On Leśniewski’s Characteristica Universalis

Leśniewski's systems deviate greatly from standard logic in some basic features. The deviant aspects are rather well known, and often cited among the reasons why Leśniewski's work enjoys little recognition. This paper is an attempt to explain why those aspects should be there at all. Leśniewski built his systems inspired by a dream close to Leibniz's characteristica universalis: a perfect system of deductive theories encoding our knowledge of the world, based on a perfect language. My main claim is that Leśniewski built his characteristica universalis following the conditions of de Jong and Betti's Classical Model of Science (2008) to an astounding degree. While showing this I give an overview of the architecture of Leśniewski's systems and of their fundamental characteristics. I suggest among others that the aesthetic constraints Leśniewski put on axioms and primitive terms have epistemological relevance. © The Author(s) 2008

Cyclic proof systems for modal fixpoint logics

This thesis is about cyclic and ill-founded proof systems for modal fixpoint logics, with and without explicit fixpoint quantifiers.Cyclic and ill-founded proof-theory allow proofs with infinite branches or paths, as long as they satisfy some correctness conditions ensuring the validity of the conclusion. In this dissertation we design a few cyclic and ill-founded systems: a cyclic one for the weak Grzegorczyk modal logic K4Grz, based on our explanation of the phenomenon of cyclic companionship; and ill-founded and cyclic ones for the full computation tree logic CTL* and the intuitionistic linear-time temporal logic iLTL. All systems are cut-free, and the cyclic ones for K4Grz and iLTL have fully finitary correctness conditions.Lastly, we use a cyclic system for the modal mu-calculus to obtain a proof of the uniform interpolation property for the logic which differs from the original, automata-based one

The Algebras of Lewis Counterfactuals

The logico-algebraic study of Lewis's hierarchy of variably strict conditional logics has been essentially unexplored, hindering our understanding of their mathematical foundations, and the connections with other logical systems. This work aims to fill this gap by providing a comprehensive logico-algebraic analysis of Lewis's logics. We begin by introducing novel finite axiomatizations for varying strengths of Lewis's logics, distinguishing between global and local consequence relations on Lewisian sphere models. We then demonstrate that the global consequence relation is strongly algebraizable in terms of a specific class of Boolean algebras with a binary operator representing the counterfactual implication; in contrast, we show that the local consequence relation is generally not algebraizable, although it can be characterized as the degree-preserving logic over the same algebraic models. Further, we delve into the algebraic semantics of Lewis's logics, developing two dual equivalences with respect to particular topological spaces. In more details, we show a duality with respect to the topological version of Lewis's sphere models, and also with respect to Stone spaces with a selection function; using the latter, we demonstrate the strong completeness of Lewis's logics with respect to sphere models. Finally, we draw some considerations concerning the limit assumption over sphere models

Evaluating the Impact of Defeasible Argumentation as a Modelling Technique for Reasoning under Uncertainty

Limited work exists for the comparison across distinct knowledge-based approaches in Artificial Intelligence (AI) for non-monotonic reasoning, and in particular for the examination of their inferential and explanatory capacity. Non-monotonicity, or defeasibility, allows the retraction of a conclusion in the light of new information. It is a similar pattern to human reasoning, which draws conclusions in the absence of information, but allows them to be corrected once new pieces of evidence arise. Thus, this thesis focuses on a comparison of three approaches in AI for implementation of non-monotonic reasoning models of inference, namely: expert systems, fuzzy reasoning and defeasible argumentation. Three applications from the fields of decision-making in healthcare and knowledge representation and reasoning were selected from real-world contexts for evaluation: human mental workload modelling, computational trust modelling, and mortality occurrence modelling with biomarkers. The link between these applications comes from their presumptively non-monotonic nature. They present incomplete, ambiguous and retractable pieces of evidence. Hence, reasoning applied to them is likely suitable for being modelled by non-monotonic reasoning systems. An experiment was performed by exploiting six deductive knowledge bases produced with the aid of domain experts. These were coded into models built upon the selected reasoning approaches and were subsequently elicited with real-world data. The numerical inferences produced by these models were analysed according to common metrics of evaluation for each field of application. For the examination of explanatory capacity, properties such as understandability, extensibility, and post-hoc interpretability were meticulously described and qualitatively compared. Findings suggest that the variance of the inferences produced by expert systems and fuzzy reasoning models was higher, highlighting poor stability. In contrast, the variance of argument-based models was lower, showing a superior stability of its inferences across different system configurations. In addition, when compared in a context with large amounts of conflicting information, defeasible argumentation exhibited a stronger potential for conflict resolution, while presenting robust inferences. An in-depth discussion of the explanatory capacity showed how defeasible argumentation can lead to the construction of non-monotonic models with appealing properties of explainability, compared to those built with expert systems and fuzzy reasoning. The originality of this research lies in the quantification of the impact of defeasible argumentation. It illustrates the construction of an extensive number of non-monotonic reasoning models through a modular design. In addition, it exemplifies how these models can be exploited for performing non-monotonic reasoning and producing quantitative inferences in real-world applications. It contributes to the field of non-monotonic reasoning by situating defeasible argumentation among similar approaches through a novel empirical comparison

