Minimal Paradefinite Logics for Reasoning with Incompleteness and Inconsistency

Paradefinite (`beyond the definite\u27) logics are logics that can be used for handling contradictory or partial information. As such, paradefinite logics should be both paraconsistent and paracomplete. In this paper we consider the simplest semantic framework for defining paradefinite logics, consisting of four-valued matrices, and study the better accepted logics that are induced by these matrices

Pure Variable Inclusion Logics

The aim of this article is to discuss pure variable inclusion logics, that is, logical systems where valid entailments require that the propositional variables occurring in the conclusion are included among those appearing in the premises, or vice versa. We study the subsystems of Classical Logic satisfying these requirements and assess the extent to which it is possible to characterise them by means of a single logical matrix. In addition, we semantically describe both of these companions to Classical Logic in terms of appropriate matrix bundles and as semilattice-based logics, showing that the notion of consequence in these logics can be interpreted in terms of truth (or non-falsity) and meaningfulness (or meaninglessness) preservation. Finally, we use Płonka sums of matrices to investigate the pure variable inclusion companions of an arbitrary finitary logic.Fil: Paoli, Francesco. Università Degli Studi Di Cagliari.; ItaliaFil: Pra Baldi, Michele. Università Degli Studi Di Cagliari.; ItaliaFil: Szmuc, Damián Enrique. Universidad de Buenos Aires. Facultad de Filosofía y Letras; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Parque Centenario. Instituto de Investigaciones Filosóficas. - Sociedad Argentina de Análisis Filosófico. Instituto de Investigaciones Filosóficas; Argentin

Pure Refined Variable Inclusion Logics

In this article, we explore the semantic characterization of the (right) pure refined variable inclusion companion of all logics, which is a further refinement of the nowadays well-studied pure right variable inclusion logics. In particular, we will focus on giving a characterization of these fragments via a single logical matrix, when possible, and via a class of finite matrices, otherwise. In order to achieve this, we will rely on extending the semantics of the logics whose companions we will be discussing with infectious values in direct and in more subtle ways. This further establishes the connection between infectious logics and variable inclusion logics

Semantics without Toil? Brady and Rush Meet Halldén

The present discussion takes up an issue raised in Section 5 of Ross Brady and Penelope Rush’s paper ‘Four Basic Logical Issues’ concerning the (claimed) triviality – in the sense of automatic availability – of soundness and completeness results for a logic in a metalanguage employing at least as much logical vocabulary as the object logic, where the metalogical behaviour of the common logical vocabulary is as in the object logic. We shall see – in Propositions 4.5–4.7 – that this triviality claim faces difficulties in the face of Halldén incompleteness, for essentially the same reasons that Halldén thought this phenomenon raised seman- tic difficulties for the modal logics of C. I. Lewis exhibiting it. To counter any inclination to dismiss the phenomenon as providing at best a marginal range of counterexamples to the triviality claim, a Postscript assembles some reminders of the extent of – and the varied considerations favouring – Halldén incompleteness

On the notion of negation in certain non-classical propositional logics

The purpose of this study is to investigate some aspects of how negation functions in certain non-classical propositional logics. These include the intuitionistic system developed by Heyting, the minimal calculus proposed by Johansson, and various intermediate logics between the minimal and the classical systems. Part I contains the new results which can be grouped into two classes: extension-criteria results and infinite chain results. In the first group criteria are given for answering the question: when do formulae added to the axioms of the minimal calculus as extra axioms extend the minimal calculus to various known intermediate logics? One of the results in this group (THEOREM 1 in Chapter II, Section 1) is a generalization of a result of Jankov. In the second group certain intermediate logics are defined which form infinite chains between well-known logical systems. One of the results here (THEOREM 1 in Chapter II, Section 2) is a generalization of a result of McKay. In Part II the new results are discussed from the viewpoint of negation. It is rather difficult, however, to draw definite conclusions which are acceptable to all. For these depend on, and are closely bound up with, certain basic philosophical presuppositions which are neither provable, nor disprovable in a strict sense. Taking an essentially classical position, it is argued that the logics appearing in the defined infinite chains are such that they diverge only in the vicinity of negation, and the notions of negation in them are simply ordered in a sense which is specified during the discussion. In Appendix I a number of conjectures are formulated in connection with the new results.<p

