Talks

Abstract: Nucleus images and conucleus images are two fundamental ways of constructing new residuated lattices from old ones. Combining these two constructions allows us to obtain many of the algebras of logic from lattice-ordered groups (l-groups). For example, every MV-algebra is a nucleus image of a particular kind (top interval) of a conucleus image of a particular kind (negative cone) of an abelian l-group. This naturally raises the following question: which residuated lattices can be obtained from l-groups if we consider more general kinds of nucleus and conucleus images? We describe some partial results in this direction, but a smoother theory is obtained if we consider partially ordered monoids (pomonoids) or join semilattice-ordered monoids (sl-monoids) instead of residuated lattices. For example, we show that the nuclear images of subpomonoids of (abelian) partially ordered groups are precisely the (commutative) integrally closed pomonoids. The key construction will be the free nuclear preimage of a pomonoid or an sl-monoid, i.e. the left adjoint to the nuclear image functor. As a by-product of this line of thought, we also obtain a syntactic characterization of which quasivarieties of pomonoids or sl-monoids are closed under nucleus images, or equivalently which quasi-inequalities are preserved under nucleus images.

Abstract: We provide sufficient conditions for a variety of residuated lattices to have surjective epimorphisms (i.e., have the *ES property*).
The lattices under consideration are square-increasing [involutive] commutative residuated lattices (S[I]RLs) that are *semilinear*, i.e., that embed into a direct product of totally ordered algebras (and that are hence distributive).

We say that an S[I]RL is *negatively generated* when it is generated by the elements beneath its monoid identity.
We shall present a representation of negatively generated semilinear S[I]RLs, based on representations already in the literature.[^4][^5][^6]

This representation is then used to show that the class of all such algebras is a locally finite variety.
Moreover, we show that epimorphisms are surjective in *all* varieties of negatively generated semilinear S[I]RLs.

This result generalizes the earlier results that every variety comprising *Sugihara monoids* or *relative Stone algebras*, respectively, has surjective epimorphisms.[^1]
It also strengthens a result that every variety of *generalized Sugihara monoids* has a weak version of the ES property.[^2]

We also show that epimorphisms are surjective in the variety of idempotent semilinear S[I]RLs (but not in all of its subvarieties), from which it follows that this variety has the strong amalgamation property, because this variety known to be amalgamable.[^3]

These results settle natural questions about Beth-style definability of a range of substructural logics.

[^1]: G. Bezhanishvili, T. Moraschini, and J. G. Raftery, ‘Epimorphisms in varieties of residuated structures’, Journal of Algera, vol. 492, pp. 185–211, 2017.

[^2]: N. Galatos and J. G. Raftery, ‘Idempotent residuated structures: Some category equivalences and their applications’, Transactions of the American Mathematical Society, vol. 367, pp. 3189–3223, 2015.

[^3]: J. Gil-Férez, P. Jipsen, and G. Metcalfe, ‘Structure theorems for idempotent residuated lattices’, Algebra Universalis, vol. 81, p. 28, 2020.

[^4]: T. Moraschini, J. G. Raftery, and J. J. Wannenburg, ‘Varieties of De Morgan monoids: Minimality and irreducible algebras’, Journal of Pure and Applied Algebra, vol. 223, pp. 2780–2803, 2019.

[^5]: T. Moraschini, J. G. Raftery, and J. J. Wannenburg, ‘Varieties of De Morgan monoids: Covers of atoms’, Review of Symbolic Logic, vol. 13, pp. 338–374, 2020.

[^6]: J. G. Raftery, ‘Representable Idempotent Commutative Residuated Lattices’, Transactions of the American Mathematical Society, vol. 359, pp. 4405–4428, 2007.

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Abstract: The question of which splitting results can be transferred from (commutative) residuated lattices to their subreducts is a natural question and also a deceptively hard one. In this talk we will survey some results about this topic, with special attention to subreducts of hoops.

Slides

Abstract: In this talk I will report on an ongoing research joint project with M. Menni and P. Jipsen on “rigs”, i.e, rings without negatives. They can be axiomatised as commutative (distribuitive) semirings with both neutral elements and such that 0 in an annihilator: x*0=0. Rigs generalise both commutative rings and distributive lattices and seem to retain a reasonably good behaviour of the dual category of “spaces”. In this joint project we are trying to explore the universal algebraic aspects of these structures. Focusing on “integral” rigs (i.e. the ones that satisfies x+1=1), I will present a characterisation of the subdirectly irreducible algebras in the variety and then move to the study of Weyl rigs, i.e., the ones that have a unique homomorphism into {0,1}.

Slides

Abstract : When it comes to information, its potential incompleteness, uncertainty, and contradictoriness needs to be dealt with adequately. Separately, these characteristics have been taken into account by various appropriate logical formalisms and (classical) probability theory. While incompleteness and uncertainty are typically accommodated within one formalism, e.g. within various models of imprecise probability, contradictoriness and uncertainty less so --- conflict or contradictoriness of information is rather chosen to be resolved than to be reasoned with. To reason with conflicting information, positive and negative support---evidence in favour and evidence against---a statement are quantified separately in the semantics. This two-dimensionality gives rise to logics interpreted over twisted-product algebras or bi-lattices, e.g. the well known Belnap-Dunn logic of First Degree Entailment.

In this talk, we introduce two-dimensional many-valued logics for uncertainty which are interpreted over twisted-product algebras based on the [0,1] real interval. They can be seen to account for the two-dimensionality of positive and negative component of (the degree of) belief based on potentially contradictory information. The logics include extensions of Łukasiewicz or Gödel logic with a de-Morgan negation which swaps between the positive and negative component.

The logics inherit completeness and decidability properties of Łukasiewicz or Gödel logic respectively. Extensions of Gödel logic [2] turn out to include extensions of Nelson's paraconsistent logic N4, or Wansing's paraconsistent logic I_4C_4, with the prelinearity axiom.

