First-order belief and paraconsistency

A first-order logic of belief with identity is proposed, primarily to give an account of possible de re contradictory beliefs, which sometimes occur as consequences of de dicto non-contradictory beliefs. A model has two separate, though interconnected domains: the domain of objects and the domain of appearances. The satisfaction of atomic formulas is defined by a particular S-accessibility relation between worlds. Identity is non-classical, and is conceived as an equivalence relation having the classical identity relation as a subset. A tableau system with labels, signs, and suffixes is defined, extending the basic language LQB by quasiformulas (to express the denotations of predicates). The proposed logical system is paraconsistent since φ ∧ ¬φ does not “explode” with arbitrary syntactic consequences

In what sense is Kantian principle of contradiction non-classical?

On the ground of Kant’s reformulation of the principle of contradiction, a non-classical logic KC and its extension KC+ are constructed. In KC and KC+, ¬(φ ∧ ¬φ),φ → (¬φ → ψ), and φ ∨ ¬φ are not valid due to specific changes in the meaning of connectives and quantifiers, although there is the explosion of derivable consequences from {φ,¬φ} (the deduction theorem lacking). KC and KC+ are interpreted as fragments of an S5-based first-order modal logic M. The quantification in M is combined with a “subject abstraction” device, which excepts predicate letters from the scope of modal operators. Derivability is defined by an appropriate labeled tableau system rules. Informally, KC is mainly ontologically motivated (in contrast, for example, to Jaśkowski’s discussive logic), relativizing state of affairs with respect to conditions such as time

Reinterpreting Rigidity : Rigid and Non-Rigid Reference of Proper Names in Alethic, Doxastic, and Mixed Contexts

Työni tarkastelee erisnimien asemaa mahdollisuuksien ja uskomuslauseiden konteksteissa. Sen lähtökohtana on nykyisin yleisesti hyväksytty teoria, jonka mukaan nimet ovat “jäykkiä” aleettisten tai metafyysisten mahdollisuuksien suhteen. Mahdollisten maailmojen semantiikan kehyksessä tämä tarkoittaa sitä, että nimi viittaa samaan olioon jokaisessa mahdollisessa maailmassa. Jäykkyys on usein käsitetty erisnimien semanttiseksi ominaisuudeksi, mikä on tuottanut uusia ongelmia konteksteissa, joissa nimet eivät käyttäydy odotusten mukaan. Tunnetuimpia esimerkkejä ovat propositionaaliset asenteet kuten uskomuslauseet. Tarkoituksenani on näyttää, että oletus jäykkyydestä nimien semanttisena ominaisuutena johtaa hajanaisuuteen, jossa poikkeamia on käsiteltävä tarpeettoman monimutkaisilla teorioilla, jotka voivat erkaantua vahvastikin aleettisen logiikan mallien yksinkertaisuudesta. Sen sijaan jäykkyyden näkeminen modaalisten kontekstien ominaisuutena mahdollistaa propositionaalisten asenteiden ja aleettisten modaalisuuksien yhtenäisemmän käsittelyn. Argumenttini jakautuu kahteen osaan: negatiiviseen teesiin, jonka tarkoitus on osoittaa, että teoria jäykkyydestä nimien semanttisena ominaisuutena on ristiriidassa monien luonnollisen kielen ilmiöiden kanssa, sekä positiiviseen teesiin siitä, miten erisnimiä voidaan kohdella yhtenevästi useissa epäsuorissa konteksteissa. Vastaesimerkkini keskittyvät erityisesti yksinkertaisiin luonnollisen kielen lauseisiin, joissa filosofit kuten Saul Kripke ja Scott Soames ovat väittäneet nimien esiintyvän jäykkinä, sekä erisnimien “demonstratiiviseen” ja “attributiiviseen” käyttöön. Oman teesini kehyksenä on Kathrin Glüerin ja Peter Paginin suhteellisten modaliteettien (relational modalities) semantiikka, joka mahdollistaa kaksitasoisen (aktuaalisen tai mahdollisen) evaluoinnin erisnimille. Tämä semantiikka on uskomuslauseille tietyssä mielessä alimäärittynyt: nimien jäykkyys yksinomaan uskomuskonteksteissa voidaan määrittää kvanttorien avulla, mutta vain puhujan intentio määrittää oikean tulkinnan. Täydennän puhujan intention roolia soveltamalla Jaakko Hintikan individuoinnin teoriaa. Hintikan mukaan nimet eivät voi suoraan viitata jäykästi: puhujan on kyettävä ensin jossain viitekehyksessä tunnistamaan se henkilö tai asia, josta hän puhuu. Nimen referentti on siis tietyssä mielessä tunnettava. Toisin kuin Hintikka, rajaan individuoinnin roolin vain propositionaalisiin asenteisiin tai muihin puhujan suhteen subjektiivisiin konteksteihin. Sen tarkoitus on tarjota pragmaattinen metodi suhteellisten modaliteettien mallien rinnalle täydentämään erisnimien semanttista alimäärittyneisyyttä uskomuslauseiden sekä muiden propositionaalisten asenteiden yhteydessä

