The ubiquity of conservative translations

We study the notion of conservative translation between logics introduced by Feitosa and D'Ottaviano. We show that classical propositional logic (CPC) is universal in the sense that every finitary consequence relation over a countable set of formulas can be conservatively translated into CPC. The translation is computable if the consequence relation is decidable. More generally, we show that one can take instead of CPC a broad class of logics (extensions of a certain fragment of full Lambek calculus FL) including most nonclassical logics studied in the literature, hence in a sense, (almost) any two reasonable deductive systems can be conservatively translated into each other. We also provide some counterexamples, in particular the paraconsistent logic LP is not universal.Comment: 15 pages; to appear in Review of Symbolic Logi

On the Concept of a Notational Variant

In the study of modal and nonclassical logics, translations have frequently been employed as a way of measuring the inferential capabilities of a logic. It is sometimes claimed that two logics are “notational variants” if they are translationally equivalent. However, we will show that this cannot be quite right, since first-order logic and propositional logic are translationally equivalent. Others have claimed that for two logics to be notational variants, they must at least be compositionally intertranslatable. The definition of compositionality these accounts use, however, is too strong, as the standard translation from modal logic to first-order logic is not compositional in this sense. In light of this, we will explore a weaker version of this notion that we will call schematicity and show that there is no schematic translation either from first-order logic to propositional logic or from intuitionistic logic to classical logic

OPERADORES DE CONSEQUÊNCIA E RELAÇÕES DE CONSEQUÊNCIA

Neste artigo analisamos o operador de consequência de Tarski e algumas relações de consequência num ambiente conjuntista de lógica universal. Neste âmbito, mostramos propriedades e axiomas relativos a estas definições que implicam e são implicadas por outras. Também mostramos a independência de algumas delas e lapidamos os conceitos envolvidos nestas noções abstratas que caracterizam lógicas em contexto universal

Logical Systems On the Concept, Expressive Power and Expressiveness Characterizations

[ES] Esta tesis es una investigación sobre los conceptos principales ocurriendo en los teoremas tipo-Lindström, esto es, el concepto de sistema lógico y el concepto de expresividad. Lindström, entre otros resultados similares, caracterizó la lógica de primer orden como siendo máximamente expresiva entre las lógicas que tienen compacidad y la propriedad de Löwenheim-Skolem. Para tal, él tenía que dar una definición precisa de qué es una lógica y qué es una relación de expresividad. Tales resultados suelen ser usados para extraer conclusiones fuertes sobre la naturaleza de la lógica y, específicamente, de la lógica de primer orden. No obstante, con la excepción de una discusión inicial en el libro sobre lógicas modelo-teóricas editado por Barwise y Feferman, hay pocas discusiones conceptuales de estos resultados de caracterización en la literatura. Específicamente, nos parece problemática la falta de justificación de por qué cierto concepto de sistema lógico y cierto concepto de expresividad fueran elegidos. Nuestro propósito es de contribuir a esta discusión y también proponer un criterio de expresividad más amplio, por medio de traducciones entre lógicas

What Can You Say? Measuring the Expressive Power of Languages

There are many different ways to talk about the world. Some ways of talking are more expressive than others—that is, they enable us to say more things about the world. But what exactly does this mean? When is one language able to express more about the world than another? In my dissertation, I systematically investigate different ways of answering this question and develop a formal theory of expressive power, translation, and notational variance. In doing so, I show how these investigations help to clarify the role that expressive power plays within debates in metaphysics, logic, and the philosophy of language

Abstract Consequence and Logics - Essays in Honor of Edelcio G. de Souza

Edelcio G. de Souza is a Brazilian logician and philosopher who has researches in the domains of abstract logic, non-classical systems, philosophy of science and the foundations of mathematics. This book is in his honor with the purpose of celebrating his 60th birthday. It contains some articles connected with the above topics and other subjects in logical investigations