Such logics can be applied to reason about belief based on evidence: In [1], a logical framework in which belief is based on potentially contradictory information obtained from multiple, possibly conflicting, sources and is of a probabilistic nature, has been suggested, using a two-layer modal logical framework to account for evidence and belief separately. The logics above are the logics used on the upper level in this framework. The lower level use Belnap-Dunn logic to model evidence, and its probabilistic extension to give rise to a belief modality.

(Based on on-going joint work with S. Frittella, D. Kozhemiachenko, O. Majer, and S. Nazari.)

[1] M. Bílková, S. Frittella, O. Majer and S. Nazari: Belief based on inconsistent information, DaLi 2020: Dynamic Logic. New Trends and Applications (M.A. Martins and I. Sedlar, editors), LNCS, vol. 12569, Springer, 2020, pp. 68–86.

[2] M. Bílková, D. Kozhemiachenko and S. Frittella, Constraint tableaux for two-dimensional fuzzy logics, accepted at TABLEAUX 2021.

Slides

Abstract : Many generalizations of the basic setting of model theory have been proposed over the years. Some of the more radical departures from the classical notion of model are the many-valued generalizations which stem from (at least) three distinct origins: Boolean-valued models, continuous model theory, and semantics of predicate many-valued logics.

The predicate symbols in such models are interpreted as functions that assign, to each tuple of elements of the domain, values from a certain set of grades (which is usually endowed with some additional structure). There are many papers studying graded models, mostly focused on particular approaches and sets of problems with different levels of generality and mathematical sophistication.

In my talk I will (1): propose a particular framework which could serve as a foundation of an abstract theory of graded models; (2): explore its particularization to models with FOL-like syntax and semantics induced by Tarski-Mostowski truth definition;(3) survey the axiomatization results for consequence relations induced by such graded models; and (4) present rudiments of their model theory.

Abstract : Shortest path problems are central in areas such as transportation and navigation, communication network optimization and social network analysis. Apart from devising efficient algorithms solving shortest path problems, representing knowledge about shortest paths and reasoning with this knowledge are crucial. In this talk we show that propositional dynamic logic with both formulas and programs evaluated in finite Łukasiewicz chains is a suitable formalism for this task. We also discuss our recent technical results concerning finite-Łukasiewicz PDL, establishing axiomatization and EXPTIME-completeness of the validity problem.

Abstract : Locally finite MV-algebras form a subclass of MV-algebras which is closed under homomorphic images, subalgebras, and finite products, but not under arbitrary ones. However, the category of locally finite MV-algebras with homomorphisms has arbitrary products in the classical categorical sense. Driven by these considerations, D. Mundici posed the following question:

Is the category of locally finite MV-algebras equivalent to an equational class? (D. Mundici. Advanced Lukasiewicz calculus. Trends in Logic Vol. 35. Springer 2011, p. 235, problem 3.)

We answer this question.

Our proofs rest upon the duality between locally finite MV-algebras and multisets established by R. Cignoli, E. J. Dubuc, and D. Mundici, and categorical characterizations of varieties established by J. Duskin, F. W. Lawvere, and others.

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Abstract : The prime spectrum of a Heyting algebra is the poset of its prime filters. The problem of describing prime spectra of Heyting algebras was raised by Esakia in 1985 and echoes similar questions in lattice and ring theory. Accordingly, a poset is said to be Esakia representable if it is isomorphic to the prime spectrum of some Heyting algebra. Therefore, Esakia's problem asks for an intelligible description of Esakia representable posets. While this problem remains open in general, in this talk we characterize Esakia representable well-ordered forests (i.e., disjoint unions of well-ordered trees). In the second half of the talk, we will draw some connection with the study of profinite Heyting algebras. More precisely, we will characterize the varieties of Heyting algebras whose profinite Heyting algebras are profinite completions. This result is achieved by combining a technique based on "forbidden configurations" with the description of Esakia representable diamond systems (a class of posets that encompasses the order duals of forests).

This talk is based on D. Fornasiere's master thesis "Representable Forests and Diamond Systems" and on the manuscript "Profiniteness and representability of spectra of Heyting algebras" by G. Bezhanishvili, N. Bezhanishvili, T. Moraschini and M. Stronkowski.

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Abstract : I intend to explain the possibility and necessity models based on the Logics of Formal Inconsistency (LFI's), which we call `credal calculi', taking advantage of their expressivity in terms of the notions of consistency and inconsistency. Some basic properties of possibility and necessity functions over LFI's are provided. A nice aspect of this talk is how logic can be connected to the treatment of information, and I discuss some examples showing how such logics attain realistic models for artificial judgement. This is a joint work with Juliana Bueno-Soler.

Reference:
W. A. Carnielli and J. Bueno-Soler. Credal Calculi, Evidence, and Consistency. Outstanding Contributions to Logic, edited by O. Arieli and A. Zamansky, Springer, 2021, in print.

Slides

Abstract : Relevant logics infamously have the variable sharing property: any of their conditional theorems share a variable between antecedent and consequent. This result has been strengthened in a variety of ways over the last half-century. Two of the more famous of these strengthenings are the strong variable sharing property (the shared variable can be assumed to have the same sign in both occurrences) and the depth relevance property (the shared variable can be assumed to occur at the same depth in both occurrences). In a talk I gave about I month ago, I showed that a certain class of relevant logics in fact has what I called the strong depth relevance property: their conditional theorems share a variable between antecedent and consequent, and this variable can be assumed to simultaneously have the same depth in both occurrences and have the same sign in both occurrences.

In today's talk, I'll prove that the logics in this same class also exhibit an interesting sort of invariance---depth substitution invariance. I'll then show that depth substitution invariance and strong variable sharing immediately entail strong depth relevance. This gives us a different (and much nicer) proof of the strong depth relevance result. It also sheds light on the sense in which these logics exhibit different `levels of entailment'.

