In 2016 Béziau, introduces a restricted notion of paraconsistency, the so-called genuine paraconsistency. A logic is genuine paraconsistent if it rejects the laws φ , ¬φ ⊢ ψ and ⊢ ¬(φ ∧ ¬φ). Using the dual properties of those above mentioned, namely ⊢ φ , ¬φ and ¬(ψ V ¬ψ)⊢, in 2021 Borja-Coniglio-Hernádez introduced the notion of genuine paracomplete logics, those rejecting the mentioned properties. In both cases, authors performed an analysis of the notions in the context of three-valued logics.
As it is well known, a paranormal logic is a logic which is both paraconsistent and paracomplete. So it is natural to try to combine the notions of genuine paraconsistency and genuine paranormality mentioned above to obtain a restricted notion of paranormality, genuine paranormality. To capture this notion by using many-valued logics, 4-valued logics will be necessary at least. In this case, we will analyze how many 4-valued genuine paracomplete logics exist, such that are enlargements of classical logic, apart from some other nice features.
Substructural modal logics (extensions of substructural logics with various modal operators) have recently been studied in a wide range of contexts. These systems are the appropriate formalisms for dealing with reasoning that is both fuzzy and uncertain. Usually, the corresponding logics of these kinds of systems are introduced either semantically via a relational semantics, or syntactically, via an axiomatization or class of algebraic structures, but general systematic accounts relating these two perspectives are mostly lacking in the literature.
In this talk we present an algebraic semantics for a system that is substructural and modal [3]. More precisely, we extend FLe-algebra with two modal operators that generalize Bezhanishivili’s pseudomonadic algebras for the modal logic KD45 [1]. It is shown that these structures can be represented by ordered pairs consisting of an FLe-algebra (or commutative pointed residuated lattice) and a subalgebra with a suitable lattice filter, extending a similar result for ‘S5-like’ logics [2]. To relate these algebraic semantics with the relational semantics that determine a possibilistc substructural logic, we prove that if the FLe-algebra reduct belongs to a variety that has the superamalgamation property, then the structure equipped with an additional constant is representable as an algebra defined by a possibilistc substructural frame. We will discuss the relationship between the algebraic and the relational semantics, the advantages of the algebraic approach for the axiomatization of the systems, and the limitations of our approach in certain specific cases of substructural systems.
References
[1] N. Bezhanishvili. Pseudomonadic algebras as algebraic models of doxastic modal logic. Math. Log. Q., 48:624–646, 2002.
[2] P. Cintula, G. Metcalfe, and N. Tokuda. One-variable fragments of first-order logics. Bull. Symb. Log, (30):253–278, 2024.
[3] L. van den Berg, M. Busaniche, M. Marcos, and G. Metcalfe. Towards an algebraic theory of KD45-like logics. In Sedlár Ciabattoni, Gabelaia, editor, Proceedings of AiML 2024, volume 15, pages 171–186. College Publications, 2024.
This talk is based on joint work in collaboration with V. Giustarini and F. Manfucci (IIIA-CSIC and Universitat de Barcelona) aimed at a better understanding of the structure of free Nelson algebras and their connection with free Heyting algebras.
Nelson algebras, introduced by Rasiowa, provide the algebraic semantics for the constructive logic with strong negation developed by Nelson and Markov. They consist of bounded distributive lattices equipped with two additional operations, which can be interpreted as an implication and a strong negation. It is well known that they form a variety and can also be regarded as residuated lattices (see [3] for a recent survey collecting various facts about Nelson algebras). Free Nelson algebras play an important role in the study of the constructive logic with strong negation, as they are, up to isomorphism, the Lindenbaum–Tarski algebras of the logic.
The twist construction establishes a strong connection between Nelson algebras and Heyting algebras. In fact, the category of Nelson algebras is equivalent to a category whose objects are pairs consisting of a Heyting algebra and a Boolean filter [5]. We will see that this equivalence can be viewed as a restriction of an adjunction involving the category of Kleene lattices, and that this perspective allows us to describe the pair corresponding to a Nelson algebra free over a Kleene lattice using Heyting algebras free over bounded distributive lattices. Since free Kleene lattices are well understood, this provides an algebraic description of free Nelson algebras in terms of Heyting algebras free over bounded distributive lattices.
The connection between Nelson algebras and Heyting algebras via the twist construction also yields a duality for Nelson algebras reminiscent of Esakia duality for Heyting algebras [5]. We will exploit this duality to provide a tangible description of free algebras in various varieties of Nelson algebras. In particular, we will use the dual description of free Gödel algebras from [2] to describe the duals of free algebras in every variety of NM algebras (the algebraic counterparts of the nilpotent minimum logic, which can be thought of as particular Nelson algebras [1]), without any restriction on the cardinality of the set of free generators. We will also utilize a similar approach, together with the well known dual description of free pseudocomplemented distributive lattices, to provide a dual description of free algebras in varieties of implication-free subreducts of Nelson algebras introduced in [6] and further studied in [4].
References
[1] Busaniche, M., & Cignoli, R., Constructive logic with strong negation as a substructural logic, Journal of Logic and Computation, 20(4) (2010), 761–793.
[2] Carai, L., Free algebras and coproducts in varieties of Gödel algebras, 2024. Manuscript. Available at arXiv:2406.05480.
[3] Järvinen, J., Radeleczki, S., & Rivieccio, U., Nelson algebras, residuated lattices and rough sets: A survey, Journal of Applied Non-Classical Logics, 34(2–3) (2024), 368–428.
[4] Gomez, C., Marcos, M. A., & San Martin, H. J., On the relation of negations in Nelson algebras, Reports on Mathematical Logic, 56 (2021), 15–56.
[5] Sendlewski, A., Nelson algebras through Heyting ones: I, Studia Logica, 49 (1990), 105–126.
[6] Sendlewski, A., Topologicality of Kleene algebras with a weak pseudocomplementation over distributive p-algebras, Reports on Mathematical Logic, 25 (1991), 13-56.
A hyperoperation is a function which produces, for a given input, a set of possible outcomes rather than a single one. Hyperstructures (a.k.a. hyperalgebras or multialgebras) are structures formed by hyperoperations over a given domain. Hyperstructures were introduced by the French mathematician Frédéric Marty in 1934 in the 8th Congress of Scandinavian Mathematicians, by means of hypergroups. Marty's pioneering work laid the foundation for the development of hypergroup theory and the broader study of hyperstructures. In the context of non-classical logics, hyperalgebras play a pivotal role by providing a flexible semantical framework which intends to expand the horizons of Abstract Algebraic Logic. In the context of ordered structures, the concept of hyperlattices was first introduced in Romania by M. Benado in 1953.
