Research Seminar of the Department of Logic


On enhanced generalization (Cancelled!)

Vitezslav Svejdar (Department of Logic) will talk about enhanced generalization (Monday 16.3).

The generalization rule in logical calculi makes it possible to 
"unsubstitute" a variable $y$ from a formula $\varphi_x(y)$ and conclude 
that $\forall x \varphi$ (or that $\exists x \varphi$ if the step 
is happening in a premise), provided that there are no unwanted free 
occurrences of the unsubstituted variable. We consider an enhanced 
generalization rule that makes it possible to simultaneously 
unsubstitute not only several variables, but also several terms 
(provided that they are pairwise different and there are no unwanted 
occurrences of their outermost symbols). While we cannot claim that this 
enhanced generalization models a step in a reasoning, and it is not 
sound in logic with equality, we show that it is sound in logic without 
equality and that it has a useful application, namely the interpolation 
theorem (for logic without equality but with function symbols).


Believing the Axioms I
Tereza Stejskalová (logika, FFUK) bude mluvit axiomech ZFC a jejich historii a motivaci. Přednáška je založena na článku P. Maddy Believing the Axioms I (JSL 1988).


Believing the Axioms II

Radek Honzik will discuss P. Maddy's paper Believing the Axioms II (JSL 1988).

Believing the Axioms II is a continuation of the first paper by Maddy and deals with the Axiom of Determinacy (AD). The paper attempts to argue that AD (or its variants) may be a good candidate for a new strong axiom of set theory. The seminar will review the basic points of the paper (prior knowledge of the paper is not necessary).


Casual inferences in statistics

Lars S. Laichter (Department of logic) will introduce basic concepts of casual inferences in statistics. CHANGE OF DATE:  16.12.,  16:30.

In my presentation, I will provide a brief introduction and a discussion of the methods of causal inference in statistics. Methods of causal inference have been argued to be an important addition to traditional statistical methods. In particular, I will introduce the notion of the Structural Causal Model (SCM), as proposed by Judeau Pearl, to illustrate some of the major advancements in the general theory of causation. These advancements include methods for inferring a model’s properties based on (1) effects of interventions, (2) probabilities of counterfactuals, and (3) direct and indirect effects. Finally, I will discuss some of the possible applications of this theory, as well as its wider implications for scientific inquiry. Overall, I aim for this presentation to provide an accessible introduction to the methods of causal inference and highlight the advantages of the methods of causal inference over traditional statistical methods.


Who's who in large cardinals, II

Radek Honzik (Department of logic) will continue talking about large cardinals. Monday 2.12., 16:30. We will discuss "larger" large cardinals, such as measurable and supercompact cardinals, and their effect on the set-theoretical universe and applications in mathematics and set theory.


THE IMPOSSIBILITY OF SQUARING THE CIRCLE Gregory, Huygens and the limits of Cartesian geometry
Davide Crippa (Institute of Philosophy, AVČR) will talk about the impossibility of squaring the circle. Monday 25. November, 16:30, Department of Logic.

With the emergence of the algebraic movement in XVIth and XVIIth century geometry, the ideal that all mathematical problems should and could be solved by the most adequate means was fostered by outstanding mathematicians (Viète, Descartes). Yet it was a matter of dispute whether certain well-known problems, like the quadrature of the circle, could be solved by geometrically acceptable methods. My talk will explore this issue, considering a controversy occurred in 1668 between the Scottish mathematician James Gregory and the Dutch mathematician Christiaan Huygens, about the possibility of solving the quadrature of central conics (which included the circle) by algebraic means. Whereas the former held it was impossible, the latter believed that the circle could be squared algebraically. This controversy is significant because it hinged upon methodological or foundational questions: which were the bounds of Cartesian geometry? Are the five algebraic operations sufficient in order to express and solve all problems concerning the objects of Euclid's geometry?


Who's who in large cardinals, I
Radek Honzik (Department of logic) will talk about motivations, history and applications behind large cardinals. 16:30, Monday 18.11.2019.