Intensional Cyberforensics

This work focuses on the application of intensional logic to cyberforensic analysis and its benefits and difficulties are compared with the finite-state-automata approach. This work extends the use of the intensional programming paradigm to the modeling and implementation of a cyberforensics investigation process with backtracing of event reconstruction, in which evidence is modeled by multidimensional hierarchical contexts, and proofs or disproofs of claims are undertaken in an eductive manner of evaluation. This approach is a practical, context-aware improvement over the finite state automata (FSA) approach we have seen in previous work. As a base implementation language model, we use in this approach a new dialect of the Lucid programming language, called Forensic Lucid, and we focus on defining hierarchical contexts based on intensional logic for the distributed evaluation of cyberforensic expressions. We also augment the work with credibility factors surrounding digital evidence and witness accounts, which have not been previously modeled. The Forensic Lucid programming language, used for this intensional cyberforensic analysis, formally presented through its syntax and operational semantics. In large part, the language is based on its predecessor and codecessor Lucid dialects, such as GIPL, Indexical Lucid, Lucx, Objective Lucid, and JOOIP bound by the underlying intensional programming paradigm.Comment: 412 pages, 94 figures, 18 tables, 19 algorithms and listings; PhD thesis; v2 corrects some typos and refs; also available on Spectrum at http://spectrum.library.concordia.ca/977460

RAMIFIED TYPE THEORY AS INTENSIONAL LOGIC

Ovaj doktorski rad sastoji se od dva glavna dijela. Prvi se dio bavi pitanjem što sustava čine funkcije u razgranatoj teoriji tipova Bertranda Russella, kako ju je izložio u filozofijskome uvodu prvoga izdanja Principia Mathematica.U tome se dijelu rada brani eliminativističko tumačenje i pokušava pokazati da Russell sam stavačne funkcije u Principia razumije samo kao izraze, kao tzv. nepotpune simbole, koji ne označavaju nikakve izvanjezične predmete poput pojmova ili atributa.This doctoral thesis consists of two main sections. The first section addresses the background ontology of Bertrand Russell’s ramified type theory as described in Principia Mathematica. More precisely, it deals with the question of the ontological status of propositional functions. The concept of a propositional function is one of the central concepts of Russell’s theory of types, both in the first draft of the theory in “Appendix B” of The Principles of Mathematics andinitsmatureformulationinthefirsteditionofPrincipia.However,howtounderstandwhat Russell meant by “propositional functions” remains controversial. What are propositional functions? Are they some sort of intensional abstract entities, like properties and relations, or just expressionsofthelanguageoftypetheory,i.e.openformulas?Aneliminativistinterpretationis proposedandclaimedthatRussell’spropositionalfunctionsaretobeunderstoodonlyasexpressions,astheso-called“incompletesymbols”,whichdonotdenoteanyextra-linguisticobjects, such as attributes, whether in realist or constructivist sense. It is argued that the ramified type theory of Principia should not be understood as an abandonment of Russell’s earlier substitutional theory, but rather as its continuation. The ramified type hierarchy is a consequence of Russell’s belief that the paradoxes of propositions that plagued the substitutional theory can only be avoided by some kind of a type differentiation of propositions. On the other hand, the elimination of propositional functions (as well as propositions) from the ontology of Principia is a consequence of Russell’s conception of logic as universal science, which must contain only one type of genuine variables – viz., completely unrestricted entity variables, with everything that exists as their values. The doctrine of the unrestricted variable has been formulated by Russell in The Principles of Mathematics and is an inseparable part of his understanding of logic. The theory of denoting phrases he developed in “On Denoting” provided the tool for the elimination of higher-order entities from the background ontology of his logic. This way, Russell managed to retain a complex type hierarchy of expressions needed to avoid the paradoxes and at the same time preserve the doctrine of the unrestricted variable. At the end of the first section, certain advantages of rejecting the doctrine of the unrestricted variable and Russell’s understanding of propositional functions as incomplete symbols are recognized, and suggested that the interpretation of the ramified hierarchy as an ontological hierarchy of concepts might be philosophically justified. Inthesecondsection,aformalsystemofcumulativeintensionalramifiedtypetheory(KIRTT) is presented, guided by a realist interpretation of a ramified type hierarchy and with semantics based on an intensional generalization of Henkin models. The aim was to formalize certain metaphysical intuitions concerning the nature of intensional entities and to sketch one possible formal theory of concept