The Portuguese mathematician José Morgado, in his book from 1962, proposed a new definition of hyperlattice, that he called 'Reticuloids'. Arguably, his notion of hyperlattice is much more natural than Benado's (and the other variants considered up to now in the literature), as it generalizes the usual lattice definitions by using pre-ordered structures instead of ordered ones. Based on Morgado hyperlattices, the Brazilian mathematician Antonio M. Sette proposed in his 1971 Master’s thesis at UNICAMP, supervised by Newton da Costa, the notion of implicative hyperlattices (here called SIHLs). Sette further extended SIHLs by defining a unary hyperoperator for interpreting a paraconsistent negation, leading to the formulation of a class of hyperalgebras (here called SHCws), providing an algebraic characterization of the paraconsistent logic Cw.
Building on recent advances in the theory of swap structures, and inspired by the ideas of Sette and Morgado, in the first part of this talk we revisit Morgado's hyperlattices and Sette's implicative hyperlattices, providing a lattice-theoretic characterization and establishing foundational results on SIHLs. We introduce a class of swap structures, an special class of hyperalgebras over the signature of Cw naturally induced by implicative lattices. It is proven that these swap structures are SHCws, being so a very intuitive class of models for these hyperalgebras. Then it is proven, for the first time, that the class of SHCws, as well as the above mentioned class of swap structures, semantically characterize the logic Cw. After this, based on the novel notion of hyper swap structures, we obtain an equivalence between the category of SIHLs and a subcategory of SHCws that satisfy additional properties.
As a second example of application of this powerful framework, we introduce a (non-normal) Ivlev-like modal expansion of Cw by means of two (independent) modalities Box and Diamond which satisfy axiom T, called CTw. Both modalities are independent, since CTw is based on positive intuitionistic logic. By defining hyper swap structures over SHCws, we obtain an equivalence between the category of SHCws and a subcategory of the category of hyperalgebraic models of CTw that satisfy additional properties. From this, soundness and completeness of CTw is obtained w.r.t. its class of hyperalgebraic models and, in particular, w.r.t. hyper swap structures.
Both equivalences of categories generalize, to the hyperstructures setting, similar results obtained by means of Kalman's functors relating categories of ordered algebras, by using twist structures constructions. This fact gives us support for the claim that swap structures correspond to non-deterministic twist structures. This is a joint work with Ana C. Golzio and Kaique M. Roberto.
How to identify a logical theory? It was common to assume that the central object of study of logic is a certain body of absolutely general facts about the world, the so-called logical truths. In this view, logical consequence, or validity, is understood as a derivative notion: an inference from statements φ1 , … , φn to statement ψ is valid just in case the conditional having the conjunction of the φis as its antecedent and ψ as its consequent is a logical truth. Logical theories, then, are identified solely by the logical truths they sanction: if two logical theories sanction the same logical truths, they are the same logical theory. As time passed, this view has slowly but surely fell into disuse. One reason for this was, of course, the increasing influence of Tarski’s works on model-theory, where logical consequence takes central stage. But another reason was that the view cannot properly account for many logical theories that have been progressively accepted as legitimate. A common example is the Asenjo-Priest logic of paradox, LP (see [1, 10]). In this logic, the reduction of logical consequence to logical truth fails, as there are logically true conditionals whose corresponding inference is invalid. More importantly, LP has the same logical truths as classical logic, CL, but it has different valid inferences. Indeed, the two logics allow for the formulation of different non-logical theories. So, identifying the two logics on the basis of their having the same logical truths seems out of the picture. Examples like these motivate the view that logical theories should be identified by the valid inferences they sanction: if two logical theories sanction the same valid inferences, they are the same logical theory. Logical truths are still accounted for in this view, but now they are understood as a limit case of logical consequence: a statement ψ is a logical truth just in case it follows from the empty collection of premises.
In the past few years, considerable attention has been paid to the strict-tolerant logic ST, championed by Cobreros et. al. [4, 3] and Ripley [11, 12]. This logic has contributed to what arguably is a new turn in our understanding of logical theories. In particular, the existence of ST has been used to argue that logical theories should not be identified by looking only at their valid inferences, but also at their valid metainferences. Intuitively, metainferences are just inferences whose premises and conclusion are themselves inferences. We often display them in a rule-like fashion. Thus, for instance, if ‘⇒’ stands for consequence, then the following expressions:
Id C φ, φ, Γ ⇒ δ
φ ⇒ φ φ, Γ ⇒ δ
Cut Γ ⇒ φ φ, Σ ⇒ δ W Γ ⇒ δ
Γ, Σ ⇒ δ φ, Γ ⇒ δ
represent metainferences. Here, Γ and Σ are collections of formulas, and labels ‘Id’, ‘C’ and ‘W’ stand for ‘Identity’, ‘Contraction’ and ‘Weakening’, respectively. It is common to distinguish between two kinds of metainferences, based on the expressions they display. Metainferences which mention some logical constant of the object language, are called operational. Metainferences which, like the above, do not mention any such constant, are called structural. It is said that the former govern some specific pieces of logical vocabulary, while the latter regulate the behavior of the very notion of consequence.
There are at least two informal readings that these beasts can receive. Under one of them, a metainference says that, if the premise-inferences are valid, the conclusion-inference is valid as well. Under the other reading it says that any counterexample to the conclusion-inference is also a counterexample to some of the premise-inferences. Each of these readings has an associated model-theoretic definition of metainferential validity; the definition associated to the former is called global, the one associated to the latter is called local (see e.g. [6],[5]). Now, usually, CL is taken as the paradigmatic case of what is known as a structural logic. In contrast, logics that in some sense invalidate some of the structural metainferences that CL validates are called substructural. The main trait of logic ST is that it has the same valid inferences as CL, but it has different valid metainferences. In particular, it is substructural, since it locally invalidates the principle of Cut, which is meant to encode the transitivity of logical consequence; this is why ST is often understood as a non-transitive counterpart of CL. Now, the difference in the valid metainferences is relevant since, just as in the case of LP, logics ST and CL allow us to formulate different non-logical theories. Crucially, ST is compatible with non-trivial, naive theories of various paradoxical phenomena that trivialize CL. And there is some agreement that, in view of this, ST and CL cannot be taken as the same logical theory (see e.g.[2, 9].)