Aspekty eliminovatelnosti řezu v různých logikách

Vítězslav Švejdar (Department of Logic, FFUK) bude mluvit o různých neklasických logikách a jejich vztahu k eleminovatelnosti řezu. 11. listopadu, 16:30.


Finitní verze Gödelových vět o neúplnosti

Martina Maxa bude mluvit o finitních verzích Gödelových vět o neúplnosti a Löbově větě. Monday November 4, 16:30.

Budeme se zabývat finitními protějšky známých vět dotýkajících se základů matematiky, jako jsou Gödelovy věty o neúplnosti či Löbova věta. Jejich finitní verze jsou již silnější než známé otevřené problémy ve výpočetní složitosti jako např. domněnka P se nerovná NP. Kromě finitní verze druhé Gödelovy věty, představíme i finitní verzy první Gödelovy věty a ukážeme vztah mezi těmito domněnkami. Také uvedeme tvrzení, jež by se dalo nazvat finitní verzí Löbovy věty. Cílem je ukázat, že otevřené problémy ve výpočetní složitosti mají blízký vztah k problémům týkajících se logiky a tímto i základů matematiky.


Most common proposals for solving the CH problem

Radek Honzik will survey old and recent attempts and proposals for solving the Continuum Hypothesis problem. Monday  October 21, 16:30.  Students interested in credits from the seminar should attend and discuss their participation in the seminar.

Students interested in credits from the seminar should attend and discuss their participation in the seminar.


Složitost Booleovských formulí

Seminář katedry: Vít Fojtík, Department of Logic, FF UK, 17:00, 29.4.2019.

Booleovské formule jsou typ Booleovských obvodů modelující výpočty, při kterých si nelze pamatovat mezivýsledky. Přestože lze ukázat, že existují funkce s exponenciální formulovou složitostí (a dokonce jich je většina), jedním z nevyřešených problémů výpočetní složitosti je najít ,,větší než polynomiální" dolní odhad pro nějakou konkrétní funkci. Snahy o sestrojení odhadu se ubírají dvěma převažujícími směry; první ja založený na abstraktním pojmu míry složitosti, zatímco druhý používá náhodně zvolené podfunkce dané funkce.


Indestructibility of Kurepa Hypothesis

Šárka Stejskalová, Department of Logic, FF UK

We will review an argument of Jensen and Schlechta form 1990 who showed that Kurepa Hypothesis can be indestructible under all ccc forcing notions. Possible generalizations will be discussed.


Abstract notion of liftability for supercompact cardinals

Radek Honzík, Department of Logic, FF UK

We will discuss generalizations of the property of kappa-directed closure which appears in the classical theorem of Levy on indestructible supercompact cardinals. Seminar takes place in C119 at 17:00.


Is there a difference in gaps between convergence and divergence in ZFC and PA?

Peter Vojtáš, Department of Software Engineering, MFF UK


(Homotopy) Type theory II.
Tomáš Lávička, Department of Logic, FF UK

Type theory is a possible alternative to set theory and first order predicate logic as a tool for mathematical foundations, which has recently been getting more and more attention especially due to the discovery of its close relation to algebraic topology. I will try to give an introductory  lecture to the basic concepts of type theory  and its formalism (not requiring any previous knowledge in the field) in such a way that we could get some idea on how one can do (not only constructive) mathematics with its framework. Along the way I will be suggesting where is the famous connection with algebraic topology coming from.  

Keywords: Types, identity types, dependent types, Univalence, constructive mathematics.


(Homotopy) Type theory

Tomáš Lávička, Department of Logic, FF UK

Type theory is a possible alternative to set theory and first order predicate logic as a tool for mathematical foundations, which has recently been getting more and more attention especially due to the discovery of its close relation to algebraic topology. I will try to give an introductory  lecture to the basic concepts of type theory  and its formalism (not requiring any previous knowledge in the field) in such a way that we could get some idea on how one can do (not only constructive) mathematics with its framework. Along the way I will be suggesting where is the famous connection with algebraic topology coming from.  

