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No Ordinary Plain: Seeing and Unseeing the Taieri with McCahon
Get PDFThere are three recorded moments of Colin McCahon encountering the Taieri Plain. In 1936, as a schoolboy, he experienced an epiphany looking over the Taieri from the coastal hills. Six years later, in 1942, he painted Sketch for landscape from Flagstaff, depicting the plain from a northern perspective. In 1966, he famously recalled his early epiphany in an essay entitled “Beginnings.” In this essay, I bring a sociohistorical and personal perspective to these three moments, arguing that knowledge of the details that McCahon was seeing and unseeing in his 1942 sketch bring nuance to our understanding of his early development
Janet Frame’s World of Books
Get PDFThe reading habits of an author are always of interest, and in the case of Janet Frame, notoriously protective about her inner life but in her autobiographies fluent and enthusiastic about her life as a reader, such a study seems promising
Understanding students' use of mathematical processes during a digital escape game
Get PDFMathematical processes have long been considered an essential component of meaningful learning in mathematics, yet these processes can sometimes be invisible in the mathematics classroom or in learning experiences. This discussion uses the context of a purpose-designed, innovative ‘digital escape’ game to illustrate how digital experiences might bring mathematical processes to the fore of student learning while offering other affordances only seen in the online space. This article reports on a pilot study conducted with 12-15-year-old school students with the aim of determining if a digital escape game could promote the use of mathematical processes. During the digital escape game, it was found that students engaged with problem-solving, reasoning, communication and made connections within, across and beyond mathematics. The preliminary findings demonstrate how digital experiences may enrich the use and development of core mathematical processes, and it is argued that teachers could use their own expertise and knowledge of their learners to design such experiences, catering to student needs and interests
Fixed-point models for paradoxical predicates
Get PDFThis paper introduces a new kind of fixed-point semantics, filling a gap within approaches to Liar-like paradoxes involving fixed-point models à la Kripke (1975). The four-valued models presented below, (i) unlike the three-valued, consistent fixed-point models defined in Kripke (1975), are able to differentiate between paradoxical and pathological-but-unparadoxical sentences, and (ii) unlike the four-valued, paraconsistent fixed-point models first studied in Visser (1984) and Woodruff (1984), preserve consistency and groundedness of truth.
Keywords: Semantic Paradoxes · Fixed-point semantics · Many-valued logic · Kripke’s theory oftrut
Some Interrelations between Geometry and Modal Logic
No full textThis is a reprinting of Ken Pledger’s PhD thesis, submitted to the University of Warsaw in 1980 with the degree awarded in 1981. It develops a one-sorted approach to the theory of plane geometry, based on the idea that the usually two-sorted theory “can be made one-sorted by keeping careful account of whether the incidence relation is iterated an even or odd number of times”.The one-sorted structures can also serve as Kripke frames for modal logics, and the thesis defines and studies two such logics that are validated by projective planes and elliptic planes respectively. It raises questions of logical completeness for these systems that are addressed in the first article of this journal issue
The Strict/Tolerant Family Continued: Quantifiers and Modalities
Get PDFThis paper continues my earlier work, which showed there is a broad family of propositional many valued logics that have a strict/tolerant counterpart. Here we generalize those results from propositional to a range of both modal and quantified many valued logics, providing strict/tolerant counterparts for all. This paper is not self-contained; some results from earlier papers are called on, and are not reproved here. The key new machinery added to earlier work, allowing modalities and quantifiers to be handled in similar ways, is the central use of bilattices that are function spaces, and more generally lattices that are function spaces. Two versions of the central proofs are considered, one at length and the other in outline
Algebra-valued models for LP-set theory
Get PDFIn this paper, we explore the possibility of constructing algebra-valued models of set theory based on Priest's Logic of Paradox. We show that we can build a non-classical model of ZFC which has as internal logic Priest's Logic of Paradox and validates Leibniz's law of indiscernibility of identicals. This is achieved by modifying the interpretation map for and in our algebra-valued model. We end by comparing our model constructions to Priest's model-theoretic strategy and point out that we have a tradeoff between desirable model-theoretic properties and the validity of ZFC and its theorems
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