RAMIFIED TYPE THEORY AS INTENSIONAL LOGIC

Ovaj doktorski rad sastoji se od dva glavna dijela. Prvi se dio bavi pitanjem što sustava čine funkcije u razgranatoj teoriji tipova Bertranda Russella, kako ju je izložio u filozofijskome uvodu prvoga izdanja Principia Mathematica.U tome se dijelu rada brani eliminativističko tumačenje i pokušava pokazati da Russell sam stavačne funkcije u Principia razumije samo kao izraze, kao tzv. nepotpune simbole, koji ne označavaju nikakve izvanjezične predmete poput pojmova ili atributa.This doctoral thesis consists of two main sections. The first section addresses the background ontology of Bertrand Russell’s ramified type theory as described in Principia Mathematica. More precisely, it deals with the question of the ontological status of propositional functions. The concept of a propositional function is one of the central concepts of Russell’s theory of types, both in the first draft of the theory in “Appendix B” of The Principles of Mathematics andinitsmatureformulationinthefirsteditionofPrincipia.However,howtounderstandwhat Russell meant by “propositional functions” remains controversial. What are propositional functions? Are they some sort of intensional abstract entities, like properties and relations, or just expressionsofthelanguageoftypetheory,i.e.openformulas?Aneliminativistinterpretationis proposedandclaimedthatRussell’spropositionalfunctionsaretobeunderstoodonlyasexpressions,astheso-called“incompletesymbols”,whichdonotdenoteanyextra-linguisticobjects, such as attributes, whether in realist or constructivist sense. It is argued that the ramified type theory of Principia should not be understood as an abandonment of Russell’s earlier substitutional theory, but rather as its continuation. The ramified type hierarchy is a consequence of Russell’s belief that the paradoxes of propositions that plagued the substitutional theory can only be avoided by some kind of a type differentiation of propositions. On the other hand, the elimination of propositional functions (as well as propositions) from the ontology of Principia is a consequence of Russell’s conception of logic as universal science, which must contain only one type of genuine variables – viz., completely unrestricted entity variables, with everything that exists as their values. The doctrine of the unrestricted variable has been formulated by Russell in The Principles of Mathematics and is an inseparable part of his understanding of logic. The theory of denoting phrases he developed in “On Denoting” provided the tool for the elimination of higher-order entities from the background ontology of his logic. This way, Russell managed to retain a complex type hierarchy of expressions needed to avoid the paradoxes and at the same time preserve the doctrine of the unrestricted variable. At the end of the first section, certain advantages of rejecting the doctrine of the unrestricted variable and Russell’s understanding of propositional functions as incomplete symbols are recognized, and suggested that the interpretation of the ramified hierarchy as an ontological hierarchy of concepts might be philosophically justified. Inthesecondsection,aformalsystemofcumulativeintensionalramifiedtypetheory(KIRTT) is presented, guided by a realist interpretation of a ramified type hierarchy and with semantics based on an intensional generalization of Henkin models. The aim was to formalize certain metaphysical intuitions concerning the nature of intensional entities and to sketch one possible formal theory of concept

