On synthesizing Skolem functions for first order logic formulae

Skolem functions play a central role in logic, from eliminating quantifiers in first order logic formulas to providing functional implementations of relational specifications. While classical results in logic are only interested in their existence, the question of how to effectively compute them is also interesting, important and useful for several applications. In the restricted case of Boolean propositional logic formula, this problem of synthesizing Boolean Skolem functions has been addressed in depth, with various recent work focussing on both theoretical and practical aspects of the problem. However, there are few existing results for the general case, and the focus has been on heuristical algorithms. In this article, we undertake an investigation into the computational hardness of the problem of synthesizing Skolem functions for first order logic formula. We show that even under reasonable assumptions on the signature of the formula, it is impossible to compute or synthesize Skolem functions. Then we determine conditions on theories of first order logic which would render the problem computable. Finally, we show that several natural theories satisfy these conditions and hence do admit effective synthesis of Skolem functions

Annotated Translation: Jeux mathématiques et vice versa (2017, Paris, pp. 33-85)

The bachelor thesis is divided into two parts. The first part consists of the translation from French to Czech language, the second part contains the commentary. Selected parts of the popular science book Jeux mathématiques et vice versa written by four French mathematicians: Gilles Dowek, Jean-Pierre Bourguignon, Jean-Christophe Novelli a Benoît Rittaud are translated. The commentary includes the translation analysis of the text, based on which the concept of translation is determined. Furthermore, the problems, which has occurred during the process of translation, are presented in conjunction with the solutions and their examples.Tato bakalářská práce se skládá ze dvou částí - první část je tvořena překladem z francouzštiny do češtiny, druhá odborným komentářem. Jako výchozí text pro překlad posloužil úryvek z populárně naučné publikace pro dospělé Jeux mathématiques et vice versa, jejímiž autory jsou čtyři francouzští matematici: Gilles Dowek, Jean-Pierre Bourguignon, Jean-Christophe Novelli a Benoît Rittaud. V odborném komentáři je provedena překladatelská analýza výchozího textu, na základě které je dále stanovena překladatelská koncepce. Následně jsou představeny problémy, které při překladu nastaly, a jejich řešení i s konkrétními příklady.Ústav translatologieInstitute of Translation StudiesFaculty of ArtsFilozofická fakult

Formalizing Chemical Physics using the Lean Theorem Prover

Chemical theory can be made more rigorous using the Lean theorem prover, an interactive theorem prover for complex mathematics. We formalize the Langmuir and BET theories of adsorption, making each scientific premise clear and every step of the derivations explicit. Lean's math library, mathlib, provides formally verified theorems for infinite geometries series, which are central to BET theory. While writing these proofs, Lean prompts us to include mathematical constraints that were not originally reported. We also illustrate how Lean flexibly enables the reuse of proofs that build on more complex theories through the use of functions, definitions, and structures. Finally, we construct scientific frameworks for interoperable proofs, by creating structures for classical thermodynamics and kinematics, using them to formalize gas law relationships like Boyle's Law and equations of motion underlying Newtonian mechanics, respectively. This approach can be extended to other fields, enabling the formalization of rich and complex theories in science and engineering

A Reflective Theorem Prover for the Connection Calculus

Rewriting logic can be used to prototype systems for automated deduction. In this paper, we illustrate how this approach allows experiments with deduction strategies in a flexible and conceptually satisfying way. This is achieved by exploiting the reflective property of rewriting logic. By specifying a theorem prover in this way one quickly obtains a readable, reliable and reasonably efficient system which can be used both as a platform for tactic experiments and as a basis for an optimized implementation. The approach is illustrated by specifying a calculus for the connection method in rewriting logic which clearly separates rules from tactics

Unser die Welt : sprachphilosophische Grundlegungen der Erkenntnistheorie ; ausgewählte Artikel

Die Weiterentwicklung der Gedanken, die Wilhelm K. Essler 1972 in seinem Buch "Analytische Philosophie I" vorgetragen hat, ist bislang nur in Artikeln erfolgt. Die hier vorgelegte Auswahl hat das Ziel, den Kern seines Philosophierens, nach Sachgebieten geordnet, darzustellen. Im Zentrum seines Philosophierens steht die Untersuchung des Reflektierens, genauer: des philosophischen Reflektierens, anhand semantischer und epistemologischer Beispiele. Er orientiert sich dabei nicht an der Untersuchung vorhandener Erkenntnisakte, die oft schwer faßbar und noch schwerer eindeutig bestimmbar sind, sondern an deren rationaler Rekonstruktion in Modellen, gemäß dem Vorgehen in experimentellen Wissenschaften, und das besagt in der Philosophie natürlich: in Modellsprachen. Dieses Vorgehen hat den Vorteil, daß unter Einsatz des Instrumentariums der modernen Logik und ihrer Metalogik definitive Ergebnisse erzielt werden können, aufbauend auf den metalogischen Resultaten Gödels und Tarskis. In der Weiterführung der Ergebnisse von Gödel und Tarski wird gezeigt, daß die methodologische Unterscheidung von Erwähnen und Verwenden genau dem Vorgehen des semantischen Reflektierens gemäß der Sprachstufentheorie Tarskis entspricht und daß diese daher das geeignete Instrument zur Darstellung des epistemologischen Reflektierens und damit auch der erfahrungswissenschaftlichen Semantik ist. Anhand solcher präziser Sprachmodelle wird die Voraussetzungshaftigkeit allen sprachgebundenen Erkennens jeweils am Beispiel nachgewiesen. Macht man eben dieses Reflektieren zum neuen Gegenstand des untersuchenden Reflektierens, so benötigt man hierzu, will man die zuvor benützte Sprache des Reflektierens nun vollständig darstellen, abermals zusätzliche, in ihr noch nicht ausdrückbare Mittel des Reflektierens, und so fort ohne Ende. Dabei zeigt sich, daß dieses "und so fort ohne Ende" zum Problem der Grenze des Sagbaren gehört, und damit a fortiori zu den Grenzen des Philosophierens. Wie bei Platon wird Denken als ein inneres Sprechen verstanden, was eine enge Verbindung von Sprachphilosophie und Philosophie des Geistes impliziert. In neueren Untersuchungen hat Wilhelm K. Essler gezeigt, daß die Grundgedanken der buddhistischen Philosophie des Geistes mit diesen Ergebnissen des Reflektierens weitgehend übereinstimmen, daß jedoch diese über zwei Jahrtausende alte buddhistische Philosophie darüber hinaus auch Instrumente zur individuellen Anwendungen einer solchen sprachphilosophisch und erkenntnistheoretisch untermauerten Philosophie des Geistes enthält, die diese dann zu einer gelebten Philosophie werden lassen können, mit dem Ziel des Mottos, das auf der Eingangspforte des Tempels von Delphi zu lesen stand, nämlich: "Erkenne Dich selbst!

