ℵ0 elements are surjective, then so are all of the epimorphisms. Using these facts, we formulate and prove manageable ‘bridge theorems’, matching the surjectivity of all epimorphisms in the algebraic counterpart of a logic ⊢ with suitable infinitary definability properties of ⊢, while not making the standard but awkward assumption that ⊢ comes furnished with a proper class of variables.The European Union’s Horizon 2020 research and innovation programme under the Marie Sklodowska-Curie Grant Agreement No. 689176 (project “Syntax Meets Semantics: Methods, Interactions, and Connections in Substructural logics”). The first author was also supported by the Project GA17-04630S of the Czech Science Foundation (GAČR). The second author was supported in part by the National Research Foundation of South Africa (UID 85407). The third author was supported by the DST-NRF Centre of Excellence in Mathematical and Statistical Sciences (CoE-MaSS), South Africa.http://link.springer.com/journal/112252020-02-07hj2019Mathematics and Applied Mathematic