Extensões de primeira ordem para a lógica do anúncio público

Tese (doutorado) - Universidade Federal de Santa Catarina, Centro de Filosofia e Ciências Humanas, Programa de Pós-Graduação em Filosofia, Florianópolis, 2015.Dentre as lógicas multimodais, a lógica epistêmica dinâmica foi desenvolvida para modelar as mudanças de estados epistêmicos em grupos de agentes. Inspirada em recursos da lógica dinâmica (concebida para lidar com programas computacionais), aquela lógica permite representar a própria transição entre estados epistêmicos, tanto individuais como grupais, dos agentes considerados. Essa transição pode ser devida a diferentes ações epistêmicas (p.ex., o compartilhamento de uma informação com uma pessoa ou grupo em privado). A versão mais simples (e inicial) dessa lógica ficou conhecida como lógica do anúncio público, que considera apenas um tipo de ação epistêmica: a divulgação pública e simultânea de uma informação para todos os agentes. Essa divulgação é referida genericamente como "anúncio público"; porém, não precisa consistir, a rigor, em um anúncio típico, podendo ser um evento percebido simultaneamente por todos os agentes, contanto que cada agente saiba que todos os agentes estão acessando essa informação juntos, e que todos saibam desse mesmo fato, etc. A lógica do anúncio público apresenta pelo menos duas especificidades, não necessariamente preservadas por suas extensões que incluam outras ações epistêmicas: trata-se de uma lógica funcional (quando um anúncio for exequível, haverá somente uma maneira de fazê-lo) e dispensa o emprego de definições e provas por dupla indução (não há necessidade de se definir fórmulas e ações como duas categorias distintas e mutuamente dependentes de expressões da linguagem). Alguns trabalhos foram publicados sobre essa lógica, explorando diferentes axiomatizações, tratamentos semânticos e extensões; entretanto, quase todos se dedicam exclusivamente ao nível proposicional. Em nossa pesquisa bibliográfica, encontramos somente um artigo que desenvolve satisfatoriamente uma versão de primeira ordem para a lógica do anúncio público (KISHIDA, K. Public announcements under sheaves. In: New Frontiers in Artificial Intelligence, p. 96-108, 2013, ISBN 978-3-642-39930-5). Contudo, apesar da extrema sofisticação e elevado rigor técnico encontrados naquele trabalho, seu tratamento considera linguagens com anúncios públicos contendo somente fórmulas fechadas (sentenças), bem como um único agente epistêmico. Além disso, sua semântica, uma combinação de semântica de vizinhanças com semântica de feixes, motivada por um interesse filosófico bem específico (a interpretação intuitiva do operador epistêmico usual como representando conhecimento verificável), pode ser vista como desnecessariamente complicada se estamos interessados em uma interpretação standard para aquele operador, além de comprometer-se com uma perspectiva (um tanto polêmica) de contrapartes individuais, segundo a qual o mesmo objeto do domínio de interpretação não pode estar associado a mais de um ponto no modelo. Nossa contribuição propõe duas famílias, por assim dizer, de extensões de primeira ordem para a lógica do anúncio público, para quaisquer conjuntos finitos não-vazios de agentes epistêmicos e para anúncios contendo quaisquer fórmulas de suas respectivas linguagens. Os sistemas da primeira família estendem os correspondentes sistemas epistêmicos estáticos, providos com os usuais operadores epistêmicos para agentes individuais; e os da segunda fazem o mesmo com seus correspondentes sistemas estáticos, os quais, além de operadores epistêmicos individuais, adotam operadores de conhecimento distribuído em grupos de agentes. Além disso, nosso framework é o tradicional (modelos relacionais), o que simplifica consideravelmente o tratamento do assunto e não se compromete com indivíduos world-bounded. Antes de construir aquelas extensões dinâmicas, dedicamos alguns capítulos ao estudo dos vários sistemas epistêmicos estáticos que servirão como lógicas de base para nossa lógica do anúncio público, inclusive detalhando sistemas epistêmicos com conhecimento distribuído, e mostramos a completude em cada caso. Em se tratando de lógica epistêmica de primeira ordem, também fazemos brevemente uma defesa filosófica do emprego de quantificadores atualistas na lógica modal, com curiosos resultados relacionados com os esquemas conhecidos como Fórmula de Barcan e sua recíproca.Abstract : Among multimodal logics, Dynamic Epistemic Logic has been developed for modelling changes in epistemic states for groups of agents. Inspired by resources from Dynamic Logic (designed for dealing with computational programs), Dynamic Epistemic Logic manages to represent the very transition between epistemic states, either individual or collective, involving relevant agents. That transition might be caused by several epistemic actions (ex., sharing privately an information with a person or group). The simplest (and earliest) version for that logic became known as Public Announcement Logic, which considers a single type of epistemic action: a public and simultaneous disclosure of an information for all agents. This disclosure is generically referred as a "public announcement"; however, it doesn't have to be a typical announcement, and it might be an event, provided that this event be simultaneously realized by every agent, and that each agent knows that everybody is accessing that information together, and that everyone know this later fact, etc. Public Announcement Logic displays at least two peculiarities, not necessarily shared with its extensions including other epistemic actions: it's a functional logic (meaning that, when an announcement is feasible, there should be a unique way of doing it), and it dispenses with definitions and proofs based on double induction (i.e, there's no need for defining formulas and actions as two distinct and mutually dependent categories of language expressions). Some results have been published on this logic, exploring different axiomatizations, semantic treatments and extensions; however, almost all of them deal exclusively with propositional level. In our bibliographical survey, we found a single paper with a satisfactory study on first-order Public Announcement Logic (KISHIDA, K. Public announcements under sheaves. In: New Frontiers in Artificial Intelligence, p. 96-108, 2013, ISBN 978-3-642-39930-5). Nevertheless, in spite of its extreme refinement and high technical rigor, its approach is restricted to single-agent languages where announcements contain only closed formulas (sentences). Besides, its semantics, a combination of neighborhood semantics with sheaf semantics, motivated by a peculiar philosophical interest (the intuitive interpretation of the usual epistemic operator as representing verifiable knowledge), might be seen as unnecessarily complicated, especially if we are interested in the standard interpretation for that operator, and it forcibly commits itself with the (somewhat controversial) perspective of individual counterparts, according to which the same object in the interpretation domain must be uniquely associated to a point in the model. Our contribution provides two families, so to speak, of first-order extensions for Public Announcement Logic, for any non-empty finite sets of agents, as well as for announcements containing any formulas from their respective languages. The systems in the first family extend their correspondent static epistemic systems, containing the usual epistemic operators for individual agents; and the systems in the second one, do the same with their correspondent static systems, which, beside individual epistemic operators, include operators for distributed knowledge in groups of agents. Moreover, our framework is traditional (relational models), which considerably simplifies the approach and doesn't commit itself to world-bounded individuals. Before presenting those dynamic extensions, we dedicate a few chapters to the study of those static epistemic systems which shall be the basis for our Public Announcement Logic, also detailing some epistemic systems with distributed knowledge, and we prove completeness for each case. Concerning first-order epistemic logic, we additionally make a brief philosophical defense for actualist quantification in modal logic, with interesting results related to the schemes known as Barcan Formula and its converse

