Intuitionism and the Modal Logic of Vagueness

Intuitionistic logic provides an elegant solution to the Sorites Paradox. Its acceptance has been hampered by two factors. First, the lack of an accepted semantics for languages containing vague terms has led even philosophers sympathetic to intuitionism to complain that no explanation has been given of why intuitionistic logic is the correct logic for such languages. Second, switching from classical to intuitionistic logic, while it may help with the Sorites, does not appear to offer any advantages when dealing with the so-called paradoxes of higher-order vagueness. We offer a proposal that makes strides on both issues. We argue that the intuitionist’s characteristic rejection of any third alethic value alongside true and false is best elaborated by taking the normal modal system S4M to be the sentential logic of the operator ‘it is clearly the case that’. S4M opens the way to an account of higher-order vagueness which avoids the paradoxes that have been thought to infect the notion. S4M is one of the modal counterparts of the intuitionistic sentential calculus and we use this fact to explain why IPC is the correct sentential logic to use when reasoning with vague statements. We also show that our key results go through in an intuitionistic version of S4M. Finally, we deploy our analysis to reply to Timothy Williamson’s objections to intuitionistic treatments of vagueness

The Nature and Logic of Vagueness

The PhD thesis advances a new approach to vagueness as dispersion, comparing it with the main philosophical theories of vagueness in the analytic tradition

Concurrent program schemes and their logics

AbstractWe define and investigate several classes of concurrent program schemes, including goto schemes and two versions of structured schemes, based on extensions of the regular expressions to trees. The schemes are studied on the first-order, Boolean-variable and propositional levels. We also define and study the dynamic logics based on these classes of schemes, including issues of decidability and axiomatization

Modelo de inferência para a classificação dinâmica de monografias de ní- vel superior

La inadecuada visibilidad de documentos de grado en modalidad monografía impide la consulta y acceso por parte de nuevos integrantes de la comunidad académica, limitando significativamente la posibilidad de dar continuidad y complementar versiones de trabajos ya finalizados, así como la aplicación de resultados tanto a nivel social como industrial. Este artículo propone un modelo de base de conocimiento soportado en una ontología para mejorar la pertinencia de documentos presentados a nivel digital como resultado de una búsqueda sobre aspectos de interés específico. Para la obtención de nuevo conocimiento se hace uso de la lógica de predicados aplicada sobre el modelo ontológico de representación de monografías, con el fin de establecer de manera dinámica y confiable inferencias o llegar a realizar modificaciones en las situaciones donde sean requeridas.The inappropriate visibility of degree documents in monograph mode prevents new members of the academic community from querying and accessing them, significantly limiting the possibility of giving continuity and complementing versions of already completed work, such as applying the results both industrial  and socially. This article proposes a knowledge base model supported on ontology to improve the relevance of documents in the digital level as a result of a search on specific interest aspects. To produce new knowledge, predicate logic is used and applied to the ontological model of the representation of monographs, in order to establish dynamic and reliable inferences or to make changes in situations where required.A inadequada visibilidade de trabalhos de grau na modalidade de monografia impede a consulta e acesso de novos membros da comunidade acadêmica, limitando significativamente a possibilidade de continuar e complementar as versões dos trabalhos já concluídos, bem como a aplicação dos resultados a nível social e industrial. Este artigo propõe um modelo de base de conhecimento suportado por uma ontologia para melhorar a relevância de trabalhos apresentados digitalmente, como resultado de uma pesquisa sobre aspectos de interesse específico. Para obter um novo conhecimento utiliza-se a lógica de predicados aplicada ao modelo ontológico de representação de monografias, a fim de estabelecer inferências de maneira dinâmica e fiável ou fazer alterações em situações em que sejam necessárias

Hybrid - a definitional two-level approach to reasoning with higher-order abstract syntax

Combining higher-order abstract syntax and (co)-induction in a logical framework is well known to be problematic.We describe the theory and the practice of a tool called Hybrid, within Isabelle/HOL and Coq, which aims to address many of these difficulties. It allows object logics to be represented using higher-order abstract syntax, and reasoned about using tactical theorem proving and principles of (co)induction. Moreover, it is definitional, which guarantees consistency within a classical type theory. The idea is to have a de Bruijn representation of \u3bb-terms providing a definitional layer that allows the user to represent object languages using higher-order abstract syntax, while offering tools for reasoning about them at the higher level. In this paper we describe how to use Hybrid in a multi-level reasoning fashion, similar in spirit to other systems such as Twelf and Abella. By explicitly referencing provability in a middle layer called a specification logic, we solve the problem of reasoning by (co)induction in the presence of non-stratifiable hypothetical judgments, which allow very elegant and succinct specifications of object logic inference rules. We first demonstrate the method on a simple example, formally proving type soundness (subject reduction) for a fragment of a pure functional language, using a minimal intuitionistic logic as the specification logic. We then prove an analogous result for a continuation-machine presentation of the operational semantics of the same language, encoded this time in an ordered linear logic that serves as the specification layer. This example demonstrates the ease with which we can incorporate new specification logics, and also illustrates a significantly more complex object logic whose encoding is elegantly expressed using features of the new specification logic