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Intuitionistic logic as a useful tool
Vachková, Eva ; Švejdar, Vítězslav (advisor) ; Bílková, Marta (referee)
This work deals with intuitionistic logic and completness of Gentzen calculus with respect to its semantics. The completness proof uses saturated sequents. The language considered is at most countable. Furthermore, our work investigates one of the generalizations of intuitionistic logic, namely intuitionistic logic with constant domain, or Grzegorczyk's logic. We deal with Markov's principle and use it to prove that Gentzen calculus adapted to this logic is not cut-free complete with respect to Grzegorczyk's logic. Part of the work deals with Heyting algebras-one of the possible semantics of intuitionistic propositional logic. We show that the Rieger-Nishimura lattice is a Heyting algebra, too. For Heyting algebras, filters and prime filters are defined and used to obtain Kripke's frames. It is shown that the same formulas hold in these frames and in Heyting algebras.
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Interpolation in modal logics
Bílková, Marta ; Pudlák, Pavel (advisor) ; Švejdar, Vítězslav (referee) ; Iemhoff, Rosalie (referee)
Since Craig's landmark result on interpolation for classical predicate logic, proved as the main technical lemma in [14], interpolation is considered one of the centra! concepts in pure logic. Various interpolation properties find their applications in computer science and have many deep purely logical consequences. We focus on two propositional versions of Craig interpolation property: Craig Interpolation Property: for every provable implication (A -+ B) there is an interpolant I containing only only common variables of A and B such that both implications (A -+ I) and (I-+ B) are provable. Craig interpolation, although it seems rather technical, is a deep logical property. It is dosely related to expressive power of a logic - as such it entails Beth's definability property, or forces functional completeness. It is also related to Robinson's joint consistency of two theories that agree on the common language. Craig interpolation has an important algebraic counterpart - it entails amalgamation or superamalgamation property of appropriate algebraic structures. In case of modal provability logics, Craig interpolation entails fixed point theorem. There are other interpolation properties, defined w.r.t. a consequence relation rather then w.r.t. a provable implication. In presence of deduction theorem the two...
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Explicit fixed-points in provability logic
Chvalovský, Karel ; Bílková, Marta (referee) ; Švejdar, Vítězslav (advisor)
The aim of this diploma thesis is to discuss the explicit calculations of xed-points in provability logic GL. The xed-point theorem reads: For every modal formula A(p) such that each occurrence of p is under the scope of ¤, there is a formula D containing only sentence letters contained in A(p), not containing the sentence letter p, such that GL proves D ' A(D). Moreover, D is unique up to the provable equivalence. Firstly, we establish some special cases of the theorem and then we will look more closely at the full theorem. We show one semantic and two syntactic full xed-point constructions and prove their correctness. We also discuss some complexity aspects connected with the constructions and present basic upper bounds on length and modal depth of the constructed xed-points.
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Categories of fuzzy sets
Luhan, Ondřej ; Bílková, Marta (referee) ; Běhounek, Libor (advisor)
Category theory provides very useful tools for studying mathematical structures and phenomena. One of the structures that is studied in a category-theoretical manner are fuzzy sets. If we consider fuzzy sets as objects and set up certain kind of structure preserving mappings as morphisms, we can obtain a suitable category for our purposes. Goal of this work is to give an overview of preferably all important category-theoretical approaches to fuzzy sets that were done throughout relatively short history of category-theoretical modelling of fuzzy sets.
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Tableaux in non-classical logics
Dančák, Michal ; Peliš, Michal (referee) ; Bílková, Marta (advisor)
Dalo by se ríct, že jako dukazová metoda jsou sémantické stromy v Cesku nepríliš používané, a to i presto, že ve svete je to nejoblíbenejší dukazový systém pro modální logiku [1]. Vedle základního Hilbertova kalkulu se v ceské literature nejcasteji objevují sekventové kalkuly, prípadne kalkul prirozené dedukce. Presto má metoda sémantických stromu nekolik nezanedbatelných predností a zajímavých témat. Jak už název napovídá, tento kalkul vychází ze sémantiky - dukazy mají predevším sémantický charakter a pro "jednodušší" logiky jsou i velmi intuitivní. Dokazování je zároven i vyvracení. Pri dokazování metodou sémantických stromu vlastne hledáme protipríklad. Jestliže ho nenajdeme, a pokud jsme postupovali správne, tak neexistuje. Na poradí použití pravidel také nezáleží (až na nekolik vyjímek v nekterých logikách, které si pozdeji ukážeme). I díky temto výhodám je tato metoda také velmi vhodná pro strojové zpracování. V této práci jsem se rozhodl zamerit na to, jak se metoda sémantických stromu chová v substrukturální logice BCK (nekdy též FLew). Zacneme základními definicemi a tím, co to vlastne sémantické stromy jsou, dále bude následovat nekolik príkladu, definice logiky BCK a príslušných odvozovacích pravidel. Celá práce bude završena dukazem úplnosti a korektnosti tohoto kalkulu vuci kripkovské...
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Bisimulation
Arazim, Pavel ; Švejdar, Vítězslav (referee) ; Bílková, Marta (advisor)
This work is about bisimulation in modal logic as well as in intuicionistic logic without contraction. Bisimulation is a relation between models, which is weaker than isomorphism, yet still guarantees equivalence in the in the selected logic. It is typically used to demonstrate that some properties of models cannot be distinguished. For instance the cardinality cannot be distinguished, which is shown by disjunct union. Bisimulation also helps to clarify the relation between modal and classical logic. Apart from bisimulation, the related notion of bounded morphism is studied because it enables to elevate the unde- nability and undistinguishability to the discourse of frames. The part about modal logic is basically a compilation of well known facts, yet their interaction is made more clear and some proofs, which are usually disregarded as obvious, are presented in an explicit manner. Talking about the second part, mere reasonable de nition of the semantics is an honest work. Yet even here the bisimulation and bounded isomorphism are introduced and some examples are shown in order to illustrate their utility.
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Semantics of some unusual modal logics
Punčochář, Vít ; Bílková, Marta (referee) ; Peregrin, Jaroslav (advisor)
The rst part deals with Carnap s contribution to the modal logic. The Carnap s work is included in the historical context. His reaction to the Lewis calculi of the strict implication is discussed and also his anticipation of the Kripkean possible worlds semantics, which the contemporary modal logic is based on. The main aim of the second part was to consider some kinds of modalities. These kinds of modalities have epistemic character because they always depend on certain knowledge. The main result of the diploma work is the introduction of four new logics. Their semantics is set up in the similar fashion in which Carnap de ned his own modal logic. Some basic features of these logics are shown and their axiomatization and relationship to some other more usual logics is investigated.
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Fuzzification of simple systems of deontic logic
Vostrá, Nelly ; Bílková, Marta (referee) ; Běhounek, Libor (advisor)
Deontické logiky bývají formalizovány jako druh modálních logik. V této práci aplikuji fuzzy modální logiku na dvojí systémy monadick ých deontických logik - systémy deontické logiky v užším smyslu a systémy alethické logiky s výrokovou konstantou Q. Pro tyto nové fuzzy deontické logiky dokazuji lokální větu o dedukci, korektnost vřuči příslušným fuzzy rámcřum a definovatelnost deontick ých systémřu v alethických.
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