Slides

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Abstract: Last time, we discussed Kripke and algebraic semantics for an extension of the intuitionistic propositional calculus with strict implication (called variously the Heyting-Lewis or the Lewis-Brouwer logic). We also had a look at examples, motivations and instances. Now it is time to roll up our sleeves and dig deeper into representation, duality, decidability, fmp and transfer results. We begin by using a representation theorem for Lewis-Brouwer algebras to define a suitable class of "descriptive frames", extending it to a full categorical duality presentable either Esakia-style or Priestley-style (in the latter case, we obtain a subduality of one studied previously by Celani and Jansana). We then adapt a transformation by Wolter and Zakharyaschev to translate Lewis-Brouwer Logic to classical modal logic with two unary operators. This allows us to prove an analogue of the Blok-Esakia theorem that we then use to obtain both known and new canonicity and correspondence theorems, as well as the finite model property and decidability for a large family of Lewis-Brouwer logics. This is a joint work with Jim de Groot and Dirk Pattinson (ANU). While I will try to keep the talk reasonably self-contained, having a look beforehand at the material presented in February can be helpful.

Abstract: A variety of MTL-algebras is called linear whenever its lattice of subvarieties, ordered by inclusion, is linearly ordered. In this talk we will describe some properties of the linear varieties of MTL-algebras, and we will provide a full classification of the linear varieties of BL-algebras and WNM-algebras. We will also discuss some additional topics and open problems.

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Abstract: In this seminar we investigate AGM belief contraction operators by using the tools of algebraic logic. We generalize the notion of contraction to arbitrary finitary propositional logics, and we show how to switch from a syntactic-based approach to a semantic one. This allows to build a solid bridge between the validity of AGM postulates in a propositional logic and specific algebraic properties of its intended algebraic counterpart. Some applications to substructural logics are provided.

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Abstract: In this seminar we investigate AGM belief contraction operators by using the tools of algebraic logic. We generalize the notion of contraction to arbitrary finitary propositional logics, and we show how to switch from a syntactic-based approach to a semantic one. This allows to build a solid bridge between the validity of AGM postulates in a propositional logic and specific algebraic properties of its intended algebraic counterpart. Some applications to substructural logics are provided.

Abstract: It is always possible to associate to an arbitrary propositional logic, two substitution-invariant consequence relations, which satisfies, respectively, a left and a right variable inclusion constraint. In the former case, the requirement of variable inclusion goes from premises into conclusions, while, in the latter, in the opposite direction, namely from variables of the conclusion of an inference into the variables of the set of premisses.

Prototypical examples of variable inclusion companions are found in the realm of three-valued logics. For instance, the left and the right variable inclusion companions of classical (propositional) logic are, respectively, paraconsistent weak Kleene logic -- PWK for short -- and Bochvar logic.

It is recent discovery the fact that the algebraic counterpart of PWK is played by the class of Plonka sums of Boolean algebras. This observation led us to investigate the relations between left and right variable inclusion companions and Plonka sums in full generality.

The starting point consists in generalizing the construction of Plonka sums from algebras to logical matrices. This allows us to characterize the matrix models for variable inclusion logics by performing appropriate Plonka sums over direct systems of models of a logic. As a matter of fact, variable inclusion companions are especially well-behaved in case the original logic has a specific kind of partition function, a feature shared by the vast majority of non pathological logics in the literature.

(Based on joint works with T. Moraschini and M. Pra Baldi)

Abstract: This talk is an introduction to what one might call the Heyting-Lewis calculus of strict implication over the intuitionistic propositional base; the names "constructive strict implication" or "Brouwer-Lewis implication/calculus" have also been used. The corresponding class of algebras can be seen as the fusion of Heyting algebras and weak Heyting algebras (Celani and Jansana) over the shared bounded lattice reduct. (Super)intuitionistic modal logics with unary box are a limiting case, but in the intuitionistic setting there are many examples where strict implication is not reducible to box. Its variants arise, e.g., in the context of preservativity in Heyting Arithmetic (where it was first invented by Visser), in the inhabitation logic of simple type theory extended with Haskell-style arrows, and in a generalization of Intuitionistic Epistemic Logic of Artemov and Protopopescu. The move to the intuitionistic propositional base also throws interesting light on the complex fate of Lewis' original systems. The Heyting-Lewis calculus enjoys a natural Kripke semantics (first studied by Iemhoff and coauthors), which also allows defining an appropriate notion of descriptive frame and Esakia-style dualities. Furthermore, one can follow the Wolter-Zakharyaschev idea of generalizing the Gödel-McKinsey-Tarski translation, reducing the metatheory of Heyting-Lewis logics to suitable bimodal logics over the classical propositional base, obtaining a suitable variant of the Blok-Esakia theorem, and (re)proving many correspondence, completeness, decidability and fmp results in an uniform way. However, it seems that ultimately one will have to drop one of the axioms, losing the natural Kripke semantics. In the final part of the talk, I am going to discuss alternative semantics for the weakened system and its position in the broader landscape of intuitionistic logics with an additional implication-like connective. This talk involves joint work with Albert Visser (Utrecht University), Jim de Groot and Dirk Pattinson (ANU), Igor Sedlar and the Prague group, and Miriam Polzer (Google).

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Abstract: The amalgamation property and its variants are in strong relationship with various syntactic interpolation properties of substructural logics, hence its investigation in varieties of residuated lattices is of particular interest. The amalgamation property will be investigated in some classes of non-divisible, non-integral, and non-idempotent involutive commutative residuated lattices. It will be proved that the classes of odd and even involutive, commutative residuated chains fail the amalgamation property for the same reason as the reason for the failure of this property in the class of discrete abelian o-groups with positive, normal homomorphisms. It is also proved that the variety of semilinear, idempotent-symmetric, odd, involutive, commutative, residuated lattices has the amalgamation property, and hence also the transferable injections property.

