Past talks delivered at the Seminar on Applied Mathematical Logic (a.k.a. the Hájek Seminar) are listed here.

• 27.3.2019
  Igor Sedlár (ICS CAS): Two approaches to non-classical modal logic
  Abstract: Two prominent approaches to non-classical modal logic are the lattice-valued one, using Kripke frames and valuation functions mapping formula-state pairs to a lattice of truthvalues, and the relational one, extending frame semantics for non-classical logics--usually given by means of so-called Routley-Meyer frames--by additional accessibility relations corresponding to modal operators. In this talk, I outline some preliminary results on the relationship between these two approaches. Using elementary dualities between residuated lattices and Routley-Meyer frames, I show that the logic of all modal associative Routley-Meyer frames is the logic of all Kripke frames with valuations in complete distributive FL-algebras.


• 20.3.2019
  Joan Betrran-San Millán (FLU CAS): Frege's Begriffsschrift and second-order logic
  Abstract: Gottlob Frege developed in Begriffsschrift (1879) the concept-script, the fi rst formal system in the history of modern logic. The similarities between Frege's system and some contemporary formal systems have been taken for granted as evidence for a contemporary interpretation of the concept-script. In fact, the most common and traditional interpretation of Begriffsschrift's concept-script claims that it consists of a formal language of second-order logic and a deductive system for that language. In this talk, I offer a detailed analysis of Begriffsschrift's deductive system and justify that it must not be interpreted as a formal system of second-order logic. Specifically, I defend that a reformulation of the calculus of the concept-script in terms of a second-order calculus distorts its nature and, moreover, that some proofs of Begriffsschrift are not reproducible by means of this reformulation.


• 13.3.2019
  Pavel Hrubeš (MI CAS): Compression schemes and the Continuum Hypothesis
  Abstract: The Continuum Hypothesis is a conjecture about the cardinality of the set of real numbers. As such, it is a classical problem known to be undecidable from the usual ZFC axioms. We will show that some problems arising in the context of machine learning are equivalent to variants of CH - and hence undecidable over ZFC. We will focus on the problem of compressing finite strings of real numbers. (Based on joint work with S. Ben-David, S.Moran, A. Shpilka, A. Yehudayoff)


• 27.2.2019
  Ansten Klev (FLU CAS): Identity and definition in natural deduction
  Abstract: Recall that in natural deduction each primitive constant is equipped with introduction and elimination rules. Such rules can be given not only for the logical connectives and the quantifiers, but also for the identity predicate: its introduction rule is the reflexivity axiom, t = t, and its elimination rule is the indiscernibility of identicals. Although a normalization theorem can be proved for the resulting system, one might not be entirely satisfied with this treatment of identity, especially if one adheres to the idea - going back to Gentzen - that the meaning of a primitive constant is determined by its introduction rule(s). Firstly, it is not obvious that the introduction rule for the identity predicate is strong enough to justify its elimination rule. Secondly, it is not clear what to say about definitions taking the form of equations. Such definitions are usually regarded as axioms, hence they must be additional introduction rules for the identity predicate. Since definitions are particular to theories, it follows that the meaning of the identity predicate changes from one theory to the other. I will show that by enriching natural deduction with a theory of definitional identity we can answer both of these worries: we can justify the elimination rule on the basis of the introduction rule, and we can extend any theory with definitions while keeping the reflexivity axiom as the only introduction rule for the identity predicate.


• 20.2.2019
  Luca Reggio (ICS CAS): Uniform interpolation for IPC via an open mapping theorem for Esakia spaces
  Abstract: The uniform interpolation property of the intuitionistic propositional calculus (IPC) wasfirst proved by Pitts in 1992 by means of proof-theoretic methods. We prove an open mapping theorem for the topological spaces dual to finitely resented Heyting algebras. In turn, this yields a short, self-contained semantic proof of Pitts result. Our proof is based on the methods of Ghilardi & Zawadowski. However, it does not require sheaves nor games, only basic duality theory for Heyting algebras. This is joint work with Sam van Gool.


• 6.2.2019
  Marco Abbadini (Universita degli Studi di Milano): The dual of compact ordered spaces is a variety
  Abstract: Last year (2018), Hofmann, Neves and Nora proved that the dual of the category of partially ordered compact spaces and monotone continuous maps is an infinitary quasi-variety. One of the open questions was: is it also a variety? We show that the answer is: yes, it is an infinitary variety.