One potential concern about the above line of argument is its limited scope. After all, CL and ST constitute just one pair of logics where the approach based on inferences underperforms. Absent any additional cases of this nature, there may not be sufficient evidence to conclude that, in general, logical theories should be identified partly by their valid metainferences. Instead, one may prefer to stick to the familiar approach based on inferences, and take the pair of CL and ST as a mere anomaly, which the less familiar creatures called metainferences help to address. Recently, the works of Fitting [7, 8] and Szmuc [13] provided results that help to address this concern. Specifically, a slight correction of Szmuc’s result shows that, for every non-trivial Tarskian logic, there is a non-transitive counterpart whose notion of consequence resembles that of ST, and which validates the same inferences but invalidates Cut. So, the case of CL is not an anomaly: there is a wide range of logics that fall within the ST phenomenon. This bolsters the argument in favor of taking metainferences seriously when it comes to the identification of logical theories.
In this talk we widen the scope of the argument much further, by showing that transitivity is not the only structural rule that can be ‘dropped’ from Tarskian logics in general, and classical logic in particular. Indeed, we prove that for every system in the class of what we call coherent Tarskian logics—and this class includes classical logic among many others—there is a ‘maximally substructural’ counterpart: a system that validates the same inferences, but locally invalidates the structural principles of Cut, Contraction, and Weakening. We claim that our results put to rest any remaining doubts about the relevance of metainferences for identifying logical theories. The valid inferences of a logical theory strongly underdetermine the valid metainferences, and the latter have an impact on the non-logical applications the theory might have. So, metainferences must be part of our regular assessment of logical theories.
References
[1] Florencio G Asenjo. A calculus of antinomies. Notre Dame Journal of Formal Logic, 7(1):103–105, 1966.
[2] Eduardo Barrio, Federico Pailos, and Damian Szmuc. Substructural Logics, Pluralism and Collapse. Synthese, 198(20):4991–5007, 2021.
[3] P. Cobreros, P. Egré, E. Ripley, and R. Van Rooij. Reaching Transparent Truth. Mind, 122(488):841–866, 2013.
[4] Pablo Cobreros, Paul Egré, Ellie Ripley, and Robert van Rooij. Tolerant, Classical, Strict. Journal of Philosophical Logic, 41(2):347–385, 2012.
[5] Bruno Da Ré, Damian Szmuc, and Paula Teijeiro. Derivability and Metainferential Validity. Journal of Philosophical Logic, pages 1–27, 2021.
[6] Bogdan Dicher and Francesco Paoli. ST, LP and tolerant metainferences. In Graham Priest on dialetheism and paraconsistency, pages 383–407. Springer, 2019.
[7] Melvin Fitting. A family of strict/tolerant logics. Journal of Philosophical Logic, 50:363–394, 2021.
[8] Melvin Fitting. The strict/tolerant idea and bilattices. In Arnon Avron on Semantics and Proof Theory of Non-Classical Logics, pages 167–191. Springer, 2021.
[9] Brian Porter. Supervaluations and the strict-tolerant hierarchy. Journal of Philosophical Logic, pages 1–20, 2021.
In this talk, we present recent results obtained by the research group at the Universidad Nacional del Sur, which I currently lead. More precisely:
Term-Defined Nuclei on Residuated Lattices. In our recent work “Terms that define nuclei on residuated lattices: A case study of BL-algebras” (Fuzzy Sets and Systems), we provide conditions under which a nucleus can be defined by a term. We also give a complete description of the subvarieties of BL-algebras that admit nontrivial nuclei, together with the corresponding defining terms.
Predicate Calculi Based on Continuous t-Norms. In the paper “Strong completeness for the predicate logic of the continuous t-norms” (Fuzzy Sets and Systems), we present an infinitary axiomatic system that is complete with respect to continuous t-norms, achieved via a modified Henkin construction along with embeddings of BL-chains into continuous t-norms.
Standard Completeness for S5-Modal Łukasiewicz Logics. In “Strong standard completeness theorems for S5-modal Łukasiewicz logics” (Annals of Pure and Applied Logic), we establish both strong and finitely strong completeness theorems for monadic Łukasiewicz logics and some of their extensions with respect to the Łukasiewicz t-norm, using an infinite amalgamation theorem and the finite embeddability property.
References
S. Buss, D. Castaño, J. P. Díaz Varela. “Terms that define nuclei on residuated lattices: A case study of BL-algebras.” Fuzzy Sets and Systems, vol. 519, 2025.
D. Castaño, J. P. Díaz Varela, G. Savoy. “Strong completeness for the predicate logic of the continuous t-norms.” Fuzzy Sets and Systems, vol. 500, 2025.
D. Castaño, J. P. Díaz Varela, G. Savoy. “Strong standard completeness theorems for S5-modal Łukasiewicz logics.” Annals of Pure and Applied Logic, vol. 176(3):103529, 2025.
D. Castaño, C. R. Cimadamore, J. P. Díaz Varela, L. A. Rueda. “Completeness for monadic fuzzy logics via functional algebras.” Fuzzy Sets and Systems, vol. 407, 2021, pp. 161–174.
In ‘Logical metainferentialism’ (Ergo, forthcoming), Dicher & Paoli develop a theory of harmony for metainferential calculi in the FDE family, which includes the identification of a certain normal form, called there ‘structurally atomic-analytic synthetic’ (SAAS) normal form as the mark of harmony. In this paper I will explore the possibility of using the same normal form as a mark of harmony in other metainferential proof systems and discuss some of the philosophical consequences of this investigation.
Fuzzy logics are logics of graded truth that have been proposed as a suitable tool for reasoning with imprecise information, in particular for reasoning with propositions containing vague predicates. Their main feature is that they allow to interpret formulas in a linearly ordered scale of truth-values, and this is specially suited for representing the gradual aspects of vagueness. In particular, systems of fuzzy logic have been in-depth developed within the frame of mathematical fuzzy logic [3] (MFL). Most well known and studied systems of mathematical fuzzy logic are the so-called t-norm based fuzzy logics, corresponding to formal many-valued calculi with truth-values in the real unit interval [0, 1] and with a conjunction and an implication interpreted respectively by a (left-) continuous t-norm and its residuum, and thus, including e.g. the well-known Łukasiewicz and Godel infinitely-valued logics, corresponding to the calculi defined by Łukasiewicz and min t-norms respectively. The most basic t-norm based fuzzy logic is the logic MTL (monoidal t-norm based logic). In logical systems in MFL, the usual notion of deduction is defined by requiring the preservation of the truth-value 1 (full truth-preservation), which is understood as representing the absolute truth. For instance, let L be any extension of MTL, which we assume to be complete w.r.t. the family CL = {[0, 1]∗ | [0, 1]∗ is a L-algebra} of standard L-algebras. Then the typical notion of logical consequence is the following for every set of formulas Γ ∪ {φ}:
Γ ⊨L φ if, for any [0, 1]∗ ∈ CL and any [0, 1]∗-evaluation e, if e(ψ) = 1 for any ψ ∈ Γ, then e(φ) = 1 as well.