Keywords: Types, identity types, dependent types, Univalence, constructive mathematics. 


Definability of stationary subsets of $\omega_1$

Stefan Hoffelner, Department of Logic, FF UK



Prediction principles in set theory: It's OK to guess

Miha Habić, Department of Logic, FF UK

Diagonalization arguments play a central role in logic and specifically set theory, reaching from Cantor's proof of the uncountability of the reals to complicated forcing arguments. Guessing principles provide a substitute in cases where there are too many objects to diagonalize against in a naive way. They work by providing small approximations to the objects and guaranteeing that the approximations are correct "often enough". In the talk I shall give a short introduction to some simple guessing principles, after which I shall examine joint guessing principles, which allow us to guess many objects at once.  


The rearrangement number

William Brian, University of North Carolina at Charlotte, USA

Recall the classical result of Riemann that every conditionally convergent series can be rearranged to get a series which is divergent to infinity, oscillates or converges to an arbitrary real number. A natural question to ask is how many rearrangements are needed to witness the validity of this theorem. This leads to the definition of the rearrangement number which is a cardinal invariant of the continuum. We present recent results related to this number. The talk is based on the preprint by several authors including the speaker. 


Nestandardní metody v konstrukci modelů slabých aritmetik
Jana Glivická

Představím novou metodu konstrukce modelů slabých aritmetik. Použijeme nestandardních metod ke konstrukci jisté elementární extenze standardního modelu aritmetiky, která je uzavřena na operaci *. Tento fakt nám umožní definovat na univerzu této extenze tzv. gradované verze aritmetických operací sčítání a násobení. Ukážeme, jak volbou různých gradací můžeme ovlivňovat platnost některých aritmetických tvrzení ve vzniklých strukturách. Speciálně předvedeme, jak lze touto metodou získat model Robinsonovy aritmetiky, v němž platí hypotéza prvočíselných dvojčat.


Birkhoff's subdirect representation from a broader perspective

Tomáš Lávička

It is well known that varieties (resp.. quasi-varieties) are generated by its (relatively) subdirectly irreducible members (Birkhoff's representation). As we will see this representation theorem will not hold in general in the setting of generalized quasi-varieties (classes of algebras described by quasi-equations with countably many premises). I will present some characterization results of Birhoff's representability, which I then use to prove that the proper generalized quasi-variety generated by the [0,1] Lukasiewicz chain is representable in the above sense - the core idea of the proof uses a modification of the well-known topological proof of compactness for classical logic into higher cardinalities.


Gentzen's cut elimination strategy and Tait's cut elimination strategy in propositional sequent calculus
Anna Horská

Nowadays, we usually eliminate cuts by decreasing the cut-rank of the derivation (Tait's strategy). That is, a cut with the greatest complexity is chosen such that there are only smaller cuts above it and this one is then decomposed into smaller cuts. Gentzen applies another strategy in his article of 1935. He eliminates the highest cuts, i.e., cuts that there are no other cuts above them. Hence, this procedure does not pay attention to the complexity of the chosen cut. The main property of cut elimination that we are interested in is the growth of the height of the derivation during the elimination, especially we want to know the height of the cut-free derivation. I want to show that both strategies mentioned above give us the same cut-free derivation in propositional logic. Not only is the height of the cut-free derivations the same, but they also have the same structure. I will define a procedure for eliminating a single cut (according to Buss) which makes global changes to the derivation. Then, I use Church-Rosser property to obtain that cut-free derivations are the same, when the only difference during the elimination is the way we choose the next cut to eliminate.


