We investigate a recently devised polyhedral semantics for intermediate logics, in which formulas are interpreted in n-dimensional polyhedra. An intermediate logic is polyhedrally complete if it is complete with respect to some class of polyhedra. The first main result of this paper is a necessary and sufficient condition for the polyhedral completeness of a logic. This condition, which we call the Nerve Criterion, is expressed in terms of Alexandrov’s notion of the nerve of a poset. It affords a purely combinatorial characterisation of polyhedrally complete logics. Using the Nerve Criterion we show, easily, that there are continuum many intermediate logics that are not polyhedrally complete but which have the finite model property. We also provide, at considerable combinatorial labour, a countably infinite class of logics axiomatised by the Jankov–Fine formulas of ‘starlike trees’ all of which are polyhedrally complete. The polyhedral completeness theorem for these ‘starlike logics’ is the second main result of this paper.

An iterative square root of a self-map f is a self-map g such that $g(g(\cdot ))=f(\cdot )$ . We obtain new characterizations for detecting the non-existence of such square roots for self-maps on arbitrary sets. They are used to prove that continuous self-maps with no square roots are dense in the space of all continuous self-maps for various topological spaces. The spaces studied include those that are homeomorphic to the unit cube in ${\mathbb R}^{m}$ and to the whole of $\mathbb {R}^{m}$ for every positive integer $m.$ However, we also prove that every continuous self-map on a space homeomorphic to the unit cube in $\mathbb {R}^{m}$ with a fixed point on the boundary can be approximated by iterative squares of continuous self-maps.

The goal of much observational research is to identify risk factors that have a causal effect on health and social outcomes. However, observational data are subject to biases from confounding, selection and measurement, which can result in an underestimate or overestimate of the effect of interest. Various advanced statistical approaches exist that offer certain advantages in terms of addressing these potential biases. However, although these statistical approaches have different underlying statistical assumptions, in practice they cannot always completely remove key sources of bias; therefore, using design-based approaches to improve causal inference is also important. Here it is the design of the study that addresses the problem of potential bias – either by ensuring it is not present (under certain assumptions) or by comparing results across methods with different sources and direction of potential bias. The distinction between statistical and design-based approaches is not an absolute one, but it provides a framework for triangulation – the thoughtful application of multiple approaches (e.g. statistical and design based), each with their own strengths and weaknesses, and in particular sources and directions of bias. It is unlikely that any single method can provide a definite answer to a causal question, but the triangulation of evidence provided by different approaches can provide a stronger basis for causal inference. Triangulation can be considered part of wider efforts to improve the transparency and robustness of scientific research, and the wider scientific infrastructure and system of incentives.

Optical tracking systems typically trade off between astrometric precision and field of view. In this work, we showcase a networked approach to optical tracking using very wide field-of-view imagers that have relatively low astrometric precision on the scheduled OSIRIS-REx slingshot manoeuvre around Earth on 22 Sep 2017. As part of a trajectory designed to get OSIRIS-REx to NEO 101955 Bennu, this flyby event was viewed from 13 remote sensors spread across Australia and New Zealand to promote triangulatable observations. Each observatory in this portable network was constructed to be as lightweight and portable as possible, with hardware based off the successful design of the Desert Fireball Network. Over a 4-h collection window, we gathered 15 439 images of the night sky in the predicted direction of the OSIRIS-REx spacecraft. Using a specially developed streak detection and orbit determination data pipeline, we detected 2 090 line-of-sight observations. Our fitted orbit was determined to be within about 10 km of orbital telemetry along the observed 109 262 km length of OSIRIS-REx trajectory, and thus demonstrating the impressive capability of a networked approach to Space Surveillance and Tracking.

Dans Donald Davidson's Triangulation Argument: A Philosophical Inquiry, Robert H. Myers et Claudine Verheggen offrent une élucidation ainsi qu'une édification de l'argument de la triangulation avancé par Donald Davidson. Cet article est une introduction à un symposium consacré à leur développement de cet argument. Le symposium a débuté en 2018 en tant que table ronde réunissant les auteurs et des critiques lors du congrès annuel de l'Association canadienne de philosophie, et comprend les réponses de trois critiques, Kirk Ludwig, Alexander Miller et Paul Hurley, suivies des répliques de Verheggen et Myers. J'offre ici un bref aperçu de chacune des deux parties du livre et une esquisse des questions posées par les critiques.

In response to Paul Hurley, I argue that Donald Davidson's triangulation argument can be applied to normative beliefs only if such beliefs are answerable to properties that are at once normative and causal. The argument thus commits Davidson to a non-reductive and strikingly non-revisionary form of naturalism. In response to Kirk Ludwig, I argue that Davidson had good reason to abandon Humean accounts of pro-attitudes because he had good reason to welcome the non-reductive and non-revisionary form of naturalism that comes into view once the triangulation argument is applied to normative beliefs.

This paper is a reply to Kirk Ludwig's and Alexander Miller's comments on the first part of Donald Davidson's Triangulation Argument: A Philosophical Inquiry. It addresses concerns Ludwig expresses about the triangulation argument's success in establishing the social character of language and thought. It answers Miller's invitation to compare Davidson's non-reductionism with that of Crispin Wright, as well as the social aspect of Davidson's view with the social aspect of Saul Kripke's. And it addresses Miller's worries concerning my claims about the normativity of meaning.