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An increasing amount of contemporary philosophy of mathematics posits, and theorizes in terms of special kinds of mathematical modality. The goal of this paper is to bring recent work on higher-order metaphysics to bear on the investigation of these modalities. The main focus of the paper will be views that posit mathematical contingency or indeterminacy about statements that concern the 'width' of the set theoretic universe, such as Cantor's continuum hypothesis. Within a higher-order framework I show that contingency about the width of the set-theoretic universe refutes two orthodoxies concerning the structure of modal reality: the view that the broadest necessity has a logic of S5, and the 'Leibniz biconditionals' stating that what is possible, in the broadest sense of possible, is what is true in some possible world. Nonetheless, I suggest that the underlying picture of modal set-theory is coherent and has attractions. [ABSTRACT FROM AUTHOR]

Purpose: Due to the risk of intracranial aneurysm (IA) recurrence and the potential requirement for re-treatment following endovascular treatment (EVT), radiological follow-up of these aneurysms is necessary. There is little evidence to guide the duration and frequency of this follow-up. The aim of this study was to establish the current practice in neurosurgical units in the UK and Ireland. Methods: A survey was designed with input from interventional neuroradiologists and neurosurgeons. Neurovascular consultants in each of the 30 neurosurgical units providing a neurovascular service in the UK and Ireland were contacted and asked to respond to questions regarding the follow-up practice for IA treated with EVT in their department. Results: Responses were obtained from 28/30 (94%) of departments. There was evidence of wide variations in the duration and frequency of follow-up, with a minimum follow-up duration for ruptured IA that varied from 18 months in 5/28 (18%) units to 5 years in 11/28 (39%) of units. Young patient age, previous subarachnoid haemorrhage and incomplete IA occlusion were cited as factors that would prompt more intensive surveillance, although larger and broad-necked IA were not followed-up more closely in the majority of departments. Conclusions: There is a wide variation in the radiological follow-up of IA treated with EVT in the UK and Ireland. Further standardisation of this aspect of patient care is likely to be beneficial, but further evidence on the behaviour of IA following EVT is required in order to inform this process. [ABSTRACT FROM AUTHOR]

Is there a business, service and technology case for flying constellations lower? This study describes work by Thales Alenia Space UK and Thales Alenia Space France and the Satellite Applications Catapult to select an attractive telecommunications application for Very Low Earth Orbit (VLEO) satellites. Through a trade process comparing 8 different applications, mobile 5G internet access was selected as a financially compelling prospect for a VLEO constellation. This work develops a conceptual design of the payload and platform for the satellites in the constellation. A constellation was selected which will provide coverage between latitudes of ± 55° (as these latitudes cover > 95% of the Earth's population). This consists of 33 planes of 70 satellites each inclined at 55° which will cover many potential users, with extended service in northern highly populated areas. The constellation will be able to provide an average 3.8 Mbps downlink data rate per beam to a conventional mobile phone handset, with each satellite supporting 320 beams. The flexibility of the payload allows this rate to be increased if not all beams are in use. [ABSTRACT FROM AUTHOR]

We develop a theory of necessity operators within a version of higher-order logic that is neutral about how fine-grained reality is. The theory is axiomatized in terms of the primitive of being a necessity, and we show how the central notions in the philosophy of modality can be recovered from it. Various questions are formulated and settled within the framework, including questions about the ordering of necessities under strength, the existence of broadest necessities satisfying various logical conditions, and questions about their logical behaviour. We also wield the framework to probe the conditions under which a logicist account of necessities is possible, in which the theory is completely reducible to logic. [ABSTRACT FROM AUTHOR]

Individuals play a prominent role in many metaphysical theories. According to an individualistic metaphysics, reality is determined (at least in part) by the pattern of properties and relations that hold between individuals. A number of philosophers have recently brought to attention alternative views in which individuals do not play such a prominent role; in this paper I will investigate one of these alternatives. [ABSTRACT FROM AUTHOR]

Generalizing on some arguments due to Arthur Prior and Dmitry Mirimanoff, we provide some further limitative results on what can be thought. [ABSTRACT FROM AUTHOR]

In this paper the logic of broad necessity is explored. Definitions of what it means for one modality to be broader than another are formulated, and it is proven, in the context of higher-order logic, that there is a broadest necessity, settling one of the central questions of this investigation. It is shown, moreover, that it is possible to give a reductive analysis of this necessity in extensional language (using truth functional connectives and quantifiers). This relates more generally to a conjecture that it is not possible to define intensional connectives from extensional notions. This conjecture is formulated precisely in higher-order logic, and concrete cases in which it fails are examined. The paper ends with a discussion of the logic of broad necessity. It is shown that the logic of broad necessity is a normal modal logic between S4 and Triv, and that it is consistent with a natural axiomatic system of higher-order logic that it is exactly S4. Some philosophical reasons to think that the logic of broad necessity does not include the S5 principle are given. [ABSTRACT FROM AUTHOR]

In this paper a principle of substitutivity of logical equivalents salve veritate and a version of Leibniz's law are formulated and each is shown to cause problems when combined with naive truth theories. [ABSTRACT FROM AUTHOR]