Logics of formal inconsistency

Orientadores: Walter Alexandre Carnielli, Carlos M. C. L. CaleiroTexto em ingles e portuguesTese (doutorado) - Universidade Estadual de Campinas, Instituto de Filosofia e Ciencias HumanasTese (doutorado) - Universidade Tecnica de Lisboa, Instituto Superior TecnicoResumo: Segundo a pressuposição de consistência clássica, as contradições têm um cará[c]ter explosivo; uma vez que estejam presentes em uma teoria, tudo vale, e nenhum raciocínio sensato pode então ter lugar. Uma lógica é paraconsistente se ela rejeita uma tal pressuposição, e aceita ao invés que algumas teorias inconsistentes conquanto não-triviais façam perfeito sentido. A? Lógicas da Inconsistência Formal, LIFs, formam uma classe de lógicas paraconsistentes particularmente expressivas nas quais a noção meta-teónca de consistência pode ser internalizada ao nível da linguagem obje[c]to. Como consequência, as LIFs são capazes de recapturar o raciocínio consistente pelo acréscimo de assunções de consistência apropriadas. Assim, por exemplo, enquanto regras clássicas tais como o silogismo disjuntivo (de A e {não-,4)-ou-13, infira B) estão fadadas a falhar numa lógica paraconsistente (pois A e (nao-A) poderiam ambas ser verdadeiras para algum A, independentemente de B), elas podem ser recuperadas por uma LIF se o conjunto das premissas for ampliado pela presunção de que estamos raciocinando em um ambiente consistente (neste caso, pelo acréscimo de (consistente-.A) como uma hipótese adicional da regra). A presente monografia introduz as LIFs e apresenta diversas ilustrações destas lógicas e de suas propriedades, mostrando que tais lógicas constituem com efeito a maior parte dos sistemas paraconsistentes da literatura. Diversas formas de se efe[c]tuar a recaptura do raciocínio consistente dentro de tais sistemas inconsistentes são também ilustradas Em cada caso, interpretações em termos de semânticas polivalentes, de traduções possíveis ou modais são fornecidas, e os problemas relacionados à provisão de contrapartidas algébricas para tais lógicas são examinados. Uma abordagem formal abstra[cjta é proposta para todas as definições relacionadas e uma extensa investigação é feita sobre os princípios lógicos e as propriedades positivas e negativas da negação.Abstract: According to the classical consistency presupposition, contradictions have an explosive character: Whenever they are present in a theory, anything goes, and no sensible reasoning can thus take place. A logic is paraconsistent if it disallows such presupposition, and allows instead for some inconsistent yet non-trivial theories to make perfect sense. The Logics of Formal Inconsistency, LFIs, form a particularly expressive class of paraconsistent logics in which the metatheoretical notion of consistency can be internalized at the object-language level. As a consequence, the LFIs are able to recapture consistent reasoning by the addition of appropriate consistency assumptions. So, for instance, while classical rules such as disjunctive syllogism (from A and (not-A)-or-B, infer B) are bound to fail in a paraconsistent logic (because A and (not-.4) could both be true for some A, independently of B), they can be recovered by an LFI if the set of premises is enlarged by the presumption that we are reasoning in a consistent environment (in this case, by the addition of (consistent-/!) as an extra hypothesis of the rule). The present monograph introduces the LFIs and provides several illustrations of them and of their properties, showing that such logics constitute in fact the majority of interesting paraconsistent systems from the literature. Several ways of performing the recapture of consistent reasoning inside such inconsistent systems are also illustrated. In each case, interpretations in terms of many-valued, possible-translations, or modal semantics are provided, and the problems related to providing algebraic counterparts to such logics are surveyed. A formal abstract approach is proposed to all related definitions and an extended investigation is carried out into the logical principles and the positive and negative properties of negation.DoutoradoFilosofiaDoutor em Filosofia e Matemátic

Abstraktní studium úplnosti pro infinitární logiky

V této dizertační práci se zabýváme studiem vlastností úplnosti infinitárních výrokových logik z pohledu abstraktní algebraické logiky. Cílem práce je pochopit, jak lze základní nástroj v důkazech uplnosti, tzv. Lindenbaumovo lemma, zobecnit za hranici finitárních logik. Za tímto účelem studujeme vlastnosti úzce související s Lindenbaumovým lemmatem (a v důsledku také s vlastnostmi úplnosti). Uvidíme, že na základě těchto vlastností lze vystavět novou hierarchii infinitárních výrokových logik. Také se zabýváme studiem těchto vlastností v případě, kdy naše logika má nějaké (případně hodně obecně definované) spojky implikace, disjunkce a negace. Mimo jiné uvidíme, že přítomnost daných spojek může zajist platnost Lindenbaumova lemmatu. Keywords: abstraktní algebraická logika, infinitární logiky, Lindenbau- movo lemma, disjunkce, implikace, negaceIn this thesis we study completeness properties of infinitary propositional logics from the perspective of abstract algebraic logic. The goal is to under- stand how the basic tool in proofs of completeness, the so called Linden- baum lemma, generalizes beyond finitary logics. To this end, we study few properties closely related to the Lindenbaum lemma (and hence to com- pleteness properties). We will see that these properties give rise to a new hierarchy of infinitary propositional logic. We also study these properties in scenarios when a given logic has some (possibly very generally defined) connectives of implication, disjunction, and negation. Among others, we will see that presence of these connectives can ensure provability of the Lin- denbaum lemma. Keywords: abstract algebraic logic, infinitary logics, Lindenbaum lemma, disjunction, implication, negationKatedra logikyDepartment of LogicFaculty of ArtsFilozofická fakult