Theory of Judgment in Edmund Husserl's Logical Investigations

Tutkielma käsittelee fenomenologisen perinteen perustajan Edmund Husserlin kognitiivisen tai tiedollisen arvostelman teoriaa vuosina 1900-1901 julkaistuissa Loogisissa tutkimuksissa ja muissa samalta Husserlin ajattelun kaudelta peräisin olevissa kirjoituksissa. Tavoitteena on esittää tarkasti ja selvässä muodossa Husserlin teorian filosofiset pääpiirteet, teorian keskeisiin väitteisiin johtavat analyysit ja niitä ohjaavat teoreettiset pyrkimykset sekä asettaa Husserlin teoria laajempaan filosofiseen kontekstiin. Työssä on kolme päälukua, joista ensimmäinen on historiallinen katsaus erilaisiin arvostelman luonnetta koskeviin käsityksiin Platonista Brentanoon. Luvun keskeisin tulos on jaottelu kahden arvostelmaa koskevan teoriaperheen tai käsityksen välillä, joihin tässä työssä viitataan platonis-aristoteelisenä ja apprehensio-myöntämis-käsityksenä, ja joiden keskeisin erottava tekijä liittyy predikaation ja myöntämisen tai kieltämisen väliseen suhteeseen yleensä perustapauksena pidetyssä kategorisessa, "S on p" -muotoisessa arvostelmassa. Platonis-aristoteelisen käsityksen mukaan myöntäminen ja kieltäminen ovat predikaation muotoja, kun taas kilpailevan käsityksen mukaan arvostelma edellyttää erillistä apprehensio- tai käsittämisaktia, johon predikoiminen sisältyy, ja arvostelmassa tällaisen käsittämisen kohteeseen suhtaudutaan hyväksyvästi tai hylkäävästi sen päälle rakentuvassa asenteessa. Luvun viimeisissä osioissa käsitellään lisäksi 1800-luvun arvostelmateorioiden keskeisiä kiistakysymyksiä ja erityisesti niiden filosofien teorioita, jotka vaikuttivat suoraan Husserlin omien käsitysten muotoutumiseen, ja joista keskeisimmät ovat Bolzano, Lotze ja Brentano. Toisessa luvussa tarkastellaan Loogisten tutkimusten laajempaa filosofista projektia, joka liittyi sen selventämiseen, mikä on logiikan peruskäsitteiden rooli tiedossa, sekä arvostelman analyysien asemaa tässä projektissa. Huomion keskipisteenä on jännite kahden Husserlin ajattelua luonnehtivan perussitoumuksen välillä, joista ensimmäinen on käsitys logiikan objektiivisuudesta ja riippumattomuudesta psykologisista ilmiöistä ja toinen taas ajatus, että filosofisten peruskäsitteiden analyysissä tietoisuuden ilmiöiden kuvailemisella on olennainen rooli. Luvussa osoitetaan, että keskeinen ajatus näiden käsitysten yhteensovittamisen kannalta Loogisissa tutkimuksissa on Husserlin teoria merkityksistä ideaalisina aktilajeina, joiden instansseja yksittäiset intentionaaliset aktit kuten arvostelmat ovat. Luvun lopussa tarkastellaan lisäksi Husserlin intentionaalisuusteorian yleisiä piirteitä erityisesti suhteessa Brentanon aiempaan teoriaan. Kolmas luku on työn temaattinen ydin, jossa keskitytään nimenomaisesti Husserlin analyyseihin arvostelman luonteesta, rakenteesta ja suhteista objekteihin maailmassa. Luvussa tarkastellaan neljää toisiinsa läheisesti