Densification is a key step in proving standard completeness of fuzzy logics. We prove in an algebraic manner that the variety of semilinear odd involutive commutative residuated lattices, and the variety of semilinear odd idempotent-symmetric involutive commutative residuated lattices admit densification.

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Slides

Abstract: Basic Logic BL, introduced by P. Hajek in 1998, is the logic of all continuous t-norms and their residua. The variety of BL-algebras forms the algebraic semantics of BL.
Let V be a variety of BL-algebras, and let L(V) be its lattice of subvarieties, ordered by inclusion.
V is called strictly join irreducible (SJI) if, whenever V is the join of a non-empty set S of varieties of BL-algebras, then V belongs to S.
Every variety in L(V) is obtained as join of SJI varieties, which may be considered as the building blocks of all the varieties in L(V). In this talk I will present the results of a recent joint work with Stefano Aguzzoli, where we provided a full classification of the SJI varieties of BL-algebras.

Slides

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Abstract: Proving 'standard completeness', that is completeness with respect to a class of algebras based on the real unit interval, has been for a long time a central problem for t-norm based (fuzzy) logics. Elaborated techniques to prove this kind of result have been developed and most of them rely on the fact that totally ordered algebras can be embedded, or just partially embedded, into standard structures. However, when we move from t-norm based logics to probabilistic modal logics based on them, these methods are no longer applicable and it is necessary to consider new ideas to prove standard completeness. In this seminar, besides clarifying what ’standard completeness’ means in the probabilistic setting, we will present the logic FP(L, L), a formalisms that allows to reason about probabilistic statements on events represented as formulas of Lukasiewicz logic, and we prove it to be standard complete. Further elaborating on the standard completeness for FP(L, L) we will also present results from an ongoing research line that allow to regard a peculiar class of projective MV-algebras as a semantics for that probability logic.

Abstract: The use of logic to reason about probability has a long tradition in science, and any ambition of surveying past work in a single talk would be ill-advised. Instead, in this light, informal, leisurely talk, I attempt to highlight selected fundamental issues that arise in the field. For example, starting from the logical side: Is "The coin probably lands heads" a sentence in classical logic? Or is it a modal sentence? Can we attach any meaning to the sentence "It is likely that the coin probably lands heads"? And how do we infer one such sentence from another? By the end of the talk, I hope to manage to indicate that convincing answers to these and other related questions are available. These answers pertain to logic and algebra, but in turn suggest new questions in probability theory that are not traditionally associated with that field; for example, is there a "free", or most general, assignment of probabilities to the sentence "The coin lands heads”?

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Abstract: Algebraic investigations into substructural logics have been flourishing in the past decades, but the focus of this research has been fairly biased towards integral or idempotent or divisible structures which were already well-understood. On the contrary, (quasi)varieties of not necessarily integral and not necessarily divisible algebras form equivalent algebraic semantics for all the main logics in the linear and in the relevant family, including Abelian logic, and it is precisely in this area where it is possible to find very interesting connections with (lattice ordered) groups and thus with classical algebra.

In this talk we address the problem of structural description of involutive commutative residuated lattices, the non-integral case. The algebras in our focus are non-divisible and non-idempotent either. Related attempts in the literature have, so far, been confined to either lattice-ordered groups (the cancellative case) or Sugihara monoids (the idempotent case). For all involutive commutative residuated chains, where either the residual complement operation leaves the unit element fixed (odd case) or the unit element is the cover of its residual complement (even case), a representation theorem will be presented in this talk by means of direct systems of abelian o-groups.

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Abstract: In this talk we will investigate projective algebras in varieties of bounded commutative integral residuated lattices, using different tools, ranging from purely algebraic methods to categorical equivalences and functional representations of free algebras (in varieties related to fuzzy logics).

In particular, we will start by identifying a large class of algebras where all finite Boolean algebras are projective, given by varieties with a Boolean retraction term.

We will then use the functional representation of free product algebras to show that finitely generated projective product algebras are the finitely presented algebras in their variety, and that (by categorical equivalence) the same holds for the variety generated by perfect MV-algebras.

Finally, we will discuss Ghilardi’s approach to unification theory for algebraizable logics, where a unification problem can be seen as a finitely presented algebra, and a solution (or unifier) is a homomorphism to a projective algebra. In particular, we will see some consequences of our work in this framework.

This talk, although self-contained, is to be considered the second part of the talk given by Paolo Aglianò on recent joint work.

Abstract: Our work is motivated by a game theoretic question, i. e. characterize the 􏲜fixed point sets of Shapley operators. These last are operators related to zero-sum games, in particular their fi􏲜xed points represent optimal strategies.

These fi􏲜xed point sets are 􏲐topological retracts􏲑 of R^n in a suitable category and they are characterized in terms of their lattice-theoretic properties.
In developing this characterization, we shall observe that for some subsets of R^n the property of being closed in the Euclidean topology can be expressed by lattice theoretic properties.

Moreover, we shall see that these sets are also related to the concept of convexity.

The previous result is later generalized in the context of conditionally complete lattices, characterizing sets which arise as 􏲜fixed point sets of an order-preserving map from a conditionally complete lattice to itself.

(Joint work with Stéphane Gaubert and Marianne Akian)

Slides

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Abstract: A well-known theorem of classical model theory due to Robinson states that the (first-order) theory of totally ordered abelian groups has a model completion, namely the theory of totally ordered divisible abelian groups with at least two elements. On the other hand, it was proved by Glass and Pierce that the theory of lattice-ordered abelian groups does not have a model completion. Similarly, Lacava and Saeli have proved that the theory of totally ordered MV-algebras has a model completion, but that this is not the case for the theory of all MV-algebras.