In [2], Bou, Esteva et al. introduced the degree preserving MTL-logics where they change the (full) truth paradigm to the degree preserving paradigm, in which a conclusion follows from a set of premises if, for all evaluations, the truth degree of the conclusion is greater or equal than those of the premises. For any extension L of MTL complete w.r.t. the family CL of standard L-algebras the degree preserving variant of L, denoted by L⩽ is defined as
Γ ⊨⩽L φ if, for any [0, 1]∗ ∈ CL, any [0, 1]∗-evaluation e and for any a ∈ [0, 1], if e(ψ) ⩾ a for any ψ ∈ Γ, then e(φ) ⩾ a.
As a matter of fact, the degree preserving logic L⩽ is strongly related to the 1-preserving logic L. Indeed, on the one hand, it holds that ⊨⩽L φ iff ⊨L φ, so both logics share the set of valid formulas. Moreover, if for any finite set of formulas Γ we let Γ^ = ∧{ψ | ψ ∈ Γ}, we can observe that
Γ ⊨⩽L φ iff ⊨L Γ^→ φ,
and hence, iff ⊨⩽L Γ^→ φ. This property can be seen as a sort of deduction theorem for ⊨⩽L.
Still, another way of defining different variants of a fuzzy logic is put forward in [1], although for the particular case somehow related to a fragment of Łukasiewicz fuzzy logic. In this approach, the notion of consequence at work is the non-falsity preservation, according to which a conclusion follows from a set of premises whenever if the premises are non-false, so must be the conclusion. See also [6] for the case of Nilpotent Minimum logic and [4] for preliminary more general results.
The purpose of this paper is to describe axiomatisations for other variants of (truth-preserving) MTL logics whose consequence relations are based on the preservation of weaker notions of truth. In particular we plan to deal with three types of variants for any extension L of MTL complete w.r.t. a family CL of standard L-algebras:
Non-falisty preserving variant nf-L: this captures when, for any evaluation, if truth degrees of the premises are above 0, then the truth-degree of the conclusion is so as well. This idea gives rise to the following notion of consequence:
Γ ⊨(0L φ if, for any [0, 1]∗ ∈ CL, any [0, 1]∗-evaluation e, if e(ψ) > 0 for any ψ ∈ Γ, then e(φ) > 0.
Acceptance preserving variant: this variant captures the idea of reasoning that preserves the acceptability of formulas, in the sense that they are evaluated higher than their negations. Fornally, we define:
Γ ⊨accL φ if, for any [0, 1]∗ ∈ CL, any [0, 1]∗-evaluation e, if e(ψ) > e(¬φ) for any ψ ∈ Γ, then e(φ) > e(¬φ).
Threshold preserving variants: with this we refer to logics preserving lower bounds of truth-values, namely, fix some positive value a ∈ (0, 1], and define the logic La as follows:
Γ ⊨aL φ if, for any [0, 1]∗ ∈ CL, any [0, 1]∗-evaluation e, if e(ψ) ⩾ a for any ψ ∈ Γ, then e(φ) ⩾ a.
Similarly one can define logics that preserve strict lower bounds ⊨(aL for a ∈ [0, 1).
Actually, the framework of threshold-preserving logics is more general, as non-falsity preserving logics are a particular case of the latter when a = 0, and acceptance logics correspond to logics ⊨aL when L is a involutive MTL logics whose negation has a fix point is a.
Acknowledgments The authors acknowledge support by the MOSAIC project (EU H2020-MSCA-RISE Project 101007627). Gispert acknowledges partial support by the Spanish project SHORE (PID2022-141529NB-C21) while Esteva and Godo by the Spanish project LINEXSYS (PID2022-139835NB-C21), both funded by MCIU/AEI/10.13039/501100011033. Gispert also acknowledges the project 2021 SGR 00348 funded by AGAUR.
References
[1] A. Avron. Paraconsistent fuzzy logic preserving non-falsity. Fuzzy Sets and Systems 292, 75-84, 2016.
[2] F. Bou, F. Esteva, J. M. Font, A. Gil, L. Godo, A. Torrens and V. Verdú. Logics preserving degrees of truth from varieties of residuated lattices. Journal of Logic and Computation, 19, 6, pp: 1031-1069, 2009.
[3] P. Cintula, C. Noguera C. A general framework for mathematical fuzzy logic. In: Cintula P, Hájek P, Noguera C (eds) Handbook of Mathematical Fuzzy Logic - Volume 1. Studies in logic, Mathematical Logic and Foundations, vol 37. College Publications, London, pp. 103-207.
[4] F. Esteva, J. Gispert, L. Godo. On the paraconsistent companions of involutive fuzzy logics that preserve non-falsity. M.J. Lesot et. al (Eds.), Proceedings of the 20th International Conference of Information Processing and Management of Uncertainty (IPMU 2024), Lecture Notes in Networks and Systems 1175, Springer, pp. 375-389, 2025.
[5] F. Esteva, J. Gispert, L. Godo. On the non-falsity and threshold preserving variants of MTL logics. In M. Baczynski et al. (Eds.), Proceedings of the 13th Conference of the European Society for Fuzzy Logic and Technology, EUSFLAT 2025. Lecture Notes in Computer Science, to appear.
[6] J. Gispert, F. Esteva, L. Godo, M.E. Coniglio. On Nilpotent Minimum logics defined by lattice filters and their paraconsistent non-falsity preserving companions. Logic Journal of the IGPL, Volume 33, Issue 3, June 2025, jzae126, https://doi.org/10.1093/jigpal/jzae126
Contemporary research on substructural logic makes heavy use of (higher-level) meta-inferences—both as a tool for constructing logics, and as a lens through which to study them. Following influential arguments by e.g. Barrio et al. (2020); Dicher & Paoli (2019) and Golan (2021), it became widely accepted that the appropriate criterion of validity for metainferences is not global, but local. Very recently, this prevailing view has come under pressure, with Cobreros (2024) and Kortenbach (2025) proposing various notions of global validity which avoid the known objections. However, thus far these new notions have only been defined for classes of metainferences which are relatively limited, compared to the wide variety that is, by now, being studied in the literature. The goal of this talk is to bring global back up to speed, by exploring how to extend the new global criteria to more complex metainferential settings.