kytkeytyvää Husserlin teorian piirrettä, joita arvioidaan suhteessa Brentanon teoriaan ja ensimmäisessä luvussa esitettyyn historialliseen jaotteluun. Ensimmäinen näistä on väite, ettei arvostelmaan sisälly erotettavissa olevaa kokemusta, jossa sen kohde ainoastaan käsitettäisiin. Toinen on, ettei suoraviivainen arvostelman tekeminen ole sellaista myöntämistä, joka edellyttää edeltävää harkintaa ja kohdistuu harkinnan kohteena olevaan väitteeseen. Kolmas väite on, ettei Brentanon kuvailema pelkkä jonkin kohteen tietoinen esittäminen tai mieltäminen ole perustavampi tai rakenteellisesti yksinkertaisempi intentionaalinen aktityyppi kuin arvostelma, vaan jälkimmäisen "modifikaatio". Neljäs Husserlin teorian pääpiirre on monitahoinen käsitys arvostelmista "propositionaalisina akteina", jotka instantioivat ideaalisia propositioita ja suuntautuvat intentionaalisesti asiaintiloihin maailmassa tavalla, joka Husserlin yksinkertaisimpana pitämässä tapauksessa rakentuu havaintokokemuksen pohjalle jäsentämällä havainnon sisällön subjekti-predikaatti-muodossa. Tutkielman keskeinen johtopäätös ensimmäisessä luvussa esitetyn jaottelun näkökulmasta on, että Husserlin arvostelmateoriaa voidaan oiketutetusti pitää kriittisesti uudelleenmuotoiltuna versiona platonis-aristoteelisestä käsityksestä, jossa kuitenkin huomioidaan apprehensio-myöntämis-käsityksen keskeiset käsitteelliset erottelut ja sisällytetään teoriaan erityistapauksina ne ilmiöt, joihin jälkimmäinen käsitys keskittyi. Erityisesti Husserlin teoriaa voidaan pitää sellaisen aristoteelisen perinteen jatkajana, jossa predikaation kaltaisten ajattelun loogisten rakenteiden ajatellaan olevan läheisessä suhteessa havaintokokemukseen ja tavallaan kasvavan tällaisen kokemuksen rakenteista

Approximate syllogistic reasoning: a contribution to inference patterns and use cases

In this thesis two models of syllogistic reasoning for dealing with arguments that involve fuzzy quantified statements and approximate chaining are proposed. The modeling of quantified statements is based on the Theory of Generalized Quantifiers, which allows us to manage different kind of quantifiers simultaneously, and the inference process is interpreted in terms of a mathematical optimization problem, which allows us to deal with more arguments that standard deductive ones. For the case of approximate chaining, we propose to use synonymy, as used in a thesaurus, for calculating the degree of confidence of the argument according to the degree of similarity between chaining terms. As use cases, different types of Bayesian reasoning (Generalized Bayes' Theorem, Bayesian networks and probabilistic reasoning in legal argumentation) are analysed for being expressed through syllogisms

Logic, ontology, and arithmetic : a study of the development of Bertrand Russell’s Mathematical Philosophy from The Principles of Mathematics to Principia Mathematica

