Working in constructive set theory we formulate notions of constructive topological space and set-generated locale so as to get a good constructive general version of the classical Galois adjunction between topological spaces and locales. Our notion of constructive topological space allows for the space to have a class of points that need not be a set. Also our notion of locale allows the locale to have a class of elements that need not be a set. Class sized mathematical structures need (...) to be allowed for in constructive set theory because the powerset axiom and the full separation scheme are necessarily missing from constructive set theory. We also consider the notion of a formal topology, usually treated in Intuitionistic type theory, and show that the category of set-generated locales is equivalent to the category of formal topologies. We exploit ideas of Palmgren and Curi to obtain versions of their results about when the class of formal points of a set-presentable formal topology form a set. (shrink)

The axiom of choice ensures precisely that, in ZFC, every set is projective: that is, a projective object in the category of sets. In constructive ZF (CZF) the existence of enough projective sets has been discussed as an additional axiom taken from the interpretation of CZF in Martin-Löf’s intuitionistic type theory. On the other hand, every non-empty set is injective in classical ZF, which argument fails to work in CZF. The aim of this paper is to shed some light on (...) the problem whether there are (enough) injective sets in CZF. We show that no two element set is injective unless the law of excluded middle is admitted for negated formulas, and that the axiom of power set is required for proving that “there are strongly enough injective sets”. The latter notion is abstracted from the singleton embedding into the power set, which ensures enough injectives both in every topos and in IZF. We further show that it is consistent with CZF to assume that the only injective sets are the singletons. In particular, assuming the consistency of CZF one cannot prove in CZF that there are enough injective sets. As a complement we revisit the duality between injective and projective sets from the point of view of intuitionistic type theory. (shrink)

Situation Theory and Its Applications, Vol. 1 . Robin Cooper, Kuniaki Mukai, and John Perry (Eds.). Lecture Notes No. 22. ...

This work is derived from the SERC "Logic for IT" Summer School Conference on Proof Theory held at Leeds University. The contributions come from acknowledged experts and comprise expository and research articles which form an invaluable introduction to proof theory aimed at both mathematicians and computer scientists.

Situation theory is the result of an interdisciplinary effort to create a full-fledged theory of information. Created by scholars and scientists from cognitive science, computer science and AI, linguistics, logic, philosophy, and mathematics, it aims to provide a common set of tools for the analysis of phenomena from all these fields. Unlike Shannon-Weaver type theories of information, which are purely quantitative theories, situation theory aims at providing tools for the analysis of the specific content of a situation. The question addressed (...) is not how much information is carried, but what information is carried. (shrink)

In this note a T1 formal space is a formal space whose points are closed as subspaces. Any regular formal space is T1. We introduce the more general notion of a formal space, and prove that the class of points of a weakly set-presentable formal space is a set in the constructive set theory CZF. The same also holds in constructive type theory. We then formulate separation properties for constructive topological spaces , strengthening separation properties discussed elsewhere. Finally we relate (...) the properties for ct-spaces with corresponding properties of formal spaces. (shrink)

Working in the weakening of constructive Zermelo-Fraenkel set theory in which the subset collection scheme is omitted, we show that the binary refinement principle implies all the instances of the exponentiation axiom in which the basis is a discrete set. In particular binary refinement implies that the class of detachable subsets of a set form a set. Binary refinement was originally extracted from the fullness axiom, an equivalent of subset collection, as a principle that was sufficient to prove that the (...) Dedekind reals form a set. Here we show that the Cauchy reals also form a set. More generally, binary refinement ensures that one remains in the realm of sets when one starts from discrete sets and one applies the operations of exponentiation and binary product a finite number of times. (shrink)

We introduce a new axiom scheme for constructive set theory, the Relation Reflection Scheme . Each instance of this scheme is a theorem of the classical set theory ZF. In the constructive set theory CZF–, when the axiom scheme is combined with the axiom of Dependent Choices , the result is equivalent to the scheme of Relative Dependent Choices . In contrast to RDC, the scheme RRS is preserved in Heyting-valued models of CZF– using set-generated frames. We give an application (...) of the scheme to coinductive definitions of classes. (shrink)