Interpolants in the context of software verification
Martin Blicha

In the first part we present the role of interpolants in some verification algorithms to demonstrate the usefulness of the concept and the motivation for our work. In the second part we examine interpolation systems for propositional logic (the algorithms for computation of interpolants from a refutation proof), namely symmetric system (Pudlák, Krajíček), McMillan's system, and their generalization, framework of Labeled Interpolation Systems (D'Silva et al.). We conclude with incorporating partial variable assignment into the computation of interpolant, done in the framework of Labeled Partial Assignment Interpolation Systems (Jančík et al.).


Nestovaný sekventový kalkulus pro intuicionistickou logiku
Eva Kolovratníková

Nejprve představíme nestovaný kalkulus pro intuicionistickou logiku a vysvětlíme, jak funguje nestování. Pak přidáme pravidla pro kvantifikátory a získáme kalkulus pro intuicionistickou predikátovou logiku s konstantním univerzem. Následně přidáme omezení, abychom získali kalkulus pro standardní intuicionistickou predikátovou logiku. Na závěr si předvedeme prefixová tabla pro obě varianty intuicionistické logiky a ukážeme, jak transformovat tabla do nestovaných sequentů. (English version of abstract) Firstly, we introduce nested sequent calculus for intuitionistic logics and show how the nesting works. Secondly, we show that standard quantification rules lead to calculus for intuitionistic predicate logic with constant domains. Then we add some restrictions to obtain calculus for standard Intuitionistic predicate logic. Finally, we introduce prefixed tableaux for both variants of Intuitionistic logic and show how we can transform tableaux to nested sequents.


Vopěnkův princip a Vopěnkovy kardinály
Radek Honzík
Vopěnkův princip formuloval Petr Vopěnka a zní takto: Je-li A vlastní třída struktur v daném (množinovém) jazyce, pak existují dvě struktury M a N v A, že M je elementárně vnořitelná do N. Petr Vopěnka navrhl tento princip spíše s představou, že se brzy ukáže jako sporný, nicméně se tak nestalo. Ukážeme, že konzistentní síla tohoto tvrzení je velmi velká; v hierarchii velkých kardinálů se tento princip nachází mezi nejsilnějšími kardinály (např. implikuje konzistenci superkompatních kardinálů). Pozn. Podle obecenstva bude přednáška buď v češtině nebo angličtině.


Fermat's last theorem and Catalan's conjecture in arithmetics with weak exponentiation
Petr Glivický

Wiles's proof of Fermat's Last Theorem (FLT) has stimulated a lively discussion on how much is actually needed for the proof. Despite the fact that the original proof uses set-theoretical assumptions unprovable in Zermelo-Fraenkel set theory with axiom of choice (ZFC) - namely, the existence of Grothendieck universes - it is widely believed that "certainly much less than ZFC is used in principle, probably nothing beyond Peano arithmetic, and perhaps much less than that." (McLarty) In this talk, I will present a joint work with V. Kala. We studied (un)provabiliy of FLT and Catalan's conjecture in arithmetical theories with weak exponentiation, i.e. in theories in the language  $L=(0,1,+,\cdot,exp,<)$ where the $(0,1,+,\cdot,<)$-fragment is usually very strong (often even the complete theory $\mbox{Th}(\mathbb N)$ of natural numbers in that language) but the exponentiation satisfies only basic arithmetical properties and not much of induction. In such theories, Diophantine problems such as FLT or Catalan's conjecture, are formalized using the exponentiation exp instead of the exponentiation definable in the $(0,1,+,\cdot,<)$-fragment. I will present a natural basic set of axioms Exp for exponentiation (consisting mostly of elementary identities) and show that the theory $T=\mbox{Th}(\mathbb N)+Exp$ is strong enough to prove Catalan's conjecture, while FLT is still unprovable in $T$. This gives an interesting separation of strengths of the two famous Diophantine problems. Nevertheless, I show that by adding just one more axiom for exponentiation (the, so called, "coprimality" of exp) the theory becomes strong enough to prove FLT, i.e. FLT is provable in T+"coprimality". (Of course, in the proof of this, we use the Wiles's result too.)