Schematic and hyperschematic global-global (sGG and hGG) are defined by Kortenbach (2025) only for single-conclusion, non-cumulative metainferences of finite levels. We will argue that in the known (Da Ré et al., 2020) choice one faces when applying global to multiple-conclusions, sGG and hGG force our hand. Secondly, we will present a serious difficulty that arises when generalizing sGG and hGG to the cumulative metainferences used by e.g. Ferguson & Ramírez-Cámara (2022), and motivate a solution. Lastly, we’ll show that once cumulative inferences are dealt with, extending sGG and hGG to transfinite levels (Scambler, 2020) is comparatively straightforward.
Cobreros’ (2024) favoured p-global is only defined for metainferences of level 2, and the challenge lies already in extending it to arbitrary finite levels. While p-global is equivalent to what Da Ré et al. (2022) call absolute global validity over the powerset, which enjoys obvious extensions to arbitrary levels, we submit that said extensions don’t fit with Cobreros’ motivating interpretation of p-global. We’ll propose an alternative generalization which does, though noting that it is technically complex, and that the extended interpretation requires eccentric linguistic assumptions.
We close with a comparative evaluation of sGG and hGG as opposed to p-global, and by drawing some general lessons for the global approach in relation to substructural logic.
References
Barrio, E., Pailos, F., & Szmuc, D. (2020). A hierarchy of classical and paraconsistent logics. Journal of Philosophical Logic, 49 (1), 93–120.
Cobreros, P. (2024). Una oda a los árboles. Pamplona: Servicio de Publicaciones de la Universidad de Navarra.
Da Ré, B., Pailos, F., Szmuc, D., & Teijeiro, P. (2020). Metainferential duality. Journal of Applied Non-Classical Logics, 30 (4), 312–334.
Da Ré, B., Szmuc, D., & Teijeiro, P. (2022). Derivability and metainferential validity. Journal of Philosophical Logic, 51 , 1521–1547.
Dicher, B., & Paoli, F. (2019). ST, LP and tolerant metainferences. In C. Başkent & T. M. Ferguson (Eds.), Graham priest on dialetheism and paraconsistency (pp. 383–407). Dordrecht: Springer.
Ferguson, T. M., & Ramírez-Cámara, E. (2022). Deep ST. Journal of Philosophical Logic, 51, 1261–1293.
Golan, R. (2021). There is no tenable notion of global metainferential validity. Analysis, 81 (3), 411–420.
Kortenbach, B. (2025). Appreciating global validity. Synthese, forthcoming.
Scambler, C. (2020). Transfinite meta-inferences. Journal of Philosophical Logic, 49 (6), 1079–1089.
Nelson's constructive logic with strong negation, denoted by N , is a well-known non-classical logic that combines the constructive approach of positive intuitionistic logic with a classical negation (see [6]). The algebraic models of N form a variety called Nelson algebras or Nelson residuated lattices (see [8]). One of the main algebraic insights on this variety came with the realisation (independently due to M. Fidel and D. Vakarelov, see [11]) that every Nelson algebra can be represented as a special binary product (called a twist-structure) of a Heyting algebra. This correspondence was formulated as a categorical equivalence (see [10]) between Nelson algebras and a category of enriched Heyting algebras, which made it possible to transfer a number of fundamental results from the more widely studied theory of intuitionistic algebras to the realm of Nelson algebras.
Subresiduated lattices, which are a generalization of Heyting algebras, were introduced by G. Epstein and A. Horn [4] as an algebraic counterpart of some logics with strong implication previously studied by C. Lewy and I. Hacking [5]. These logics are examples of subuintuitionistic logics, i.e., logics in the language of intuitionistic logic that are defined semantically by using Kripke models, in the same way as intuitionistic logic is defined, but without requiring of the models some of the properties required in the intuitionistic case. Also in relation with the study of subintuitionistic logics, Celani and Jansana [2] got these algebras as the elements of a subvariety of weak Heyting algebras (see also [1, 3]). It is known that the variety S4, whose members are the S4-algebras, is the algebraic semantics of the modal logic S4. This means that φ is a theorem of S4 if and only if the variety S4 satises φ ≈ 1. The variety of subresiduated lattices corresponds to the variety of algebras defined for all the equations φ ≈ 1 satised in the variety S4 with the connectives conjunction ∧, disjunction ∨, bottom ⊥, top ⊤ and a new connective of implication ⇒, called strict implication, defined by φ ⇒ ψ := 1 → (φ → ψ), where → denotes the classical implication.
The main goal of the talk is to extend the Vakarelov's construction, which establishes a link between Heyting algebras and Nelson algebras in the framework of subresiduated lattices. More precisely, we give a variety of algebras, whose members will be called subresiduated Nelson algebras, for which the original Vakarelov's construction can be extrapolated by considering subresiduated lattices and subresiduated Nelson algebras, respectively. We will characterize the lattice of congruences of a subresiduated Nelson algebra through some of its implicative lters. We use this characterization to describe simple and subdirectly irreducible algebras, as well as principal congruences. Moreover, we prove that the variety of subresiduated Nelson algebras has equationally denable principal congruences and also the congruence extension property. Additionally, we present an equational base for the variety generated by the totally ordered subresiduated Nelson algebras. Finally, we show that there exists an equivalence between the algebraic category of subresiduated lattices and the algebraic category of centedred subresiduated Nelson algebras. These results were published in [7].
References
[1] Castiglioni J.L., Fernández V., Mallea H.F. and San Martín H.J., On subreducts of subresiduated lattices and logic. Journal of Logic and Computation 34, No. 856-886 (2024)
[2] Celani S. and Jansana R., Bounded distributive lattices with Strict implication. Math. Log. Quart. 51, No. 3, 219-246 (2005).
[3] Celani S., Nagy A. and San Martín H.J., Dualities for subresiduated lattices. Algebra Universalis 82, article number 59 (2021).
[4] Epstein G. and Horn A., Logics wich are characterized by subresiduated lattices. Z. Math. Logik Grundlagen Math. 22, 199-210 (1976).
[5] Hacking I., What is strict implication? J. Symb. Logic 28, 51-71 (1963).
[6] Nelson D., Constructible falsity. Journal of Symbolic Logic 14, 16-26 (1949).