O presente trabalho tem por objeto de análise o desenvolvimento da Filosofia Matemática de Bertrand Russell desde os Principles of Mathematics até ­ e inlcuindo ­ a primeira edição de Principia Mathematica, tendo como fio condutor as mudanças no pensamento de Russell com respeito a três tópicos interligados, a saber: (1) a concepção de Russell da Lógica enquanto uma ciência (2) os compromissos ontológicos da Lógica e (3) a tese Russelliana de que a Matemática Pura ­ a Aritmética particular ­ é nada mais do que um ramo da Lógica. Esses três tópicos interligados formam um fio condutor que seguimos na tese para avaliar qual interpretação fornece o melhor relato da evidência textual disponível em Principia Mathematica e nos manuscritos produzidos por Russell no período relevante. A posição geral defendida é que a interpretação de Gregory Landini apresenta argumentos decisivos contra a ortodoxia de comentadores que atribuem à Principia uma hierarquia de tipos ramificada de entidades confusamente formulada, e mostramos que os três pontos apontados acima que formam o fio condutor da tese corroboram fortemente a interpretação de Landini. Os resultados que apontam para a conclusão geral de nossa investigação estão apresentados na tese dividida em duas partes. A primeira parte discute o desenvolvimento da lógica de concepção de Russell e do projeto Logicista desde sua gênese e nos Principles of Mathematics até Principia Mathematica. Esta primeira parte define o contexto para a segunda, que discute a Lógica Russeliana e o Logicismo em sua versão madura apresentada em Principia. Mostramos que, ao fim e ao cabo, o a teoria Lógica e a forma da tese Logicista apresentada em Principia é o resultado do longo processo iniciado com descoberta da Teoria dos Símbolos Incompletos que levou Russell a gradualmente reduzir os compromissos ontológicos de sua concepção da Lógica enquanto uma ciência, culminando na teoria apresentada na Introdução de Principia, na qual ele procura formular uma hierarquia dos tipos que evita o compromisso ontológico com classes, proposições e também com as assim chamadas funções proposicionais e que esse mesmo processo levou Russell a uma concepção da tese de Logicista de acordo com a qual a Matemática é uma ciência cujos compromissos ontológicos não incluem qualquer espécie de objetos (no sentido Fregeano) sejam eles particulares concretos ou abstratos.The present work has as its object of analysis the development of Bertrand Russell’s Mathematical Philosophy from the Principles of Mathematics up to ­ and including ­ the first edition of Principia Mathematica, having as a guiding thread the changes in Russell’s thought with respect to three interconnected topics, namely: (1) Russell’s conception of Logic as a science (2) the ontological commitments of Logic and (3) Russell’s thesis that Pure Mathematics ­ in particular Arithmetic ­ is nothing more than a branch of Logic. These three interconnected topics form a common thread that we follow in the dissertation to assess which interpretation offers the best account of the available textual evidence in Principia Mathematica and in the manuscripts produced by Russell in the relevant period. The general position held is that Gregory Landini’s interpretation presents decisive arguments against the orthodoxy of commentators who attribute to Principia a confusingly formulated hierarchy of ramfified types of entities, and we show that the three points indicated out above that form the main thread of the thesis strongly corroborate Landini’s interpretation. The results that point to the general conclusion of our investigation are stated in the dissertation divided into two parts. The first part discusses the development of Russell’s conception of Logic and the of Logicist project from its genesis and in Principles of Mathematics up to Principia Mathematica. This first part sets the context for a second, which discusses a Russellian Logic and Logicism in its mature version presented in Principia. We show that, in the end, the Logic theory and the form of the Logicist thesis presented in Principia is the result of a long process that started with the discovery of the theory of Incomplete Symbols which led Russell to reduce the ontological commitments of his conception of Logic as a science, culminating in the theory of types presented in Principia’s Introduction, in which Russell seeks to formulate a hierarchy of types that avoids the ontological commitment to classes, propositions and also with so­called propositional functions, and that this same process led Russell to a conception of the Logicist thesis according to Mathematics is a science with no ontological commitments to any kind of objects (in the Fregean sense) whether these are conceived as concrete or abstract particulars

Journal of Telecommunications and Information Technology, 2004, nr 3

kwartalni