Logika v souvislostech a aplikacich
Osnova sekce
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Though not as glamorous as some of its relatives, the notion of well-ordering is arguably one of the most important and useful ideas in mathematical logic and set theory, lying at the heart of a number of important breakthroughs over the last 150 years. We will survey the valuable role that well-orderings and the Well-Ordering Theorem have played in the development of the modern foundations of mathematics and in applications of logic and set theory to other areas of mathematics. We will cover over 100 years of research, from the late 19th century to the present day, along the way investigating the roles that well-orderings play in connection with constructions of pathological mathematical objects, infinitary logics, questions of definability, large cardinals, and infinite games.
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In my two lectures I will introduce you to type theory, which is a general form of language that includes propositional and predicate logic and that lies at the basis of many programming languages. My presentation will (of course) be coloured by my own interests, which tend to be philosophical. I will start by explaining, in very general terms, what a type is and then define, for the purposes of illustration, a simple type theory. The second lecture will be devoted to the Curry–Howard correspondence, by means of which logic is included in type theory. I will try to explain to you what a dependent type is and why dependent types are useful for the formalization of the universal quantifier ("for all x").
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V přednášce se seznámíme se základními vícehodnotovými logikami, jejich motivacemi a jejich algebraickou sémantikou. Podíváme se také na základní vlastnosti logických spojek, které často zůstávají v platnosti i mimo klasickou logiku.
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This lecture introduces inquisitive semantics as a contemporary framework for the logical analysis of questions. It is standard to analyse declarative sentences in terms of their truth conditions. However, truth-conditional semantics cannot be straightforwardly extended to questions, since they lack truth values. Inquisitive semantics addresses this limitation by shifting the focus from truth to informational support. This support-based perspective enables a uniform treatment of both statements and questions, and provides the means to analyse various interrogative types, including conjunctive and conditional questions. It also captures logical dependencies between questions and systematically accounts for phenomena such as presupposition. The resulting system, inquisitive logic, extends classical logic in a conservative way: it preserves classical treatment of statements but enriches the logical space to incorporate also questions. The lecture will present the basic ideas and underlying principles of this approach.
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There is no lecture on April 16th. Instead of the lecture, you can attend a talk by Prof. Peregrin at the Institute of Philosophy. You can find more information below.
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V první části přednášky se seznámíme s Belnap-Dunnovou logikou, kterou lze chápat jako základ skupiny logik zvaných relevantní. Nejprve s budeme zabývat historií a motivacemi celého projektu relevance – tzv. paradoxy implikace. Ukážeme si, jak souvisí řešení těchto paradoxů s klasickým principem Ex falso quodlibet, jehož důsledkem je, že se v deduktivním uzávěru jakéhokoli souboru tvrzení obsahujícího nekonzistence ocitnou všechna tvrzení a celý soubor se tím trivializuje. S nekonzistencemi v různých souborech informací se ovšem setkáváme denně a je proto užitečné mít nějaký logický systém, který, na rozdíl od klasické logiky, dokáže s nekonzistencemi pracovat netriviálně. Příkladem takovýchto logik, nazývaných parakonzistentní, je i Belnap-Dunnova logika. V druhé části se pak seznámíme s metodou, jak je možné definovat nad tímto systémem pravděpodobnost. Tato metoda je založena na tzv dvouúrovňových logikách, které oddělují usuzování o událostech („dolní“ úroveň) od usuzování o pravděpodobnostech těchto událostí („horní“ úroveň) a poskytují tak obecný a flexibilní rámec pro pro vytváření logických rámců pro usuzování s neurčitostmi. Závěrem pak budeme diskutovat různé další reprezentace neurčitosti, které jsou alternativami ke klasické pravděpodobnosti (belief functions, kvalitativní pravděpodobnost).
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We will explore topological semantics for modal logic and its applications in epistemic logic. In the first lecture, we will establish the completeness of modal logic S4 under interior semantics with respect to various classes of topological spaces, including all spaces, Alexandroff spaces, and T_0-spaces.
In the second lecture, we will explain how interior semantics can be interpreted epistemically — open sets can be seen as pieces of evidence. We will introduce topological evidence logics based on a coherentist account of epistemic justification and prove completeness for a simple version amounting to an extension of S4 with a universal modality. If time permits, we will show how to extend the topological evidence framework to account for the fact that accessing evidence usually requires resources. -
There is no lecture on May 21th.