In this talk, I will relate these negative results on model completions to the fact that the varieties of lattice-ordered abelian groups and MV-algebras admit “parametrically definable” but not “equationally definable” principal congruences, yielding also a general negative model completion result for varieties of commutative pointed residuated lattices. I will also explain how to rescue equationally definable principal congruences and hence positive model completion results for certain varieties by adding an additional binary operator, showing in particular that the theory of MV-Delta-algebras has a model completion.

(Joint work with Luca Reggio)

Slides

Abstract: The use of logic to reason about probability has a long tradition in science, and any ambition of surveying past work in a single talk would be ill-advised. Instead, in this light, informal, leisurely talk, I attempt to highlight selected fundamental issues that arise in the field. For example, starting from the logical side: Is "The coin probably lands heads" a sentence in classical logic? Or is it a modal sentence? Can we attach any meaning to the sentence "It is likely that the coin probably lands heads"? And how do we infer one such sentence from another? By the end of the talk, I hope to manage to indicate that convincing answers to these and other related questions are available. These answers pertain to logic and algebra, but in turn suggest new questions in probability theory that are not traditionally associated with that field; for example, is there a "free", or most general, assignment of probabilities to the sentence "The coin lands heads”?

Abstract: The ring of real-valued continuous functions over a topological space has naturally a structure of lattice-ordered R-algebra (l-algebra for short). Gelfand-Naimark-Stone duality provides a dual equivalence between the category of compact Hausdorff spaces and the category of uniformly complete bounded archimedean l-algebras. In this talk I will describe a series of results obtained in collaboration with G. Bezhanishvili and P. J. Morandi regarding modal operators on bounded archimedean l-algebras. Our definition of a modal operator is motivated by the fact that such operators correspond to continuous relations on compact Hausdorff spaces. We show that this correspondence gives rise to a dual equivalence extending Gelfand duality. Algebraic/coalgebraic methods can also be employed to obtain an alternate proof of this new duality which gives the opportunity to talk about the existence of free bounded archimedean l-algebras. We will then compare our results with the celebrated Jónsson–Tarski duality from modal logic. Some correspondence results and other future directions of research will be mentioned at the end of the talk.

Slides

Abstract: In this talk we will consider the problem of existence of saturated models for first-order many-valued logics. We employ a general notion of type as pairs of sets of formulas in one free variable which express properties that an element of a model should, respectively, satisfy and falsify. By means of an elementary chains construction, we prove that each model can be elementarily extended to a -saturated model, that is, a model where as many types as possible are realized. In order to prove this theorem we obtain, as by-products, some results on tableaux (understood as pairs of sets of formulas) and their consistency and satisfiability, and a generalization of the Tarski-Vaught theorem on unions of elementary chains. Finally, we provide a structural characterization of saturation in terms of the completion of a diagram representing a certain configuration of models and mappings.

Slides

Abstract: : The problem of determining the projective objects in a category is old and respected; here we deal with algebraic categories in their natural incarnation: varieties of algebras. More precisely we try to describe the algebras that are projective in subvarieties of (bounded) commutative integral residuated lattices.

This is the first part of a two-parts talk (the second will be given by Sara Ugolini). In this first part, after a thorough introduction of the subject, we will mainly discuss:

- general results involving projectivity and ordinal sums;

- projective Heyting Algebras and Stonean Heyting algebras;

- some observations about projectivity in varieties of hoops.

Abstract: It is a classic result in topology that the Vietoris space of a compact Hausdorff space is compact Hausdorff. The Vietoris topology is the hit-or-miss topology on the set of closed sets. This construction plays an important role not only in topology but also in other branches of mathematics, including logic as the restriction of the Vietoris functor to the category of Stone spaces is essential in the study of coalgebraic logic.

There have been several attempts to generalize the Vietoris construction to larger classes of (not necessarily Hausdorff) spaces. One such construction for locally compact (non-Hausdorff) spaces is due to Fell. This is at the origin of the notion of the Lawson topology, an important tool in domain theory (and elsewhere).

I will discuss the history of the hit-or-miss topology and the corresponding Fell compactification. This is closely related to the theory of stable compactifications (originated by Smyth). In the special case when the topology is the Alexandroff topology of a poset P, the hit-or-miss topology on the set of closed sets gives rise to the Priestley space whose lattice of clopen upsets is isomorphic to the bounded distributive lattice freely generated by (the dual of) P.

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Abstract: Algebras that contain two unary S5-modalities have been widely studied in different contexts throughout the literature, usually under the name "monadic". Examples include monadic Boolean algebras, monadic Heyting algebras, monadic Godel algebras, monadic MV-algebras, and monadic abelian l-groups. Such algebras often occur in the context, or as a consequence, of the study of one-variable fragments of first-order logics. Indeed, there is a one-to-one correspondence between one-variable first-order formulas and modal formulas, where the universal and existential quantifier correspond to the box and diamond modality, respectively.

In this talk I will present an attempt to generalize and unify some of the known results for these "monadic algebras" by introducing monadic residuated lattices. All aforementioned monadic algebras are included in this class.

We show that for such a monadic residuated lattice, the modalities are determined by a "relatively complete" subalgebra of the residuated lattice reduct. Moreover, congruences are determined by congruences of this relatively complete subalgebra. As a case study, we look at some semi-linear subvarieties.

Slides

Abstract: In this talk I will discuss some results on concrete representations of free finitely generated BL-Algebras, which I have introduced with Simone Bova a few years ago [1].

The motivation of the talk is to recall the constructions and get some feedback in order to simplify them: as a matter of fact I shall describe quite intricate objects, and, even though I am convinced that a major part of the involved complexity is structural, it may well be the case that a significant part of it could be really simplified.

In the first part I shall introduce a representation of the free n-generated BL-algebra which is recursive, in the sense that elements of this algebra are described by suitably combining elements in free BL-algebras with less than n many generators.

In the second part I show how we managed to remove the recurrence, at the cost of a more complex description of the free algebras.