Intensional Cyberforensics

This work focuses on the application of intensional logic to cyberforensic analysis and its benefits and difficulties are compared with the finite-state-automata approach. This work extends the use of the intensional programming paradigm to the modeling and implementation of a cyberforensics investigation process with backtracing of event reconstruction, in which evidence is modeled by multidimensional hierarchical contexts, and proofs or disproofs of claims are undertaken in an eductive manner of evaluation. This approach is a practical, context-aware improvement over the finite state automata (FSA) approach we have seen in previous work. As a base implementation language model, we use in this approach a new dialect of the Lucid programming language, called Forensic Lucid, and we focus on defining hierarchical contexts based on intensional logic for the distributed evaluation of cyberforensic expressions. We also augment the work with credibility factors surrounding digital evidence and witness accounts, which have not been previously modeled. The Forensic Lucid programming language, used for this intensional cyberforensic analysis, formally presented through its syntax and operational semantics. In large part, the language is based on its predecessor and codecessor Lucid dialects, such as GIPL, Indexical Lucid, Lucx, Objective Lucid, MARFL, and JOOIP bound by the underlying intensional programming paradigm

FOIL Axiomatized

In an earlier paper, [5], I gave semantics and tableau rules for a simple first-order intensional logic called FOIL, in which both objects and intensions are explicitly present and can be quantified over. Intensions, being non-rigid, are represented in FOIL as (partial) functions from states to objects. Scoping machinery, predicate abstraction, is present to disambiguate sentences like that asserting the necessary identity of the morning and the evening star, which is true in one sense and not true in another. In this paper I address the problem of axiomatizing FOIL. I begin with an interesting sublogic with predicate abstraction and equality but no quantifiers. In [2] this sublogic was shown to be undecidable if the underlying modal logic was at least K4, though it is decidable in other cases. The axiomatization given is shown to be complete for standard logics without a symmetry condition. The general situation is not known. After this an axiomatization for the full FOIL is given, which is straightforward after one makes a change in the point of view.