[7] Lubomirsky N., Menchón M.P. and San Martín H.J., Subresiduated Nelson algebars. Fuzzy Sets and Systems 498, 109-170 (2025).
[8] Rasiowa H., An algebraic approach to non-classical logics, volume 78 of Studies in Logic and the Foundations of Mathematics. North-Holland, Amsterdam (1974).
[9] Sendlewski A., Some investigations of varieties of N-lattices. Studia Logica 43, 257-280 (1984).
[10] Sendlewski A., Nelson algebras through Heyting ones. I. Studia Logica 49, 105-126 (1990).
[11] Vakarelov D., Notes on N-lattices and constructive logic with strong negation. Studia Logica 36, 109-125 (1977).
Betweenness relations—well-known from geometry—are among the most thoroughly studied ternary relations in logic and mathematics. Their systematic investigation can be traced back at least to the works of Huntington and Kline (1917; 1924), followed by the seminal contributions of Tarski (Givant and Tarski, 1999), and continued through the results of Altwegg (1950), Sholander (1952), Düvelmeyer and Wenzel (2004), and Düntsch and Urquhart (2006). In Düntsch et al. (2023, 2024), we initiated a theory of betweenness algebras as a framework to study the algebraic and logical properties of the ternary relation of betweenness. Our approach is based on a logic K# equipped with two binary modalities which in some sense complement each other. The logic and its corresponding algebraic models—Boolean algebras with two binary operators—can be seen as a binary extension of the mixed logic of Düntsch et al. (2017) and its expressive power extends vastly that of the classical modal logic.
Formally, let U be a non-empty set and B ⊆ U3. After (Düntsch and Urquhart, 2006) B will be called a betweeness relation iff it satisfies the following (universal) axioms:
B(a, a, a) (BT0)
B(a, b, c) → B(c, b, a) (BT1)
B(a, b, c) → B(a, a, b) (BT2)
B(a, b, c) ∧ B(a, c, b) → b = c . (BT3)
A ternary relation B is a weak betweenness iff it satisfies (BT0)–(BT2) and
B(a, b, a) → a = b . (BTW)
F := ⟨U, B⟩ is a betweenness frame iff it satisifies (BT0)–(BT3). F is a weak betweenness frame iff it satisfies (BT0)–(BT2) and (BTW). It is easy to see that due to (BT2) and (BT3) any betweenness relation must be a weak betweenness.
Since the class of weak betweenness relations is not modally axiomatizable (Düntsch and Orłowska, 2019), the algebraic investigation of betweenness requires suitable expansions of the standard modal algebras ⟨A, f⟩ equipped with a binary possibility operator f : A × A → A. In this setting, we consider algebras of the form ⟨A, f, g⟩, where g : A × A → A is a sufficiency operator in the sense of (Düntsch and Orłowska, 1999).
In the presentation I will introduce the class of betweenness algebras and the logic K# and will report some of its properties as developed in Düntsch et al. (2023, 2024).
References
Altwegg, M. (1950). Zur Axiomatik der teilweise geordneten Mengen. Commentarii Mathematici Helvetici, 24(1):149–155.
Düntsch, I., Gruszczyński, R., and Menchon, P. (2023). Betweenness algebras. The Journal of Symbolic Logic, pages 1–25.
Düntsch, I., Gruszczyński, R., and Menchón, P. (2024). A mixed logic with binary operators. arXiv preprint arXiv:2408.09581.
Düntsch, I. and Orłowska, E. (1999). Mixing modal and sufficiency operators. Bulletin of the Section of Logic, 28(2):99–107.
Düntsch, I. and Orłowska, E. (2019). Betweenness structures. Manuscript.
Düntsch, I., Orłowska, E., and Tinchev, T. (2017). Mixed algebras and their logics. Journal of Applied Non-Classical Logics, 27(3-4):304–320.
Düntsch, I. and Urquhart, A. (2006). Betweenness and Comparability Obtained from Binary Relations. In Schmidt, R. A., editor, Relations and Kleene Algebra in Computer Science, pages 148–161, Berlin, Heidelberg. Springer Berlin Heidelberg.
Düvelmeyer, N. and Wenzel, W. (2004). A characterization of ordered sets and lattices via betweenness relations. Results in Mathematics, 46(3-4):237–250.
Givant, S. and Tarski, A. (1999). Tarski’s system of geometry. The Bulletin of Symbolic Logic, 5(2):175–214.
Huntington, E. V. (1924). A new set of postulates for betweenness, with proof of complete independence. Transactions of the American Mathematical Society, 26(2):257–282.
Huntington, E. V. and Kline, J. R. (1917). Sets of independent postulates for betweenness. Transactions of the American Mathematical Society, 18(3):301–325.
Sholander, M. (1952). Trees, lattices, order, and betweenness. Proceedings of the American Mathematical Society, 3(3):369–381.
Despite validity usually being taken to be a bivalent property, Barrio, Fiore, and Pailos (2025) argue for the controversial thesis that it is not the case that every in- ference is either valid or invalid, and never both. The main reason for adopting this standpoint is the claim that if one endorses a many-valued semantics for the object language, then one likely has good reasons to also endorse a many-valued notion of validity. They present several logical systems based on Belnap’s algebra 4, whose notion of validity is non-bivalent—meaning that there are inferences that are both valid and invalid, and/or inferences that are neither valid nor invalid—and show that, under natural assumptions, the validity of metainferences is also non-bivalent in many of these systems. This makes room for new forms of substructurality.
Nevertheless, the scope of their approach is still limited, as every logic they explore is, in a sense, close to the one presented in Chemlá, Egré, and Spector (2017): mixed and therefore with an extensional semantics. In this talk, we will expand their approach and prove that non-mixed and inferentially non-bivalent logical systems are possible. We will support this claim by defining a new logic based on the connexive logic C (Wansing (1998)), called NBC (for “inferentially Non-Bivalent version of C”).
Both of these logics have intensional semantics defined in terms of possible worlds. In order to achieve our goal, we will not only define our validity criterion based on the truth conditions for the connexive conditional, but also an invalidity standard based on the falsity conditions of the conditional. NBC’s validity and invalidity behavior will thus be reflected in the conditional’s validities and invalidities. Moreover, its metainferential behavior will also be paired with the conditional. Therefore, NBC will not only be sententially, inferentially, and metainferentially contradictory, but also substructural and suprastructural, in just the ways to be expected from a connexive point of view.
