Syntactic characterizations of classes of first-order structures in mathematical fuzzy logic

This paper is a contribution to graded model theory, in the context of mathematical fuzzy logic. We study characterizations of classes of graded structures in terms of the syntactic form of their first-order axiomatization. We focus on classes given by universal and universal-existential sentences. In particular, we prove two amalgamation results using the technique of diagrams in the setting of structures valued on a finite MTL-algebra, from which analogues of the Los--Tarski and the Chang--Los--Suszko preservation theorems follow

An Improved Homomorphism Preservation Theorem From Lower Bounds in Circuit Complexity

Previous work of the author [Rossmann\u2708] showed that the Homomorphism Preservation Theorem of classical model theory remains valid when its statement is restricted to finite structures. In this paper, we give a new proof of this result via a reduction to lower bounds in circuit complexity, specifically on the AC0 formula size of the colored subgraph isomorphism problem. Formally, we show the following: if a first-order sentence of quantifier-rank k is preserved under homomorphisms on finite structures, then it is equivalent on finite structures to an existential-positive sentence of quantifier-rank poly(k). Quantitatively, this improves the result of [Rossmann\u2708], where the upper bound on quantifier-rank is a non-elementary function of k

Team logic : axioms, expressiveness, complexity

Team semantics is an extension of classical logic where statements do not refer to single states of a system, but instead to sets of such states, called teams. This kind of semantics has applications for example in mathematical logic, verification of dynamic systems as well as in database theory. In this thesis, we focus on the propositional, modal and first-order variant of team logic. We study the classical questions of formal logic: Expressiveness (can we formalize sufficiently interesting properties of models?), axiomatizability (can all true statements be deduced in some formal system?) and complexity (can problems such as satisfiability and model checking be solved algorithmically?). Finally, we classify existing team logics and show approaches how team semantics can be defined for arbitrary other logics.Team-Semantik ist eine Erweiterung klassischer Logik, bei der Aussagen nicht über einzelne Zustände eines Systems getroffen werden, sondern über Mengen solcher Zustände, genannt Teams. Diese Art von Semantik besitzt unter anderem Anwendungen in der mathematischen Logik, in der Verifikation dynamischer Systeme sowie in der Datenbanktheorie. In dieser Arbeit liegt der Fokus auf der aussagenlogischen, der modallogischen und der prädikatenlogischen Variante der Team-Logik. Es werden die klassischen Fragestellungen formaler Logik untersucht: Ausdruckskraft (können hinreichend interessante Eigenschaften von Modellen formalisiert werden?), Axiomatisierbarkeit (lassen sich alle wahren Aussagen in einem Kalkül ableiten?) und Komplexität (können Probleme wie Erfüllbarkeit und Modellprüfung algorithmisch gelöst werden?). Schlussendlich werden existierende Team-Logiken klassifiziert und es werden Ansätze aufgezeigt, wie Team-Semantik für beliebige weitere Logiken definiert werden kann

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

Logical Localism in the Context of Combining Logics

[eng] Logical localism is a claim in the philosophy of logic stating that different logics are correct in different domains. There are different ways in which this thesis can be motivated and I will explore the most important ones. However, localism has an obvious and major challenge which is known as ‘the problem of mixed inferences’. The main goal of this dissertation is to solve this challenge and to extend the solution to the related problem of mixed compounds for alethic pluralism. My approach in order to offer a solution is one that has not been considered in the literature as far as I am aware. I will study different methods for combining logics, concentrating on the method of juxtaposition, by Joshua Schechter, and I will try to solve the problem of mixed inferences by making a finer translation of the arguments and using combination mechanisms as the criterion of validity. One of the most intriguing aspects of the dissertation is the synergy that is created between the philosophical debate and the technical methods with the problem of mixed inferences at the center of that synergy. I hope to show that not only the philosophical debate benefits from the methods for combining logics, but also that these methods can be developed in new and interesting ways motivated by the philosophical problem of mixed inferences. The problem suggests that there are relevant interactions between connectives, justified by the philosophical considerations for conceptualising different logic systems, that the methods for combining logics should allow to emerge. The recognition of this fact is what drives the improvements on the method of juxtaposition that I develop. That is, in order to allow for the emergence of desirable interaction principles, I will propose alternative ways of combining logic systems -specifically classical and intuitionistic logics- that go beyond the standard for combinations, which is based on minimality conditions so as to avoid the so-called collapse theorems.[spa] El localismo lógico es una tesis en filosofía de la lógica según la cual diferentes sistemas lógicos son correctos en función del dominio en el que se aplican. Dicha tesis cuenta, prima facie, con cierta plausibilidad y con varios argumentos que la respaldan como mostraré. Sin embargo, el localismo se presta a un evidente y poderoso contraargumento conocido como ‘el problema de las inferencias mixtas’. El objetivo principal de esta disertación es dar respuesta a ese problema y extender la solución al problema afín de los compuestos mixtos que afecta al pluralismo alético. La manera de abordar el problema de las inferencias mixtas consistirá en analizar casos paradigmáticos en la literatura a la luz de los métodos de combinación de lógicas. En concreto, me centraré en el método de la yuxtaposición, desarrollado por Joshua Schechter. Así, ofreceré una solución al problema de las inferencias mixtas que pasará por realizar un análisis más sutil y una formalización más precisa de las mismas, para después aplicar los mecanismos de combinación como criterio de validez. Además, mostraré que el problema de las inferencias mixtas provee de multitud de ejemplos que invitan a desarrollar los métodos de combinación de lógicas de formas novedosas. Una de las aportaciones más relevantes de la disertación consistirá en modificar el método de la yuxtaposición para obtener mecanismos que van más allá del estándar de las extensiones mínimas conservativas. En concreto, propondré diferentes mecanismos para combinar la lógica clásica y la intuicionista, de manera que se permita la aparición de distintos principios puente para los que tenemos buenas razones que los justifican, sin que ello conduzca al colapso de las lógicas que se combinan

