On elimination of quantifiers in some non-classical mathematical theories

Elimination of quantifiers is shown to fail dramatically for a group of well-known mathematical theories (classically enjoying the property) against a wide range of relevant logical backgrounds. Furthermore, it is suggested that only by moving to more extensional underlying logics can we get the property back

Mathematical Explanation by Law

Call an explanation in which a non-mathematical fact is explained --- in part or in whole --- by mathematical facts: an extra-mathematical explanation. Such explanations have attracted a great deal of interest recently in arguments over mathematical realism. In this paper, a theory of extra-mathematical explanation is developed. The theory is modeled on a deductive-nomological theory of scientific explanation. A basic DN account of extra-mathematical explanation is proposed and then redeveloped in the light of two difficulties that the basic theory faces. The final view appeals to relevance logic and uses resources in information theory to understand the explanatory relationship between mathematical and physical facts

Hyperdoctrines and the Ontology of Stratified Semantics

I present a version of Kit Fine's stratified semantics for the logic RWQ and define a natural family of related structures called RW hyperdoctrines. After proving that RWQ is sound with respect to RW hyperdoctrines, we show how to construct, for each stratified model, a hyperdoctrine that verifies precisely the same sentences. Completeness of RWQ for hyperdoctrinal semantics then follows from completeness for stratified semantics, which is proved in an appendix. By examining the base category of RW hyperdoctrines, we find reason to be worried about the ontology of stratified models

Eficacia de procesamiento de los entrañamientos y de las implicaturas conversacionales generalizadas: la lógica deductiva y la lógica por defecto

Este trabajo tiene como objetivo estudiar las diferencias de procesamiento que se producen al aplicar una lógica de tipo deductivo frente a una lógica por defecto. Como los entrañamientos se fundamentan en la lógica deductiva, mientras que las implicaturas conversacionales generalizadas (ICG) se construyen a partir de la lógica por defecto, se propusieron dos cuestionarios destinados a registrar la eficacia de procesamiento entendida como el número de aciertos por minuto. Los datos recogidos muestran que existe una diferencia significativa en el procesamiento de los entrañamientos y de las ICG: los entrañamientos se procesan de forma más eficaz que las ICG y, por tanto, se concluye que la lógica por defecto exige un mayor esfuerzo cognitivo

Second-order propositional modal logic: expressiveness and completeness results

In this paper we advance the state-of-the-art on the application of second-order propositional modal logic (SOPML) in the representation of individual and group knowledge, as well as temporal and spatial reasoning. The main theoretical contributions of the paper can be summarised as follows. Firstly, we introduce the language of (multi-modal) SOPML and interpret it on a variety of different classes of Kripke frames according to the features of the accessibility relations and of the algebraic structure of the quantification domain of propositions. We provide axiomatisations for some of these classes, and show that SOPML is unaxiomatisable on the remaining classes. Secondly, we introduce novel notions of (bi)simulations and prove that they indeed preserve the interpretation of formulas in (the universal fragment of) SOPML. Then, we apply this formal machinery to study the expressiveness of Second-order Propositional Epistemic Logic (SOPEL) in representing higher-order knowledge, i.e., the knowledge agents have about other agents’ knowledge, as well as graph-theoretic notions (e.g., 3-colorability, Hamiltonian paths, etc.). The final outcome is a rich formalism to represent and reason about relevant concepts in artificial intelligence, while still having a model checking problem that is no more computationally expensive than that of the less expressive quantified boolean logic