One thousand fair causally isolated coins will be independently flipped tomorrow morning and you know this fact. I argue that the probability, conditional on your knowledge, that any coin will land tails is almost 1 if that coin in fact lands tails, and almost 0 if it in fact lands heads. I also show that the coin flips are not probabilistically independent given your knowledge. These results are uncomfortable for those, like Timothy Williamson, who take these probabilities to play a central role in their theorizing. [ABSTRACT FROM AUTHOR]

A number of authors have objected to the application of non-classical logic to problems in philosophy on the basis that these non-classical logics are usually characterised by a classical metatheory. In many cases the problem amounts to more than just a discrepancy; the very phenomena responsible for non-classicality occur in the field of semantics as much as they do elsewhere. The phenomena of higher order vagueness and the revenge liar are just two such examples. The aim of this paper is to show that a large class of non-classical logics are strong enough to formulate their own model theory in a corresponding non-classical set theory. Specifically I show that adequate definitions of validity can be given for the propositional calculus in such a way that the metatheory proves, in the specified logic, that every theorem of the propositional fragment of that logic is validated. It is shown that in some cases it may fail to be a classical matter whether a given sentence is valid or not. One surprising conclusion for non-classical accounts of vagueness is drawn: there can be no axiomatic, and therefore precise, system which is determinately sound and complete. [ABSTRACT FROM AUTHOR]

In recent years there has been a revitalised interest in non-classical solutions to the semantic paradoxes. In this paper I show that a number of logics are susceptible to a strengthened version of Curry's paradox. This can be adapted to provide a proof theoretic analysis of the ω-inconsistency in Łukasiewicz's continuum valued logic, allowing us to better evaluate which logics are suitable for a naïve truth theory. On this basis I identify two natural subsystems of ukasiewicz logic which individually, but not jointly, lack the problematic feature. [ABSTRACT FROM AUTHOR]

The 'Ice Giants' Uranus and Neptune are a different class of planet compared to Jupiter and Saturn. Studying these objects is important for furthering our understanding of the formation and evolution of the planets, and unravelling the fundamental physical and chemical processes in the Solar System. The importance of filling these gaps in our knowledge of the Solar System is particularly acute when trying to apply our understanding to the numerous planetary systems that have been discovered around other stars. The Uranus Pathfinder (UP) mission thus represents the quintessential aspects of the objectives of the European planetary community as expressed in ESA's Cosmic Vision 2015-2025. UP was proposed to the European Space Agency's M3 call for medium-class missions in 2010 and proposed to be the first orbiter of an Ice Giant planet. As the most accessible Ice Giant within the M-class mission envelope Uranus was identified as the mission target. Although not selected for this call the UP mission concept provides a baseline framework for the exploration of Uranus with existing low-cost platforms and underlines the need to develop power sources suitable for the outer Solar System. The UP science case is based around exploring the origins, evolution, and processes at work in Ice Giant planetary systems. Three broad themes were identified: (1) Uranus as an Ice Giant, (2) An Ice Giant planetary system, and (3) An asymmetric magnetosphere. Due to the long interplanetary transfer from Earth to Uranus a significant cruise-phase science theme was also developed. The UP mission concept calls for the use of a Mars Express/Rosetta-type platform to launch on a Soyuz-Fregat in 2021 and entering into an eccentric polar orbit around Uranus in the 2036-2037 timeframe. The science payload has a strong heritage in Europe and beyond and requires no significant technology developments. [ABSTRACT FROM AUTHOR]

I consider two puzzles in which an agent undergoes a sequence of decision problems. In both cases it is possible to respond rationally to any given problem yet it is impossible to respond rationally to every problem in the sequence, even though the choices are independent. In particular, although it might be a requirement of rationality that one must respond in a certain way at each point in the sequence, it seems it cannot be a requirement to respond as such at every point for that would be to require the impossible. [ABSTRACT FROM AUTHOR]

Fragility fractures of the ankle are difficult to treat by conventional fixation due to poor bone quality, compromised soft tissues, and inherent instability. Conservative management of these patients also has its problems. We retrospectively reviewed 13 patients who underwent intramedullary nailing through the tibiotalocalcaneal joints in an attempt to achieve the dual aims of fracture control and early mobilisation. The Olerud and Molander scale was used as outcome measures. There were 12 females and 1 male with a mean age of 78.9 (range 64–93). Half of the patients were discharged from hospital within the first 2 weeks after the operation. All achieved a comparable function to their pre-operative state. The mean follow-up period was 11 months (range of 2–62 months). Six are now deceased from unrelated causes. The mean Olerud and Molander score was 50 (range 30–65). All the radiographs showed evidence of fracture union with no changes in the overall alignment of the joint. Given the low survivorship of this frail group of patients the main objectives are achieving early mobilisation whilst maintaining good fracture position. In our experience, tibiotalocalcaneal nailing is a very useful and successful way of treating fragility fractures of the ankle because it has a low risk of complications and restores function with impressive patient satisfaction. The potential benefits of this technique, we believe, outweigh the disability ensued from subtalar joint fusion. [ABSTRACT FROM AUTHOR]

The original version of this article unfortunately contains mistakes. [ABSTRACT FROM AUTHOR]