Cuts and completions: Algebraic aspects of structural proof theory

In this thesis we look at different aspects of the interplay between structural proof theory and algebraic semantics for several non-classical propositional logics. Concretely, we explore connections between proof theory and algebra as they relate to structural sequent and hypersequent calculi for intermediate and substructural logics. Such connections are particularly strong for logics associated with the levels $\mathcal{P}_3$ and $\mathcal{N}_2$ of the substructural hierarchy introduced by Ciabattoni, Galatos, and Terui. Therefore, we investigate different algebraic aspects of these two levels. Among the algebraic aspects considered, completions of lattices and lattice-based algebras take on a prominent role

One-step Heyting algebras and hypersequent calculi with the bounded proof property

We investigate proof-theoretic properties of hypersequent calculi for intermediate logics using algebraic methods. More precisely, we consider a new weakly analytic subformula property (the bounded proof property) of such calculi. Despite being strictly weaker than both cut-elimination and the subformula property, this property is sufficient to ensure decidability of finitely axiomatized calculi. We introduce one-step Heyting algebras and establish a semantic criterion characterizing calculi for intermediate logics with the bounded proof property and the finite model property in terms of one-step Heyting algebras. Finally, we show how this semantic criterion can be applied to a number of calculi for well-known intermediate logics such as LC,KC and BD2

MacNeille transferability and stable classes of Heyting algebras

A lattice P is transferable for a class of lattices K if whenever P can be embedded into the ideal lattice IK of some K∈K, then P can be embedded into K. There is a rich theory of transferability for lattices. Here we introduce the analogous notion of MacNeille transferability, replacing the ideal lattice IK with the MacNeille completion ¯K¯. Basic properties of MacNeille transferability are developed. Particular attention is paid to MacNeille transferability in the class of Heyting algebras where it relates to stable classes of Heyting algebras, and hence to stable intermediate logics

Vitamin D, vitamin D binding protein, lung function and structure in COPD

SummaryRationaleVitamin D and vitamin D binding protein (DBP) have been associated with COPD and FEV1. There are limited data regarding emphysema and vitamin D and DBP.ObjectiveThis is a pilot study of a portion of the subjects in the Evaluation of COPD Longitudinally to Identify Predictive Surrogate Endpoints (ECLIPSE) study designed to examine the relationship between vitamin D status, DBP, FEV1 and emphysema in COPD patients.MethodsWe measured serum 25(OH)D and DBP in 498 ECLIPSE subjects. Subjects were distributed amongst smoker controls, non-smoker controls, and GOLD stages 2, 3 and 4. Within each GOLD stage, the subjects were equally divided amongst high and low emphysema burden. The associations between 25(OH)D, DBP, and free vitamin D with FEV1, CT-defined emphysema, biomarkers and clinical data including CT-measured bone attenuation were assessed.Measurements25(OH)D and DBP were measured using tandem mass spectroscopy and competitive enzyme-linked immunosorbent assay, respectively,Main result25(OH)D was correlated with FEV1 (p = 0.01) and with severity of emphysema (p < 0.01). 25(OH)D was also associated with six-minute walk (p = 0.02), bronchodilator response (p = 0.04), and Clara cell secretory protein (CC-16) (p = 0.01). 25(OH)D levels were not associated with CT-measured bone attenuation, however DBP was associated with bone attenuation in subjects with emphysema. DBP was not associated with FEV1 or emphysema. 25(OH)D and DBP were inversely associated (p = 0.01).ConclusionThis is the first study to demonstrate a relationship between emphysema and vitamin D. We also provide further evidence for a relationship between vitamin D and FEV1