If time allows, in a final part I shall describe a categorical duality for finite BL-algebras, introduced in a work with Simone Bova and Vincenzo Marra [2]. I shall then apply the duality to describe free algebras in some locally finite subvarieties of BL-algebras.

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Abstract: This talk is based on a joint work with Joan Gispert, Tommaso Moraschini and Michal Stronkowski. We will revisit three propositional fuzzy logics that extend Hájek's Basic logic BL, namely Łukasiewicz, Gödel, and product logic, each expanded with rational constants: the language is enriched with propositional constants for each rational number between 0 and 1 and one adds axioms capturing the behaviour of the basic BL-operations on each of the rationals. Expansions of this kind have a rich history, starting from a paper by Goguen.

Our work looks at the lattice of extensions for each of the logics RŁ (Łukasiewicz logic with rational constants, also known as Rational Pavelka logic), RG, and RP (Gödel and product logics with rational constants respectively) and at their structural completeness; the results can be viewed in juxtaposition to the already known situation for the logics Ł, G, and P. We work in the respective equivalent algebraic semantics of the expanded logics, provided by the varieties RMV, RGA, and RPA. To appreciate the difference the constants make, one can contemplate the following facts. In the BL-language, the variety MV is Q-universal (the lattice of subquasivarieties is quite complicated), whereas for product algebras, the lattice of subquasivarieties is known to be a three-element chain. (The notion of Q-universality was introduced by Sapir, and Q-universality for MV is due to Adams and Dziobiak.) Expanding with the rational constants, it turns out RMV has no nontrivial proper subquasivarieties, while the class RPA is Q-universal (relaxing the definition to accommodate infinite languages). Moreover, while RŁ is structurally complete, RP is not: we provide a base for its admissible rules and show they are decidable.

While the comparison of the case with/without constants for Łukasiewicz and product logics may take up the time allotted to the talk, the manuscript provides other results, discussing subquasivariety lattice of RGA, which has uncountable chains and antichains (we leave the problem open whether RGA is Q-universal), characterization of (hereditary) structural completeness of RG-extensions, or results on active and passive structural completeness.

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Abstract: In joint work with Olim Tuyt and Diego Valota we described the structure of finite commutative idempotent involutive residuated lattices (CIdInRL) by disjoint unions of Boolean algebras glued by partial isomorphisms over the distributive lattice of positive elements (arxiv.org/abs/2007.14483). Recent investigations with Melissa Sugimoto extend this work from CIdInRL to (nonunital) commutative idempotent involutive residuated posets. These partially ordered algebras are of the form (A, ≤, ⬝, -) such that

(A, ≤) is a poset,

(A, ⬝) is a semilattice and

x⬝y ≤ z ⇔ x ≤ -(y⬝ -z).

It follows that ⬝ is order-preserving, --x = x and x ≤ y ⇒ -y ≤ -x, hence these structures are also referred to as involutive residuated po-semilattices or irpo-semilattices. We give an efficient algorithm to construct finite irpo-semilattices from Plonka sums of generalized Boolean algebras. A dual representation of these Plonka sums is provided by semilattice direct systems of partial functions. For lattice-ordered irpo-semilattices, all partial functions are partial injections, and in the unital case we recover the structural results about finite CIdInRLs. If time permits we will also present an implementation of the algorithm.

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Abstract: In this talk I will discuss some results on concrete representations of free finitely generated BL-Algebras, which I have introduced with Simone Bova a few years ago [1].

The motivation of the talk is to recall the constructions and get some feedback in order to simplify them: as a matter of fact I shall describe quite intricate objects, and, even though I am convinced that a major part of the involved complexity is structural, it may well be the case that a significant part of it could be really simplified.

In the first part I shall introduce a representation of the free n-generated BL-algebra which is recursive, in the sense that elements of this algebra are described by suitably combining elements in free BL-algebras with less than n many generators.

In the second part I show how we managed to remove the recurrence, at the cost of a more complex description of the free algebras.

If time allows, in a final part I shall describe a categorical duality for finite BL-algebras, introduced in a work with Simone Bova and Vincenzo Marra [2]. I shall then apply the duality to describe free algebras in some locally finite subvarieties of BL-algebras.

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Abstract: Paraconsistent Weak Kleene Logic (PWK) is the 3-valued propositional logic defined on the weak Kleene tables and with 2 designated values. In this survey talk, we intend to explore some intriguing connections between this logic and the algebraic theories of regular varieties and of Plonka sums over semilattice direct systems of algebras. By a recourse to this toolbox, it is possible to discover some interesting properties of PWK from the point of view of Abstract Algebraic Logic. We also present a Gentzen system for PWK and show that PWK has only one nontrivial proper extension apart from Classical Logic. The results we present are due to S. Bonzio, J. Gil Férez, T. Moraschini, L. Peruzzi, M. Pra Baldi, and the speaker.

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Abstract: This is joint work with Robin Hirsch (Kings College), Marcel Jackson and James Koussas (both La Trobe).

Tarski's Relation Algebras (RA) and Representable Relation Algebras (RRA) have a long history and a rich theory. In one corner of this theory arose qualitative calculi, of which the best known example is Allen Interval Algebra. In applications of qualitative calculi it is often useful to consider representations weaker that the standard one. Roughly speaking, composition of relations is represented by the smallest "admissible" relation containing the true composition.

This notion of representation is called a qualitative representation in Hirsch, Jackson, TK "Algebraic foundations for qualitative calculi and networks", TCS, vol. 768, pp. 99-116 (2019). It gives rise to a natural subvariety of Maddux' Nonassociative Algebras, which we call Qualitatively Representable Algebras (QRA). I will give an overview of what we know about QRA, focussing on classical universal algebraic properties. A few focal points:
* Some classically non-representable algebras are qualitatively representable (e.g., the McKenzie algebra).
* Finite algebras have finite representations.
* QRA is non-elementary, but its quasiequational theory is decidable, indeed co-NP-complete. QRA has FEP.
* The subvariety lattice of QRA has three atoms (the same as RA and RRA). One of them has continuum covers.