The aim of this paper is to formulate and study two weak axiom systems for the conceptual framework of constructive set theory . Arithmetical CST is just strong enough to represent the class of von Neumann natural numbers and its arithmetic so as to interpret Heyting Arithmetic. Rudimentary CST is a very weak subsystem that is just strong enough to represent a constructive version of Jensenʼs rudimentary set theoretic functions and their theory. The paper is a contribution to the study (...) of formal systems for CST that capture significant stages in the development of constructive mathematics in CST. (shrink)

We present a generalisation of the type-theoretic interpretation of constructive set theory into Martin-Löf type theory. The original interpretation treated logic in Martin-Löf type theory via the propositions-as-types interpretation. The generalisation involves replacing Martin-Löf type theory with a new type theory in which logic is treated as primitive. The primitive treatment of logic in type theories allows us to study reinterpretations of logic, such as the double-negation translation.

The first example of a simultaneous inductive-recursive definition in intuitionistic type theory is Martin-Löf's universe á la Tarski. A set U 0 of codes for small sets is generated inductively at the same time as a function T 0 , which maps a code to the corresponding small set, is defined by recursion on the way the elements of U 0 are generated. In this paper we argue that there is an underlying general notion of simultaneous inductive-recursive definition which is (...) implicit in Martin-Löf's intuitionistic type theory. We extend previously given schematic formulations of inductive definitions in type theory to encompass a general notion of simultaneous induction-recursion. This enables us to give a unified treatment of several interesting constructions including various universe constructions by Palmgren, Griffor, Rathjen, and Setzer and a constructive version of Aczel's Frege structures. Consistency of a restricted version of the extension is shown by constructing a realisability model in the style of Allen. (shrink)

Some propositions add more information to bodies of propositions than do others. We start with intuitive considerations on qualitative comparisons of information added . Central to these are considerations bearing on conjunctions and on negations. We find that we can discern two distinct, incompatible, notions of information added. From the comparative notions we pass to quantitative measurement of information added. In this we borrow heavily from the literature on quantitative representations of qualitative, comparative conditional probability. We look at two ways (...) to obtain a quantitative conception of information added. One, the most direct, mirrors Bernard Koopman’s construction of conditional probability: by making a strong structural assumption, it leads to a measure that is, transparently, some function of a function P which is, formally, an assignment of conditional probability (in fact, a Popper function). P reverses the information added order and mislocates the natural zero of the scale so some transformation of this scale is needed but the derivation of P falls out so readily that no particular transformation suggests itself. The Cox–Good–Aczél method assumes the existence of a quantitative measure matching the qualitative relation, and builds on the structural constraints to obtain a measure of information that can be rescaled as, formally, an assignment of conditional probability. A classical result of Cantor’s, subsequently strengthened by Debreu, goes some way towards justifying the assumption of the existence of a quantitative scale. What the two approaches give us is a pointer towards a novel interpretation of probability as a rescaling of a measure of information added. (shrink)

In this groundbreaking editorial and curatorial project, more than 100 writers, artists, and philosophers rethink what politics is about. In a time of political turmoil and anticlimax, this book redefines politics as operating in the realm of things. Politics is not just an arena, a profession, or a system, but a concern for things brought to the attention of the fluid and expansive constituency of the public. But how are things made public? What, we might ask, is a republic, a (...) res publica, a public thing, if we do not know how to make things public? There are many other kinds of assemblies, which are not political in the usual sense, that gather a public around things—scientific laboratories, supermarkets, churches, and disputes involving natural resources like rivers, landscapes, and air. The authors of Making Things Public—and the ZKM show that the book accompanies—ask what would happen if politics revolved around disputed things. Instead of looking for democracy only in the official sphere of professional politics, they examine the new atmospheric conditions—technologies, interfaces, platforms, networks, and mediations that allow things to be made public. They show us that the old definition of politics is too narrow; there are many techniques of representation—in politics, science, and art—of which Parliaments and Congresses are only a part. The authors include such prominent thinkers as Richard Rorty, Simon Schaffer, Peter Galison, Richard Powers, Lorraine Daston, Richard Aczel, and Donna Haraway; their writings are accompanied by excerpts from John Dewey, Shakespeare, Swift, La Fontaine, and Melville. More than 500 color images document the new idea of what Bruno Latour and Peter Weibel call an "object-oriented democracy.". (shrink)

