Systematic Verification of the Modal Logic Cube in Isabelle/HOL

We present an automated verification of the well-known modal logic cube in Isabelle/HOL, in which we prove the inclusion relations between the cube's logics using automated reasoning tools. Prior work addresses this problem but without restriction to the modal logic cube, and using encodings in first-order logic in combination with first-order automated theorem provers. In contrast, our solution is more elegant, transparent and effective. It employs an embedding of quantified modal logic in classical higher-order logic. Automated reasoning tools, such as Sledgehammer with LEO-II, Satallax and CVC4, Metis and Nitpick, are employed to achieve full automation. Though successful, the experiments also motivate some technical improvements in the Isabelle/HOL tool.Comment: In Proceedings PxTP 2015, arXiv:1507.0837

Forward refutation for Gödel-Dummett Logics

We propose a refutation calculus to check the unprovability of a formula in Gödel-Dummett logics. From refutations we can directly extract countermodels for unprovable formulas, moreover the calculus is designed so to support a forward proof-search strategy that can be understood as a top-down construction of a model

Hypersequent calculi for non-normal modal and deontic logics: Countermodels and optimal complexity

We present some hypersequent calculi for all systems of the classical cube and their extensions with axioms $T$, $P$, $D$, and, for every $n\geq 1$, rule $RD^+_n$. The calculi are internal as they only employ the language of the logic, plus additional structural connectives. We show that the calculi are complete with respect to the corresponding axiomatisation by a syntactic proof of cut elimination. Then we define a terminating root-first proof search strategy based on the hypersequent calculi and show that it is optimal for coNP-complete logics. Moreover, we obtain that from every saturated leaf of a failed proof it is possible to define a countermodel of the root hypersequent in the bi-neighbourhood semantics, and for regular logics also in the relational semantics. We finish the paper by giving a translation between hypersequent rule applications and derivations in a labelled system for the classical cube

Intuitionistic S4 is decidable

In this paper we demonstrate decidability for the intuitionistic modal logic S4 first formulated by Fischer Servi. This solves a problem that has been open for almost thirty years since it had been posed in Simpson's PhD thesis in 1994. We obtain this result by performing proof search in a labelled deductive system that, instead of using only one binary relation on the labels, employs two: one corresponding to the accessibility relation of modal logic and the other corresponding to the order relation of intuitionistic Kripke frames. Our search algorithm outputs either a proof or a finite counter-model, thus, additionally establishing the finite model property for intuitionistic S4, which has been another long-standing open problem in the area.Comment: 13 pages conference paper + 26 pages appendix with examples and proof

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