We explore certain algebraic structures that naturally emerge within the framework of logics of synonymy, analytic containment, and hyperintensionality. In particular, we argue that Angell’s logic AC, one of the earliest and most successful attempts to analyse the properties of logical constants with a topic-transformative character, can be better understood through a direct algebraic study of De Morgan bisemilattices. Inter alia, we show that a certain 9-element algebra introduced by Ferguson generates De Morgan bisemilattices as a quasivariety, making it the most adequate semantics for AC, as opposed to other 7-element and 16-element algebras considered in the literature.
Methods for probability updating, of which Bayesian conditionalization is the most well-known and widely used, are modeling tools that aim to represent the process of modifying an initial epistemic state. This state is typically represented by a prior probability function P, which is adjusted in light of new information. Notably, updating methods and conditional sentences seem to intuitively share a deep connection, as is evident in the case of conditionalization. Starting with a probability P, the question of what the probability of an event B is, given that A holds (i.e., P(B | A)), appears to involve conditional reasoning of the form "if A then B". Indeed, Adams [1] argues that the assertability of (indicative) conditionals, represented as "if A then B", aligns with the corresponding conditional probability P(B | A). However, this intuitive connection between conditionals and updating methods has been challenged by Lewis’s celebrated triviality result [18]. This result demonstrates that conditional probability P(B | A) cannot generally be equated with the probability of the corresponding conditional, P(A ▷B), without leading to trivializing constraints on the initial probability functions.
Building upon these foundational works, research on the relationship between updating methods and conditionals has been and continues to be prolific. This research has primarily focused on understanding whether and how updating methods can be semantically represented in terms of conditional connectives, or conversely, what kind of updating methods are associated with specific semantic conditional operators (see, for instance, [20, 9, 14, 10, 7, 21]).
The present work contributes to this line of research and aims at shedding new light on the relationship between updating methods and conditional connectives. Departing from previous literature that often focused on a specific type of conditional or a particular updating method, our goal is to prove general results concerning the connection between conditionals and their probabilities. This will allow us to characterize the probabilities of certain conditional connectives and to understand what class of updating procedures can be represented using specific conditional connectives. Broadly, we adopt a general perspective that encompasses a large class of conditionals and a wide range of updating methods, enabling us to prove some general results concerning their interrelation.
Let us begin by providing some background. As observed in [13] (see also [9]), conditionalization updating can be situated within the broader context of imaging updating methods. These latter methods were initially introduced by Lewis [18] as a way to circumvent his triviality result and it has been extended and generalized by several authors, notably by Gärdenfors [12]. While conditionalization is an updating method that requires only an algebra of events and a (positive) probability function, imaging methods update an a prior probability by employing a closeness relation among possible worlds (the atoms of the algebra), often interpreted as a measure of similarity. This closeness relation imposes a certain structure on the underlying set of possible worlds that is employed within the context of imaging updating methods to determine which possible worlds the probability mass should be redistributed to, typically the closest (most similar) ones to a given one.
Such a closeness relation can also be used to specify the truth conditions of a wide range of conditional connectives, such as variably strict conditionals [17] and preferential conditionals [3, 19, 11]. The informal idea is that a conditional a ▷ b is true at a world w if and only if b is true in the closest a-worlds to w, i.e., those closest worlds that make a true. The structure of closeness has been represented through various model-theoretic frameworks, among which ordered models and sphere models are particularly noteworthy. Sphere models were prominently used by Lewis [17] as the intended semantic device for specifying the truth conditions of variably strict conditionals (including counterfactual conditionals), while ordered models have been used, for example, to specify the truth conditions of preferential conditionals. Consequently, this closeness relation of possible worlds has been employed both in specifying imaging-like updating methods and in establishing a semantics for a large class of conditional connectives. Hence, intuitively, one might expect a deep relationship between imaging-like updating methods and conditionals based on a closeness relation, given that both rely on the same structure.
Our investigation will specifically focus on these two dimensions: imaging-like updating methods and conditionals, both having a semantics grounded on a closeness relation. Given our aim for generality, we will slightly deviate from the standard model-theoretic approach to conditionals and updating methods that relies on sphere models or ordered models to represent closeness relations. In particular, we will utilize a selection function tool, which generalizes many models for conditionals, including sphere models. Indeed, a closeness relation between worlds can also be represented more abstractly using a selection function. Notably, certain classes of ordered models and sphere models have equivalent representations in terms of selection functions (see, e.g., [16]). One of the main advantages of the selection function is its technical utility: it is a very general tool for representing a general structure over worlds. However, one of its main shortcomings is conceptual: while orders and spheres offer an intuitive representation of a certain closeness relation, the relation induced by a selection function can sometimes be obscure and abstract.
In the present contribution, however, we aim to provide a further generalization of imaging methods by relying on this more general setting of selection functions. Simultaneously, we aim to present results that conceptually enhance the understanding of certain updating procedures and selection function models. Inspired by this generality, we will primarily work within an algebraic setting, which also generalizes possible worlds models and imaging updating methods. Our basic framework will consist of a finite Boolean algebra A, whose domain is denoted by the corresponding non-bold italic letter A, equipped with a probability distribution P. The structure on A that we need to establish our general updating methods is represented by two basic tools (more technical details will be provided in the next sections):
a selection function f : A×at(A) → A that, for every atoms α ∈ at(A) and every a ∈ A, specifies an element of A that correspond to the set of worlds closest to α according to a;
a function λ : A×at(A) → [0,1]|at(A)| that will be used to determine how to distribute the probability mass of each α ∈ at(A) to its closest worlds f(a,α).
References
[1] Ernest W. Adams (1975): The Logic of Conditionals. Reidel, Dordrecht.
[3] John P. Burgess (1981): Quick Completeness Proofs for Some Logics of Conditionals. Notre Dame Journal of Formal Logic 22(1), pp. 76–84, doi:10.1305/ndjfl/1093883341.
[7] Paul Égré, Lorenzo Rossi & Jan Sprenger (forthcoming): Certain and Uncertain Inference with Indicative Conditionals. Australasian Journal of Philosophy.
[9] Tommaso Flaminio, Giuliano Rosella & Lluis Subirana (2025): From conditionalization to imaging through the triviality glass. Submitted Manuscript.
[10] Bas C. van Fraassen (1976): Probabilities of Conditionals. In: Foundations of Probability Theory, Statistical Inference, and Statistical Theories of Science, Springer Netherlands, pp. 261–308, doi:10.1007/978-94-010-1853-1_10.
[11] Marianna Girlando, Sara Negri & Nicola Olivetti (2021): Uniform labelled calculi for preferential conditional logics based on neighbourhood semantics. Journal of Logic and Computation 31(3), pp. 947–997, doi:10.1093/logcom/exab019.