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Abstract: A logic is said to admit an equational completeness theorem when it can be interpreted into the equational consequence of some class of algebras. Even if the vast majority of completeness theorems in the literature are of this form, it is known that there are logics lacking any equational completeness theorem. Despite the simplicity of this concept, intrinsic characterizations of logics with admitting an equational completeness theorem have proved elusive. This is partly because equational completeness theorem can take unexpected forms, e.g., in view of Glivenko's Theorem, classical propositional logic is related to the variety of Heyting algebras by an equational completeness theorem. As it happens, nonstandard equational completeness theorems of this form are ubiquitous.

In this talk, we will present a characterization of logics with at least one tautology (resp. locally tabular logics) admitting an equational completeness theorem. For a protoalgebraic logic, this amounts to the demand that the algebraic counterpart of the logic validates a non-trivial equation. While the problem of determining whether a logic admits an algebraic semantics will be shown to be decidable for logics (resp. locally tabular logics) presented by a finite set of finite matrices (resp. by a finite Hilbert calculus), we shall see that it becomes undecidable for arbitrary logics presented by finite Hilbert calculi.

A draft collecting these observations is available at http://uivty.cs.cas.cz/~moraschini/files/submitted/equational.pdf

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Abstract: The central theme of this talk is the isomorphism problem for logical consequence relations, namely the examination of general conditions under which any isomorphism between the (expanded) lattices of the theories of two consequence relations is induced by an equivalence between them. A categorical and order-theoretical solution of the isomorphism problem for consequence relations on formulas was developed in [2] and was presented in my July 14 webinar talk. The present talk gives an account of the main results of [1] and shows that the approach in [2] can be suitably modified to provide a solution of the isomorphism problem in the multiset setting. The primary difference between the two approaches is the choice of the categories of modules under consideration. Our work is motivated by and owes considerable debt to [3], which develops an alternative but related approach to the study of multiset consequence relations.

References

[1] Z. Khanjanzadeh, A. Madanshekaf, F. Paoli and C. Tsinakis, A Categorical Approach to Logical Consequence Relations: The Multiset Case, in preparation.

[2] N. Galatos and C. Tsinakis, Equivalence of consequence relations: an order-theoretic and categorical perspective, J. Symbolic Logic 74 (3) (2009), 780-810.

[3] P. Cintula, J. Gil-Férez, Tommaso Moraschini and Francesco Paoli, An abstract approach to consequence relations, Review of Symbolic Logic 12(2) (2019), 331-371.

Abstract: This is a joint work with Paolo Aglianò. The twist-product of a lattice L is the cartesian product of L with its order dual, equipped with the natural order involution ~(x,y) = (y,x). The idea of considering this kind of construction to deal with order involutions on lattices goes back to Kalman's 1958 paper, but the denomination "twist" appeared thirty years later. The extension of this concept to residuated lattices is due to Tsinakis and Wille, and applying their construction to integral residuated lattices, Busaniche and Cignoli defined Kalman lattices (or just K-lattices) as the variety of 1-involutive residuated lattices that can be represented by a twist-product.

In this work we explore the lattice of subvarieties of K-lattices, specifically the bottom part, as well as the full lattice for some particular subvarieties.

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Abstract: The aim of this talk is to present an order-theoretic and categorical treatment of various constructions and concepts connected with the study of logical consequence relations. Its central theme is the isomorphism problem for consequence relations, namely the examination of general conditions under which any isomorphism between the (expanded) lattices of the theories of two consequence relations is induced by an equivalence between them.

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Further material: B. Jónsson lecture series , J.B. Nation Universal algebra without equality

Abstract: A residuated lattice is called conic if it is the union of its positive and negative cone. We give an overview of conic idempotent residuated lattices and their decomposition as ordinal sums of Brouwerian algebras and pre-lattices along a idempotent residuated chain. In particular, we investigate the action of the two inverse operations and how they connect to the order and the multiplication. To complete the analysis we pay even closer attention to the idempotent residuated chains that serve as skeletons for the decomposition. Among other things we consider the enhanced monoidal preorder, nested and ordinal sums, and flow diagrams of the inverses.

We observe that the amalgamation property fails even for chains (therefore also for conic) but our analysis identifies the key condition of weak involutivity. We make use of all of the above in establishing the strong amalgamation property for conic idempotent residuated lattices that are weakly involutive and De Morgan. (The condition of being De Morgan ensures that the pre-lattices are actually lattices and without it amalgamation also fails.) The situation is substantially more complicated compared to the commutative case.

The above algebras generate the corresponding variety of semiconic algebras; we extend the strong amalgamation to this variety. The proof requires, among other things, a version of the congruence extension property which is typical for commutative structures. We show that even though we do not assume commutativity, and therefore iterated conjugates are needed in the congruence generation, we can still provide ideal terms without parameters, therefore establishing the congruence extension property. As a side result, this further leads to a single one-variable equation that axiomatizes semiconic residuated lattices in the presence of idempotency.

This is joint work with Wesley Fussner.

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Abstract: We present the class of Modules with Fusion and Implication based over distributive lattices (FIDL -modules, for short). These structures are intended to be a suitable generalization of both Distributive Lattices with Fusion and Implication and Modal Distributive Lattices.

The aim of this talk is to develop a representation theory for FIDL-modules in terms of FI-Frames and moreover, by extending the well known duality between distributive lattices and Priestley spaces, exhibit a topological bi-space duality between the category of FIDL-modules and the category of Urquhart spaces.

