A Note on the Relevance of Semilattice Relevance Logic

A propositional logic has the variable sharing property if φ →’ ψ is a theorem only if φ and ψ share some propositional variable(s). In this note, I prove that positive semilattice relevance logic (R+u) and its extension with an involution negation (R¬u) have the variable sharing property (as these systems are not subsystems of R, these results are not automatically entailed by the fact that R satisfies the variable sharing property). Typical proofs of the variable sharing property rely on ad hoc, if clever, matrices. However, in this note, I exploit the properties of rather more intuitive arithmetical structures to establish the variable sharing property for the systems discussed

Connexive Extensions of Regular Conditional Logic

The object of this paper is to examine half and full connexive extensions of the basic regular conditional logic CR. Extensions of this system are of interest because it is among the strongest well-known systems of conditional logic that can be augmented with connexive theses without inconsistency resulting. These connexive extensions are characterized axiomatically and their relations to one another are examined proof-theoretically. Subsequently, algebraic semantics are given and soundness, completeness, and decidability are proved for each system. The semantics is also used to establish independence results. Finally, a deontic interpretation of one of the systems is examined and defended

A Note on the Relevance of Semilattice Relevance Logic

A propositional logic has the variable sharing property if φ →’ ψ is a theorem only if φ and ψ share some propositional variable(s). In this note, I prove that positive semilattice relevance logic (R+u) and its extension with an involution negation (R¬u) have the variable sharing property (as these systems are not subsystems of R, these results are not automatically entailed by the fact that R satisfies the variable sharing property). Typical proofs of the variable sharing property rely on ad hoc, if clever, matrices. However, in this note, I exploit the properties of rather more intuitive arithmetical structures to establish the variable sharing property for the systems discussed

Frontiers of Conditional Logic

Conditional logics were originally developed for the purpose of modeling intuitively correct modes of reasoning involving conditional—especially counterfactual—expressions in natural language. While the debate over the logic of conditionals is as old as propositional logic, it was the development of worlds semantics for modal logic in the past century that catalyzed the rapid maturation of the field. Moreover, like modal logic, conditional logic has subsequently found a wide array of uses, from the traditional (e.g. counterfactuals) to the exotic (e.g. conditional obligation). Despite the close connections between conditional and modal logic, both the technical development and philosophical exploitation of the latter has outstripped that of the former, with the result that noticeable lacunae exist in the literature on conditional logic. My dissertation addresses a number of these underdeveloped frontiers, producing new technical insights and philosophical applications. I contribute to the solution of a problem posed by Priest of finding sound and complete labeled tableaux for systems of conditional logic from Lewis\u27 V-family. To develop these tableaux, I draw on previous work on labeled tableaux for modal and conditional logic; errors and shortcomings in recent work on this problem are identified and corrected. While modal logic has by now been thoroughly studied in non-classical contexts, e.g. intuitionistic and relevant logic, the literature on conditional logic is still overwhelmingly classical. Another contribution of my dissertation is a thorough analysis of intuitionistic conditional logic, in which I utilize both algebraic and worlds semantics, and investigate how several novel embedding results might shed light on the philosophical interpretation of both intuitionistic logic and conditional logic extensions thereof. My dissertation examines deontic and connexive conditional logic as well as the underappreciated history of connexive notions in the analysis of conditional obligation. The possibility of interpreting deontic modal logics in such systems (via embedding results) serves as an important theoretical guide. A philosophically motivated proscription on impossible obligations is shown to correspond to, and justify, certain (weak) connexive theses. Finally, I contribute to the intensifying debate over counterpossibles, counterfactuals with impossible antecedents, and take—in contrast to Lewis and Williamson—a non-vacuous line. Thus, in my view, a counterpossible like If there had been a counterexample to the law of the excluded middle, Brouwer would not have been vindicated is false, not (vacuously) true, although it has an impossible antecedent. I exploit impossible (non-normal) worlds—originally developed to model non-normal modal logics—to provide non-vacuous semantics for counterpossibles. I buttress the case for non-vacuous semantics by making recourse to both novel technical results and theoretical considerations