The Dedekind cuts in an ordered set form a set in the sense of constructive Zermelo—Fraenkel set theory. We deduce this statement from the principle of refinement, which we distill before from the axiom of fullness. Together with exponentiation, refinement is equivalent to fullness. None of the defining properties of an ordering is needed, and only refinement for two—element coverings is used. In particular, the Dedekind reals form a set; whence we have also refined an earlier result by Aczel (...) and Rathjen, who invoked the full form of fullness. To further generalise this, we look at Richman's method to complete an arbitrary metric space without sequences, which he designed to avoid countable choice. The completion of a separable metric space turns out to be a set even if the original space is a proper class; in particular, every complete separable metric space automatically is a set. (shrink)

Berkeley’s likeness principle is the claim that “an idea can be like nothing but an idea”. The likeness principle is intended to undermine representationalism: the view (that Berkeley attributes to thinkers like Descartes and Locke) that all human knowledge is mediated by ideas in the mind which represent material objects. Yet, Berkeley appears to leave the likeness principle unargued for. This has led to several attempts to explain why Berkeley accepts it. In contrast to ‘metaphysical’ and ‘epistemological’ interpretations available in (...) the literature, in this paper I defend a ‘conceptual’ interpretation. I argue that Berkeley accepts the likeness principle on the basis of (i) his commitment to the transparency of ideas, and (ii) his account of resemblance which he sets out in his works on vision. Thus, I provide an explanation for Berkeley’s reasons for accepting the likeness principle which, appropriately, focuses on his views concerning ideas and likeness. (shrink)

When do two sentences say the same thing, that is, express the same content? We defend two-component (2C) semantics: the view that propositional contents comprise (at least) two irreducibly distinct constituents, (1) truth-conditions, and (2) subject-matter. We contrast 2C with one-component (1C) semantics, focusing on the view that subject-matter is reducible to truth- conditions. We identify exponents of this view and argue in favor of 2C. An appendix proposes a general formal template for propositional 2C semantics.

From the end of the 19th century until his death, one of history's most brilliant mathematicians languished in an asylum. The Mystery of the Aleph tells the story of Georg Cantor (1845-1918), a Russian-born German who created set theory, the concept of infinite numbers, and the "continuum hypothesis," which challenged the very foundations of mathematics. His ideas brought expected denunciation from established corners - he was called a "corruptor of youth" not only for his work in mathematics, but for his (...) larger attempts to meld spirituality and science. (shrink)

Margaret Macdonald was at the institutional heart of analytic philosophy in Britain in the mid-twentieth century. Yet, her views on the nature of philosophical theories diverge quite considerably from those of many of her contemporaries. In this paper, I focus on her 1953 article ‘Linguistic Philosophy and Perception’, a provocative paper in which Macdonald argues that the value of philosophical theories is more akin to that of poetry or art than science or mathematics. I do so for two reasons. First, (...) because it reveals just how far Macdonald’s meta- philosophical views diverged from those of many of her contemporaries. Second, because the discussion in LPP pre-empts recent literature on the nature of philosophical inquiry and the efficacy of philosophical arguments. Indeed, Macdonald’s paper is just as likely to provoke discussion today as it was in the 1950s. (shrink)

It is argued that causal understanding originates in experiences of acting on objects. Such experiences have consistent features that can be used as clues to causal identification and judgment. These are singular clues, meaning that they can be detected in single instances. A catalog of 14 singular clues is proposed. The clues function as heuristics for generating causal judgments under uncertainty and are a pervasive source of bias in causal judgment. More sophisticated clues such as mechanism clues and repeated interventions (...) are derived from the 14. Research on the use of empirical information and conditional probabilities to identify causes has used scenarios in which several of the clues are present, and the use of empirical association information for causal judgment depends on the presence of singular clues. It is the singular clues and their origin that are basic to causal understanding, not multiple instance clues such as empirical association, contingency, and conditional probabilities. (shrink)

In this important new book in the IPPP series, a group of leading thinkers in psychiatry, psychology, and philosophy offer alternative perspectives that address both the scientific and clinical aspects of psychiatric validation, emphasizing throughout their philosophical and historical considerations.