• 12. 12. 2018
  Berta Grimau (UTIA CAS): The Instability of Plural Scepticism towards Superplural Logic
  Abstract: Plural Logic is an extension of First-Order Logic which has, as well as singular terms and quantifiers, their plural counterparts. Analogously, Superplural Logic is an extension of Plural Logic which has, as well as plural terms and quantifiers, superplural ones. The basic idea is that superplurals stand to plurals like plurals stand to singulars (they are pluralized plurals). Allegedly, Superplural Logic enjoys the expressive power of a simple type theory while committing us to nothing more than the austere ontology of First-Order Logic. Were this true, Superplural Logic would be a useful tool, with various applications in the philosophy of mathematics, metaphysics and formal semantics. However, while the notions of plural reference and quantification enjoy widespread acceptance today, their superplural counterparts have been received with a lot of scepticism. In this talk, I will argue for the legitimacy of a face value interpretation of Superplural Logic by showing that some ordinary languages display clear cases of superplural reference and that they do so in an indispensable manner. Since the arguments I will put forward are of the same sort friends of Plural Logic have employed to defend their position, I will conclude that the (commonly held) view that Plural Logic is legitimately interpreted at face value but not so its superplural extensions is likely to suffer from internal tensions.


• 5. 12. 2018
  Guillermo Badia (Kepler University, Linz): Maximality of first-order logics based on finite MTL-chains
  


• 28. 11. 2018
  Andrew Tedder (ICS CAS): Residuals and conjugates in positive substructural logic
  Abstract: While the relations between an operation and its residuals play an essential role in substructural logic, a closely related relation between operations is that of conjugation - so closely related that with Boolean negation, the conjugates and residuals of an operation are interde_nable. In this talk extensions of the Lambek Calculus including conjugates of fusion (without negation) are investigated. Some interesting properties of the conjugates are discussed, a proof system is presented, its adequacy questioned, and some applications are considered.


• 21. 11. 2018
  Carles Noguera (UTIA CAS): General neighborhood and Kripke semantics for modal many-valued logics
  Abstract: Frame semantics, given by Kripke or neighborhood frames, do not give completeness theorems for all modal logics extending, respectively, K and E. Such shortcoming can be overcome by means of general frames, i.e. frames equipped with a collection of admissible sets of worlds (which is the range of possible valuations over such frame). We export this approach from the classical paradigm to modal many-valued logics by defining general A-frames over a given residuated lattice A (i.e., the usual frames with a collection of admissible A-valued sets). We describe in details the relation between general Kripke and neighborhood A-frames and prove that, if the logic of A is finitary, all extensions of the corresponding logic E of A are complete w.r.t. general neighborhood frames. Our work provides a new approach to the current research trend of generalizing relational semantics for non-classical modal logics to circumvent axiomatization problems.


• 14. 11. 2018
  Luca Reggio (ICS CAS): Duality, definability and continuous functions
  Abstract: Weierstrass approximation theorem states that any continuous real-valued function defined on a closed real interval can be approximated by polynomials. In 1937 Marshall Stone proved a vast generalisation of this result: nowadays known as the Stone-Weierstrass theorem, this is a fundamental result of functional analysis with far-reaching consequences. We show how, through duality theory, the Stone-Weierstrass theorem can be seen as an instance of the Beth definability property of a certain logic.


• 19. 10. 2018
  Special session -- Amanda Vidal (ICS CAS): Generalizing Geiger's result to two valued versions
  Abstract: We will go through an introduction to the algebraic approach to CSP and VCSP. We will then explore a notion of order-homomophism over valued structures that gives place to certain polymorphisms, and we show that these polymorphisms, considered over a small transformation of the original structure, characterize up to preservation the set of pp-definable formulas in the structure. When the polymorphisms preserve both strong or weak conjunction, they will characterize the corresponding definable formulas in locally finite structures. This can be seen as a natural generalization of Geigers characterization result for usual CSP to valued cases. Moreover, we will study how these kind of polymorphisms can be modified to preserve positive formulas with strong conjunction only (in the sense of the usual VCSP studied in the literature).