Inconsistent Models (and Infinite Models) for Arithmetics with Constructible Falsity

An earlier paper on formulating arithmetic in a connexive logic ended with a conjecture concerning C♯ , the closure of the Peano axioms in Wansing’s connexive logic C. Namely, the paper conjectured that C♯ is Post consistent relative to Heyting arithmetic, i.e., is nontrivial if Heyting arithmetic is nontrivial. The present paper borrows techniques from relevant logic to demonstrate that C♯ is Post consistent simpliciter, rendering the earlier conjecture redundant. Given the close relationship between C and Nelson’s paraconsistent N4, this also supplements Nelson’s own proof of the Post consistency of N4♯ . Insofar as the present technique allows infinite models, this resolves Nelson’s concern that N4♯ is of interest only to those accepting that there are finitely many natural numbers

The development of freshwater science in Britain, and British contributions abroad, 1900-2000

This article outlines the development of freshwater science between 1900 and 2000 and in particular traces British contributions, both to a deepened knowledge of specifics and to their interrelation as environmental and ecological science. The author provides a selected bibliography of important publications relevant to the topic of the article

Rethinking inconsistent mathematics

This dissertation has two main goals. The first is to provide a practice-based analysis of the field of inconsistent mathematics: what motivates it? what role does logic have in it? what distinguishes it from classical mathematics? is it alternative or revolutionary? The second goal is to introduce and defend a new conception of inconsistent mathematics - queer incomaths - as a particularly effective answer to feminist critiques of classical logic and mathematics. This sets the stage for a genuine revolution in mathematics, insofar as it suggests the need for a shift in mainstream attitudes about the rolee of logic and ethics in the practice of mathematics

Usuzování s nekonzistentními informacemi

Tato dizertační práce studuje extenze čtyřhodnotové Belnapovy-Dunnovy logiky, tzv. superbelnapovské logiky, z pohledu abstraktní algebraické logiky. Popisujeme v ní globální strukturu svazu superbelnapovských logik a ukazu- jeme, že tento svaz lze zcela popsat pomocí tříd konečných grafů splňujících jisté uzávěrové podmínky. Také zde zavádíme teorii tzv. explozivních extenzí a používáme ji k důkazu nových vět o úplnosti pro superbelnapovské logiky. Poté rozvíjeme gentzenovskou teorii důkazů pro tyto logiky a použijeme ji k důkazu věty o interpolaci pro mnoho z těchto logik. Nakonec také studujeme rozšíření Belnapovy-Dunnovy logiky o operátor pravdivosti ∆. Klíčová slova: abstraktní algebraická logika, Belnapova-Dunnova logika, parakonzistentní logika, superbelnapovské logikyThis thesis studies the extensions of the four-valued Belnap-Dunn logic, called super-Belnap logics, from the point of view of abstract algebraic logic. We describe the global structure of the lattice of super-Belnap logics and show that this lattice can be fully described in terms of classes of finite graphs satisfying some closure conditions. We also introduce a theory of so- called explosive extensions and use it to prove new completeness theorems for super-Belnap logics. A Gentzen-style proof theory for these logics is then developed and used to establish interpolation for many of them. Finally, we also study the expansion of the Belnap-Dunn logic by the truth operator ∆. Keywords: abstract algebraic logic, Belnap-Dunn logic, paraconsistent logic, super-Belnap logicsKatedra logikyDepartment of LogicFaculty of ArtsFilozofická fakult

Pointfree bispaces and pointfree bisubspaces

This thesis is concerned with the study of pointfree bispaces, and in particular with the pointfree notion of inclusion of bisubspaces. We mostly work in the context of d-frames. We study quotients of d-frames as pointfree analogues of the topological notion of bisubspace. We show that for every d-frame L there is a d-frame A(L) such that it plays the role of the assembly of a frame, in the sense that it has the analogue of the universal property of the assembly and that its spectrum is a bitopological version of the Skula space of the bispace dpt(L), the spectrum of L. Furthermore, we show that this bitopological version of the Skula space of dpt(L) is the coarsest topology in which the d-sober bisubspaces of dpt(L) are closed. We also show that there are two free constructions in the category of d-frames Act(L) and A_(L), such that they represent two variations of the bitopological version of the Skula topology. In particular, we show that in dpt(Act) the positive closed sets are exactly those d-sober subspaces of dpt(L) that are spectra of quotients coming from an increase in the con component, and that the negative closed ones are those that come from increases in the tot component. For dpt(A_(L)), we show that the positive closed sets are exactly those bisubspaces of dpt(L) that are spectra of quotients coming from a quotient of L+, and that the negative closed sets come in the same way from quotients of