William Molyneux was born in Dublin, studied in Trinity College Dublin, and was a founding member of the Dublin Philosophical Society (DPS), Ireland’s counterpart to the Royal Society in London. He was a central figure in the Irish intellectual milieu during the Early Modern period and – along with George Berkeley and Edmund Burke – is one of the best-known thinkers to have come out of that context and out of Irish thought more generally. In 1688, when Molyneux wrote the (...) letter to Locke in which he posed the now famous question about a man born blind made to see, he was an active member of the DPS and was on familiar terms with several other key figures in Irish philosophy at the time. For the most part, Irish thinkers in the seventeenth and eighteenth centuries answered “no” to Molyneux’s question of whether a blind man made to see could distinguish, by sight alone, between a cube and a sphere. (shrink)

This paper provides a philosophical evaluation of the concept of handedness prevalent but largely unspoken in the scientific literature. This literature defines handedness as the preference or ability to use one hand rather than the other across a range of common activities. Using the philosophical discipline of phenomenology, I articulate and critique this conceptualization of handedness. Phenomenology shows defining a concept of handedness by focusing on hand use leads to a right hand biased concept. I argue further that a phenomenological (...) model based in spatial orientation rather than hand use provides a more inclusive concept of handedness. (shrink)

In my contribution to this Concise Collection on Margaret Cavendish, I focus on teaching Cavendish’s work in the context of philosophy (and, more specifically, Early Modern Philosophy). I have three aims. First, to explain why teaching women from philosophy’s history is crucially important to the discipline. Second, to outline my own reflections on teaching Cavendish’s philosophy. Third, to defend a specific claim about the benefits of teaching Cavendish to philosophy students; namely, that introducing biographical detail alongside philosophical ideas enriches the (...) learning experience. (shrink)

Since its original publication in 1975, this groundbreaking work has awakened millions of people to the existence of "speciesism"—our systematic disregard of nonhuman animals—inspiring a worldwide movement to transform our attitudes to animals and eliminate the cruelty we inflict on them. In Animal Liberation, author Peter Singer exposes the chilling realities of today’s "factory farms" and product-testing procedures—destroying the spurious justifications behind them, and offering alternatives to what has become a profound environmental and social as well as moral issue. (...) An important and persuasive appeal to conscience, fairness, decency, and justice, it is essential reading for the supporter and the skeptic alike. (shrink)

ABSTRACTThe general assumption that people fail to notice discrepancy between their answer and the normative answer in the conjunction fallacy task has been challenged by the theory of Logical Intuition. This theory suggests that people can detect the conflict between the heuristic and normative answers even if they do not always manage to inhibit their intuitive choice. This theory gained support from the finding that people report lower levels of confidence in their choice after they commit the conjunction fallacy compared (...) to when their answer is not in conflict with logic. In four experiments we asked the participants to give probability estimations to the options of the conflict and no-conflict versions of the tasks in the original set-up of the experiment or in a three-option design. We found that participants perceive probabilities for the options of the conflict version less similar than for the no-conflict version. As people are less confident when choosing between more similar options, this simil.. (shrink)

Prospect Theory: For Risk and Ambiguity, provides a comprehensive and accessible textbook treatment of the way decisions are made both when we have the statistical probabilities associated with uncertain future events and when we lack them. The book presents models, primarily prospect theory, that are both tractable and psychologically realistic. A method of presentation is chosen that makes the empirical meaning of each theoretical model completely transparent. Prospect theory has many applications in a wide variety of disciplines. The material in (...) the book has been carefully organized to allow readers to select pathways through the book relevant to their own interests. With numerous exercises and worked examples, the book is ideally suited to the needs of students taking courses in decision theory in economics, mathematics, finance, psychology, management science, health, computer science, Bayesian statistics, and engineering. (shrink)