Logical Localism in the Context of Combining Logics

Programa de Doctorat en Ciència Cognitiva i Llenguatge[eng] Logical localism is a claim in the philosophy of logic stating that different logics are correct in different domains. There are different ways in which this thesis can be motivated and I will explore the most important ones. However, localism has an obvious and major challenge which is known as ‘the problem of mixed inferences’. The main goal of this dissertation is to solve this challenge and to extend the solution to the related problem of mixed compounds for alethic pluralism. My approach in order to offer a solution is one that has not been considered in the literature as far as I am aware. I will study different methods for combining logics, concentrating on the method of juxtaposition, by Joshua Schechter, and I will try to solve the problem of mixed inferences by making a finer translation of the arguments and using combination mechanisms as the criterion of validity. One of the most intriguing aspects of the dissertation is the synergy that is created between the philosophical debate and the technical methods with the problem of mixed inferences at the center of that synergy. I hope to show that not only the philosophical debate benefits from the methods for combining logics, but also that these methods can be developed in new and interesting ways motivated by the philosophical problem of mixed inferences. The problem suggests that there are relevant interactions between connectives, justified by the philosophical considerations for conceptualising different logic systems, that the methods for combining logics should allow to emerge. The recognition of this fact is what drives the improvements on the method of juxtaposition that I develop. That is, in order to allow for the emergence of desirable interaction principles, I will propose alternative ways of combining logic systems -specifically classical and intuitionistic logics- that go beyond the standard for combinations, which is based on minimality conditions so as to avoid the so-called collapse theorems.[spa] El localismo lógico es una tesis en filosofía de la lógica según la cual diferentes sistemas lógicos son correctos en función del dominio en el que se aplican. Dicha tesis cuenta, prima facie, con cierta plausibilidad y con varios argumentos que la respaldan como mostraré. Sin embargo, el localismo se presta a un evidente y poderoso contraargumento conocido como ‘el problema de las inferencias mixtas’. El objetivo principal de esta disertación es dar respuesta a ese problema y extender la solución al problema afín de los compuestos mixtos que afecta al pluralismo alético. La manera de abordar el problema de las inferencias mixtas consistirá en analizar casos paradigmáticos en la literatura a la luz de los métodos de combinación de lógicas. En concreto, me centraré en el método de la yuxtaposición, desarrollado por Joshua Schechter. Así, ofreceré una solución al problema de las inferencias mixtas que pasará por realizar un análisis más sutil y una formalización más precisa de las mismas, para después aplicar los mecanismos de combinación como criterio de validez. Además, mostraré que el problema de las inferencias mixtas provee de multitud de ejemplos que invitan a desarrollar los métodos de combinación de lógicas de formas novedosas. Una de las aportaciones más relevantes de la disertación consistirá en modificar el método de la yuxtaposición para obtener mecanismos que van más allá del estándar de las extensiones mínimas conservativas. En concreto, propondré diferentes mecanismos para combinar la lógica clásica y la intuicionista, de manera que se permita la aparición de distintos principios puente para los que tenemos buenas razones que los justifican, sin que ello conduzca al colapso de las lógicas que se combinan