Semantics for Counterpossibles

The object of this paper is to examine two approaches to giving non-vacuous truth conditions for counterpossibles, counterfactuals with impossible antecedents. I first develop modifications of a Lewis-style sphere semantics with impossible worlds. I argue that this approach sanctions intuitively invalid inferences and is supported by philosophically problematic foundations. I then develop modifications of certain ceteris paribus conditional logics with impossible worlds. Tableaux are given for each of these in an appendix and soundness and completeness results are proved. While certain of the latter systems are shown to have similar problems to logics from the first approach, at least one relatively weak system appears to offer an adequate uniform semantics for counterpossibles and counterfactuals

Revisiting Constructive Mingle: Algebraic and Operational Semantics

Among Dunn’s many important contributions to relevance logic was his work on the system RM (R-mingle). Although RM is an interesting system in its own right, it is widely considered to be too strong. In this chapter, I revisit a closely related system, RM0 (sometimes known as ‘constructive mingle’), which includes the mingle axiom while not degenerating in the way that RM itself does. My main interest will be in examining this logic from two related semantical perspectives. First, I give a purely operational bisemilattice semantics for it by adapting previous work of Humberstone. Second, I examine a more conventional algebraic semantics for it and discuss how this relates to the operational semantics. A novel operational semantics for J (intuitionistic logic) as well as its conventional Heyting algebraic semantics emerge as special cases of the corresponding semantics for RM0. The results of this chapter suggest that RM0 is a more interesting logic than has been appreciated and that Humberstone’s operational semantic framework similarly deserves more attention than it has received

Looking Upstream: An Analysis of Low Water Levels in Lake Powell and the Impacts on Water Supply, Hydropower, Recreation, and the Environment: A Companion Report to The Bathtub Ring

viii, 110 pages : color illustrations, color mapshttps://scholar.law.colorado.edu/books_reports_studies/1173/thumbnail.jp

Looking Upstream: An Analysis of Low Water Levels in Lake Powell and the Impacts on Water Supply, Hydropower, Recreation, and the Environment: A Companion Report to The Bathtub Ring

viii, 110 pages : color illustrations, color mapshttps://scholar.law.colorado.edu/books_reports_studies/1173/thumbnail.jp

Tracking smell loss to identify healthcare workers with SARS-CoV-2 infection

Introduction Healthcare workers (HCW) treating COVID-19 patients are at high risk for infection and may also spread infection through their contact with vulnerable patients. Smell loss has been associated with SARS-CoV-2 infection, but it is unknown whether monitoring for smell loss can be used to identify asymptomatic infection among high risk individuals. In this study we sought to determine if tracking smell sensitivity and loss using an at-home assessment could identify SARS-CoV-2 infection in HCW. Methods and findings We performed a prospective cohort study tracking 473 HCW across three months to determine if smell loss could predict SARS-CoV-2 infection in this high-risk group. HCW subjects completed a longitudinal, behavioral at-home assessment of olfaction with household items, as well as detailed symptom surveys that included a parosmia screening questionnaire, and real-time quantitative polymerase chain reaction testing to identify SARS-CoV-2 infection. Our main measures were the prevalence of smell loss in SARS-CoV-2-positive HCW versus SARS-CoV- 2-negative HCW, and timing of smell loss relative to SARS-CoV-2 test positivity. SARS-CoV-2 was identified in 17 (3.6%) of 473 HCW. HCW with SARS-CoV-2 infection were more likely to report smell loss than SARS-CoV-2-negative HCW on both the at-home assessment and the screening questionnaire (9/17, 53% vs 105/456, 23%, P < .01). 6/9 (67%) of SARS-CoV-2-positive HCW reporting smell loss reported smell loss prior to having a positive SARS-CoV-2 test, and smell loss was reported a median of two days before testing positive. Neurological symptoms were reported more frequently among SARS-CoV-2-positive HCW who reported smell loss compared to those without smell loss (9/9, 100% vs 3/8, 38%, P < .01). Conclusions In this prospective study of HCW, self-reported changes in smell using two different measures were predictive of SARS-CoV-2 infection. Smell loss frequently preceded a positive test and was associated with neurological symptoms