Dos Primórdios da Matemática aos Sistemas Formais da Computação

Livro de apoio ao ensino de graduação em disciplinas de matemática discreta e lógica, em cursos de ciência da computação.(0) Introdução (1) As Bases da Ciência da Computa cão,(2) As Origens: A Aritmética, (3) Os Números, (4) Os Números Primos, (5) Congruência e Aritm ética Modular, (6) Álgebra na Europa (7) A Lógica: de Leibniz a Boole, (8) Século XIX: Frege e a Lógica dos Predicados, (9) A Teoria dos Conjuntos, (10) Rela ções e Funcões, (11) Grupos e Corpos (12) Conjuntos e Enumera cão, (13) A Aritm ética nos Séculos XIX e XX, (14) Hilbert - Formalismo e os Sistemas Axiom áticos, (15) Gödel e os Limites dos Sistemas Formais, (16) Dos Fundamentos da Matemática aos Sistemas Formais, e (17) Os Sistemas Formais da Computa cão.recursos próprios; recursos do Departamento de Informática e Estatística da UFS

Paradoksy

Książka "Paradoksy" jest monografią poświęconą tym argumentacjom logicznym, które prowadzą, albo do sprzeczności, albo do wniosków niezgodnych z oczekiwaniami. Prawie wszystkie są do dziś źródłem kontrowersji i sporów zarówno wśród logików jak i filozofów. Szczególnie interesującymi są paradoksy o starożytnym rodowodzie. Książka zawiera prezentację wszystkich najważniejszych pradoksów logiki wraz z krytyczną analizą najpopularniejszych propozycji ich rozwiązań. W przypadku antynomii kłamcy, Newcomba, kół Arystotelesa, Trójcy Świętej, Protagorasa, kamienia, Kata i krokodyla autor proponuje własne, oryginalne propozycje rozwiązań. Układ treści monografii "Paradoksy" reprezentuje zaproponowaną przez autora klasyfikację paradoksów

Logic from Russell to Church.

This volume is number five in the eleven volume Handbook of the History of Logic. It covers the first fifty years of the development of mathematical logic in the twentieth century, and concentrates on the achievements of the great names of the period; Russell, Post, Gödel, Tarski, Church, and the like. This was the period in which mathematical logic gave mature expression to its four main parts ; set theory, model theory, proof theory and recursion theory. Collectively this work ranks as one of the greatest achievements of our intellectual history. Written by leading researchers in the field, both this volume and the Handbook as a whole are definitive reference tools for senior undergraduates, graduate students and researchers in the history of logic, the history of philosophy, and any discipline, such as mathematics, computer science, artificial intelligence, for whom the historical background of his or her work is a salient consideration.This volume is number five in the eleven volume Handbook of the History of Logic. It covers the first fifty years of the development of mathematical logic in the twentieth century, and concentrates on the achievements of the great names of the period; Russell, Post, Gödel, Tarski, Church, and the like. This was the period in which mathematical logic gave mature expression to its four main parts ; set theory, model theory, proof theory and recursion theory. Collectively this work ranks as one of the greatest achievements of our intellectual history. Written by leading researchers in the field, both this volume and the Handbook as a whole are definitive reference tools for senior undergraduates, graduate students and researchers in the history of logic, the history of philosophy, and any discipline, such as mathematics, computer science, artificial intelligence, for whom the historical background of his or her work is a salient consideration.Russell's Logic / (Andrew D. Irvine) -- Logic for Meinongian Object Theory Semantics / (Dale Jacquette) -- The Logic of Brouwer and Heyting / (Joan Rand Moschovakis) -- Thoralf Albert Skolem / (Jens Erik Fenstad and Hao Wang) -- The Logic of the Tractatus / (Michael Potter) -- Lesniewski's Logic / (Peter Simons) -- Hibert's Proof Theory / (Wilfried Sieg) -- Hilbert's Epsilon Calculus and its Successors / (Hartly Slater) -- Gödel's Logic / (Mark van Atten and Juliette Kennedy) -- Tarski's Logic / (Keith Simmons) -- Emil Post / (Alasdair Urquhart) -- Gentzen's Logic / (Jan von Plato) -- Lambda-calculus and Combinators in the 20th Century / (Felice Cardone and J. Roger Hindley) -- The Logic of Church and Curry / (Jonathan P. Seldin) -- Paradoxes, Self-reference and Truth in the Twentieth Century / (Andrea Cantini).Includes bibliographical references and index.Print version record.Elsevie