The development of freshwater science in Britain, and British contributions abroad, 1900-2000

The nineteenth century was a period in which the framework of sciencedeveloped rapidly and internationally. At its close there were, in Britain,the background and many active ingredients of future freshwater science.Geology and natural history had prospered. Organic evolution wasestablished on a Darwinian foundation. The form of river basins and lakeshad been shown to be influenced by Pleistocene glaciations; the flow of ariver could be related to terms of a water budget; the dissolved mineralcontent of surface waters was broadly characterized. Freshwater organisms- at least many of them - were named and classified. They attracted muchinterest in local societies of naturalists, whose largely amateur enthusiasmcomplemented the work of the fewer professionals in the field - such asL.C. Miall at the Yorkshire College, later University of Leeds. Fishes,amphibians, birds and flowering plants had never been neglected by thewider public. The schoolboy caught and reared tadpoles, his father cast'flies' for trout, his aunt pressed flowers of the water crowfoot andbogbean. There was the conspicuous spread in waterways of the introducedCanadian pondweed. However, naturalists became aware of a newmicrobial assembly - with often bizarre forms - when 'evenings at themicroscope' and books on microscopy became not infrequent in theirVictorian homes

Truth, Semantic Closure, and Conditionals

Almost all theories of truth place limits on the expressive power of languages containing truth predicates. Such theories have been criticized as inadequate on the grounds that these limitations are illegitimate. These criticisms set up several requirements on theories of truth. My initial focus is on the criticisms and why their requirements should be accepted. I argue that an adequate theory of truth should validate intuitive arguments involving truth and respect intuitive evaluations of the semantic statuses of sentences. From this starting point, I analyze the arguments in favor of several common requirements on theories of truth and formulate some minimal requirements on theories of truth. One is a logic neutrality requirement that says that a theory must be compatible with a range of logical resources, such as different negations. Another is the requirement that the theory validate certain laws governing truth, such as the T-sentences. These two requirements rule out many theories of truth. The main problem is that many theories lack an adequate conditional, the addition of which is, in fact, precluded by those theories. I argue that the revision theory of truth can satisfy my criteria when augmented with a pair of conditionals, which are defined using a modification of the framework of circular definitions of the revision theory. I distinguish two roles for conditionals in theories of truth and argue that the conditionals of the proposed theory fill those roles well. The conditionals are interdefinable with a modal operator. I prove a completeness theorem for the calculus $C_0$ of \emph{The Revision Theory of Truth} modified with rules for this operator. I examine the modal logic of this operator and prove a Solovay-type completeness theorem linking the modal logic and a certain class of circular definitions. I conclude by examining Field's recent theory of truth with its new conditional. I argue that Field's theory does not meet my requirements and that it fails to vindicate some of Field's own philosophical views. I close by proposing a framework for studying Field's conditional apart from his canonical models

Asymmetric Magnetohydrodynamic Waves of the Solar Atmosphere

Over 50 years of solar magnetohydrodynamic wave theory has focussed on waveguides in symmetric plasma environments. Yet the Sun's inhomogeneous atmosphere supports waveguides held in asymmetric equilibrium. In this thesis, we break this symmetry by studying a slab waveguide model embedded in an asymmetric external plasma with three approaches: - Eigenvalue problem: We derive the dispersion relation and show that asymmetric eigenmodes have mixed properties of the traditional sausage and kink modes. - Ray theory: We demonstrate how a ray theoretic approach can be used to derive this dispersion relation, giving an intuitive description of asymmetric leaky modes. - Initial value problem: An initial perturbation of an asymmetric slab evolves, in general, through a series of three phases: the initial phase, the period before collective modes are excited; the impulsive phase, where leaky modes can dominate; and the stationary phase, where trapped modes dominate for an indefinite time period. We show that, in general, the impulsive phase for a slab is significantly shorter than for a magnetic flux tube. We then show that an asymmetric slab of cold plasma does not have a stationary phase because the principle kink mode in an asymmetric slab is leaky. Next, we derive two magneto-seismology techniques to estimate the magnetic field strength in asymmetric solar waveguides. We apply this novel technique to a series of solar chromospheric fibrils as a proof of concept with estimated values of the Alfven speed that agree with estimates using traditional techniques