[12] Peter Gärdenfors (1982): Imaging and Conditionalization. Journal of Philosophy 79, pp. 747–760.
[13] Mario Günther (2022): Probabilities of Conditionals and Imaged Probabilities. Draft manuscript.
[14] Alan Hájek (1989): Probabilities of conditionals - Revisited. Journal of Philosophical Logic 18(4), doi:10.1007/bf00262944.
[16] David Lewis (1971): Completeness and decidability of three logics of counterfactual conditionals 1. Theoria 37(1), pp. 74–85, doi:10.1111/j.1755-2567.1971.tb00061.x.
[17] David Lewis (1973): Counterfactuals. Blackwell, Oxford.
[18] David Lewis (1976): Probabilities of Conditionals and Conditional Probabilities. Philosophical Review 85(3), pp. 297–315.
[19] Sara Negri & Nicola Olivetti (2015): A Sequent Calculus for Preferential Conditional Logic Based on Neighbourhood Semantics, pp. 115–134. Springer International Publishing, doi:10.1007/978-3-319-24312-2_9.
[20] Giuliano Rosella, Tommaso Flaminio & Stefano Bonzio (2023): Counterfactuals as Modal Conditionals, and Their Probability. Artificial Intelligence 323(C), p. 103970, doi:10.1016/j.artint.2023.103970.
[21] Paolo Santorio (forthcoming): Probabilities of Counterfactuals are Counterfactual Probabilities. Journal of Philosophy.
This article aims to dualize several results that Lahav has proved concerning various types (including possibly Cut-free and Identity-free systems) of canonical sequent calculi, systems equipped with well-behaved forms of left and right introduction rules for logical expressions. Instead, we focus on such kinds but rather of co-canonical sequent calculi, systems equipped with well-behaved forms of left and right elimination rules for logical expressions---which, simply put, proceed from sequents featuring complex formulae to their component sub-formulas. Furthermore, this work aims to explore combinations of canonical and co-canonical rules within a sequent calculus. Our main goals are to prove soundness and completeness results for the target systems' consequence relation in terms of their characteristic 3- or 4-valued non-deterministic matrices.
Inferences, in the most abstract sense, are a very general relation between sets of formulas. Sequents, on the other hand, are a proof-theoretic representation of inferences, held by syntactic restrains, and are thus usually considered to be finite. This restriction is not problematic because sets of inferences are usually compact, that is, if all of their finite subsets are satisfiable, the set itself is satisfiable, no matter if it is infinitely large. Metainferences started out as a reification of sequent to sequent rules. Taken instrumentally, it was unproblematic to assume them to be finite objects. However, if they are to be considered bona fide inferences of their own kind, one should not take for granted that they can be reduced to a finite representation. This problem, however, was brushed off in the literature. In this work, I will take some of the many metainferential relations that have been discussed in the past decade, and show which of them are finitary and which of them are infinitary.
Some authors (e.g. Cobreros et al. (2020), Scambler (2020), Pailos and Barrio (2022)) discussed and developed a two-sided conception of what a logic is, distinguishing between its positive and its negative sides, the former being the set of its valid inferences, the latter being the set of its valid anti-inferences. As usual, an inference Γ ⊨ ∆ is valid if it is satisfied by all valuations, while an anti-inference from Γ to ∆ is said to be valid if there’s no valuation satisfying Γ ⊨ ∆. Building on this idea, in this work we dualize the general construction developed by Szmuc (2023), who showed that for any Tarskian logic L, one can define a non-transitive counterpart of it, namely a logic sharing exactly the same positive part, though differing from L as it locally invalidates transitivity. Here, we provide a dual result: for every Tarskian logic L, we can construct a logic which shares with L its negative part, but differs from it as reflexivity fails. Our construction makes use of the so-called infectious extension of an algebra and of Malinowski’s q-matrices (Malinowski (1990)), namely a generalization of the usual notion of logical matrices where premises and conclusions are given (possibly) different sets of designated values, the latter being a subset of the former. Finally, we offer some commentaries about the duality of both constructions.
The primitive notion of classical Euclidean geometry is the notion of a point. In contrast, the region-based theory of space (RBTS) takes regions (abstracted physical bodies) as primitives, along with basic mereological relations such as part-of, overlap, and underlap. Algebraic models of RBTS are Contact Algebras (CA), which are Boolean algebras endowed with a binary relation called the Contact Relation. This approach has been very fruitful since it allows successfully defining mereological relations such as part-of, overlap, and underlap, as well as interesting mereotopological relations such as external contact, tangential part-of, non-tangential part-of, and self-connectedness. Nevertheless, this language has certain expressivity constraints when dealing with some interesting mereotopological notions, such as n-ary contact and internal connectedness, which cannot be defined within it. To overcome this issue, Vakarelov [4] introduces Sequent Algebras (SA) as a generalization of CA, replacing binary contact relations with finitary closure relations over Boolean algebras, satisfying some formal properties of Tarskian consequence relations. This approach is interesting because it connects several concrete areas that appear unrelated at first glance: RBTS, mereotopology, algebra, and logic.
So far, only topological representation theorems exist for SA in terms of subalgebras of the regular closed sets of a topological space. To fill this gap, in this paper, we take advantage of the well-established connection between closure operators and closure relations [1] in order to develop a full topological duality for SA, adapting some of the ideas from the topological duality for CA obtained by Goldblatt and Grice [2]. Moreover, the points of the spaces we obtain coincide with those used in [4]. In addition, we show that the category of SA is topological over the category of Boolean algebras (BAs). This implies that much of the categorical structure of SAs can be understood through that of BAs.
This is an ongoing project carried out within the MOSAIC Project 101007627, funded by the European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie Actions.
References
[1] Font, J. M., Abstract algebraic logic: An introductory textbook. College Publications, ISBN-10 1848902077, 2016.
[2] Goldblatt R. and Grice M., Mereocompactness and duality for mereotopological spaces. Katalin Bimbo (ED)J. Michael Dunn on Information based Logics, Outstanding Contributions in Logic, vol 8, 2016.
[3] F.W. Lawvere, Some thoughts on the future of category theory, in: Category Theory. Proceedings of the International Conference held in Como, Italy, July 22-28, 1990, Lecture Notes in Mathematics volume. 1488. Springer, Berlin, 1990.
[4] Vakarelov D., A mereotopology based on sequent algebras, Journal of Applied Non-Classical Logics, DOI: 10.1080/11663081.2017.1420590, 2018.