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Hyperintensional Modal Logic: Motivation, Semantic Frameworks, and Basic Theory.
Dragonová, Ivana ; Sedlár, Igor (advisor) ; Punčochář, Vít (referee)
A modal operator is hyperintensional if it does not respect the Equivalence Rule (RE), according to which if two formulas are logically equivalent, then so are the results of applying the modal operator to them. Typically, this happens when dealing with topics finer-grained than propositions, such as the notions of knowledge and belief. This thesis discusses the class of modal logics not closed under (RE) called hyperintensional modal logics and gives an overview of the semantic approaches one can use to give a suitable interpretation for this class of logics. We discuss a state-based approach first introduced by Rantala(1982) and later developed by Wansing(1990) and a structuralist approach proposed by Cresswell(1975). In the final part, we discuss a recent approach by Sedlár (2021), Pascucci and Sedlár (2023), and show that the above-mentioned state-based and structuralist approaches can both be modeled within Sedlár's hyperintensional models. We prove completeness results for the discussed hyperintensional semantic frameworks - all of them are sound and complete with respect to the smallest (hyperintensional) modal logic. 1
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Naive set theory with exclusive interpretation of quantifiers
Blahynka, Martin ; Punčochář, Vít (advisor) ; Stejskalová, Šárka (referee)
Naive set theory can be formalised in first-order logic as a theory with one axiom (of extensionality) and one axiom schema (of unrestricted comprehension). It is widely known that this theory is inconsistent. What is less known is that a mere reinterpretation of the quantifiers in the schema of unrestricted comprehension blocks all the well-known paradoxes of naive set theory. This is the case when the quantifiers are interpreted exclusively, which is an idea that originates in Wittgenstein's Tractatus in the context of elimination of identity from logic. In the context of set theory, the idea was first used by Jaakko Hintikka thirty five years later. This thesis introduces and investigates the possibility of using exclusive interpretation of quantifiers to avoid paradoxes of naive set theory. The main criterion of success is consistency of the resulting theory. The main result of this thesis is the proof that the set theories, which use the idea of exclusive interpretation and which Hintikka left as possibly consistent, are inconsistent. The inconsistency is discussed in the context of Russell's vicious circle principle, which is found to be inadequate.
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Semantics of some unusual modal logics
Punčochář, Vít ; Peregrin, Jaroslav (advisor) ; Bílková, Marta (referee)
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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A general framework for logics of questions
Punčochář, Vít
This paper provides an overview of basic inquisitive semantics and its generalization proposed in (Punčochář, submitted). It is shown that the generalization allows to model questions over any of a large class of non-classical logics and so avoids paradoxes of material implication and irrelevance in the logic of questions. Moreover, it is advocated that the general framework does not lose any of the characteristic features of basic inquisitive semantics that are needed for modeling of questions.
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Substructural logics for pooling information
Sedlár, Igor ; Punčochář, Vít
This paper puts forward a generalization of the account of pooling information – offered by standard epistemic logic – based on intersection of sets of possible worlds. Our account is based on information models for substructural logics and pooling is represented by fusion of information states. This approach yields a representation of\npooling related to structured communication within groups of agents. It is shown that the generalized account avoids some problematic features of the intersection-based approach. Our main technical result is a sound and complete axiomatization of a substructural epistemic logic with an operator expressing pooling.\n
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Knowledge is a Diamond
Punčochář, Vít
In the standard epistemic logic, the knowledge operator is represented as a box operator, a universal quantifier over a set of possible worlds. There is an alternative approach to the semantics of knowledge, according to which an agent a knows a proposition iff a has a reliable (e.g. sensory) evidence that supports the proposition. In this interpretation, knowledge is viewed rather as an existential, i.e. a diamond modality. In this paper, we will propose a formal semantics for substructural logics that allows to model knowledge on the basis of this intuition. The framework is strongly motivated by a similar semantics introduced by (Bílková, Majer, Peliš, 2016). However, as we will argue, our framework overcomes some unintuitive features of the semantics from (Bílková, Majer, Peliš, 2016). Most importantly, knowledge does not distribute over disjunction in our logic.
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Hypothetical Judgements, Truth and Assertibility
Punčochář, Vít ; Kolman, Vojtěch (advisor) ; Sedlár, Igor (referee) ; Bílková, Marta (referee)
Vít Punčochář Dissertation: Hypothetical Judgements, Truth and Assertibility Abstract: The main topic of this thesis is the logic of indicative conditionals, i.e. sentences of the form If A then B. In classical logic, these sentences are analysed with the help of the so- called material implication. However, the analysis is problematic in many respects. Some chapters of the thesis are devoted to the explanation of the problems, which one necessarily faces when analysing conditionals with the apparatus of standard classical logic. The stress is laid upon the fact that here we are led to a paradoxical situation: some general principles of classical logic (e.g. the principle according to which one can infer If not-A then B from A or B) seem to be unquestionable, but they have very controversial consequences. In the thesis, attempts are presented to defend classical logic as well as to revise it. The approaches to the logical analysis of conditionals are classified into two basic kinds: the first one might be called ontic and the second one epistemic. The ontic approach defines all crucial semantic notions in terms of the concept of truth that is modelled in logic as a relation between sentences of a given language and states of affairs. In contrast, the epistemic approach is not based on the concept of truth...
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Comparison of logical and psychological perspectives on the concept of number.
Kuncová, Alexandra ; Punčochář, Vít (advisor) ; Kůrka, Petr (referee)
This paper is divided into three main parts. In the first part, we propose a logical approach to the concept of number based on Frege's Foundations of Arithmetic. Besides the main attempt to define and classify number per se, we also discuss Husserl's struggle with psychologism, Frege's logicism, and the construction of the series of natural numbers. In the second part, we look at a psychological approach to the concept of number through theories and experiments of cogni- tive science. We focus on infants' understanding of numbers and amounts, their counting abilities and later conventional skills. In the third part, we summarise differences as well as similarities of these two approaches. Keywords: cognitive science, concept, Frege, Husserl, identity, logicism, num- ber, one-to-one correspondence, Piaget, psychologism. "As far as the laws of mathematics refer to reality, they are not certain; and as far as they are certain, they do not refer to reality." (Albert Einstein, Geometry and Experience, 1921)
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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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