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Abstract: I will discuss an ongoing line of research in the relational semantics of non-distributive logics. These developments are technically rooted in dual characterization results and insights from unified correspondence theory. These developments also have broader, conceptual ramifications for the intuitive meaning of non-distributive logics. I will present two types of relational semantics of non-distributive logics that arise from dual characterization: the polarity-based frames and the graph-based frames.

Although polarity-based and graph-based semantics are tightly connected and stem from the application of the same methodology, they give rise to two radically different intuitive interpretations of non-distributive logics: namely, the polarity-based semantics supports the interpretation of non-distributive logics as logics of formal concepts; the graph-based semantics supports the idea that non-distributive logics can be viewed as hyper-constructivist logics, i.e. logics in which the principle of excluded middle fails at the meta-linguistic level, and hence their propositional base generalizes intuitionistic logic in the same way in which intuitionistic logic generalizes classical logic.

Finally, time permitting, by way of examples, I will discuss the epistemic interpretations of lattice-based normal modal logics supported by the polarity-based semantics (i.e. lattice-based normal modal logics as epistemic logics of categories and concepts) and by the graph-based semantics (i.e. lattice-based normal modal logics as epistemic logics of informational entropy.)

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Abstract : The (ambitious) aim of this talk is to give a general answer to the question: "What is the logic of a class of t-norms?". To substantiate our answer we will construct several assertional logics that can be considered as such, with particular emphasis on logics coming from t-norms with a special kind of negations, which we call nuclear.

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Abstract : BL-algebras are the algebraic counterpart of Hajek's fuzzy logic system BL. They constitute the most important tool to understand and get results of the system. During the talk we will focus on subvarieties of BL-algebras generated by a single chain and we will study functions in the finitely generated free algebras of those varieties. The idea is to use the information of the regular and dense elements of the generating chain to characterize the elements of the free algebra.

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Abstract : We present a structure theorem for idempotent commutative residuated lattices in which every element is comparable to the monoid identity e. The negation operation x* = x -> e plays a central role in our analysis: Our structure theorem describes the aforementioned algebras as ordinal sums of certain posets, indexed by an e-involutive subalgebra (which comprises a Sugihara monoid). The algebras in the class satisfying both De Morgan laws with respect to * are especially well behaved, and as an application of our structure theorem we obtain that this subclass enjoys the strong amalgamation property. We extend this result also to the variety generated by such algebras, showing that this variety has the strong amalgamation property and hence also that epimorphisms between its members are surjections. This is joint work with Nick Galatos.

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Abstract : In this talk, we broach an algebraic canonicity theorem for normal LE-logics of arbitrary signature in the generalized setting of slanted algebras, i.e. lattices expansions in which the non-lattice operations map tuples of elements of the given lattice to closed or open elements of its canonical extension. Interestingly, the syntactic shape of LE-inequalities which guarantees canonicity in this generalized setting turns out to coincide with the shape of analytic inductive inequalities, which guarantees LE-inequalities to be equivalently captured by analytic structural rules of a proper display calculus.

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Abstract : In this talk we delve into the Kripke semantics of modal logics defined over distributive lattices, possibly expanded by implication or co-implication. One prominent logic of this kind is intuitionistic modal logic (introduced by Fischer-Servi in the 1970's), another is positive modal logic (introduced by Dunn in the 1990's as the negation-free fragment of classical logic). The Kripke semantics of these modal logics comes in two flavors: one has a single relation interpreting both the box and the diamond, while the other has two relations satisfying some compatibility conditions. We take a look at the relation between these two types of semantics and formulate a basic distributive modal logic from which the systems of Fischer-Servi and Dunn (among others) can be obtained in a modular way. In doing so, we try to allow for a range of possible modal signatures, including negative box and diamond operators.

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Abstract : Mundici proved that the categories of (unital Abelian) lattice-ordered groups and of MV-algebras are equivalent. We provide a generalization: the category of (unital commutative totally distributive) lattice-ordered monoids is equivalent to a variety whose algebras we call MMV-algebras (for Monoidal MV-algebras). Roughly speaking, lattice-ordered monoids are lattice-ordered groups without the unary operation that maps x to –x, and, analogously, MMV-algebras are MV-algebras without the negation. We will mention one reason of interest for these algebraic structures: in a sense, MMV-algebras are to MV-algebras what distributive lattices are to Boolean algebras. From a duality perspective, in the same way in which lattice-ordered groups and MV-algebras are algebras of continuous functions over a compact Hausdorff space (up to infinitesimals), lattice-ordered monoids and MMV-algebras are algebras of continuous monotone functions over a compact ordered space (up to infinitesimals).

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Lukasiewcz logic carries many connections with other areas of mathematics, and in particular with probability theory. In this talk we shall see how an infinitary variety of MV-algebras can be used to obtain a new framework for many aspects of probability. To do so, we will define a “well behaved” notion of random variable in the setting of nonclassical logic. After laying the ground with the algebraic results, we will discuss how conditioning of random variables, stochastic processes and exchangeable probabilities can be described from this new point of view.

Betting methods, of which de Finetti’s Dutch Book is by far the most well-known, are uncertainty modelling devices which accomplish a twofold aim. Whilst providing an (operational) interpretation of the relevant measure of uncertainty, they also provide a formal definition of coherence. The main purpose of this talk is to present de Finetti’s foundational work and to explore on a refinement of his betting method that nowadays goes under the name of strict coherence. For both coherence and strict coherence we shall presents characterization theorems in terms of probability theory, geometry and logic.

Algebraic logic and probability theory have had a deep connection from the start: in fact, in the work of George Boole, what we now call Boolean algebras provided a first common abstraction of the notions of logical statement as well as of probabilistic event. We will talk about the probability theory of so-called many-valued events (events that can be neither true nor false, but true to some degree), and we will see how a notion of probability map has been defined on structures we are familiar with, such as lattice ordered groups, MV-algebras and other residuated structures. We will see how the algebraic approach has proven to be powerful, and in particular we’ll explore its interesting connections to measure theory.