In 1931, the young Kurt Gödel published his First Incompleteness Theorem, which tells us that, for any sufficiently rich theory of arithmetic, there are some arithmetical truths the theory cannot prove. This remarkable result is among the most intriguing in logic. Gödel also outlined an equally significant Second Incompleteness Theorem. How are these Theorems established, and why do they matter? Peter Smith answers these questions by presenting an unusual variety of proofs for the First Theorem, showing how to prove the (...) Second Theorem, and exploring a family of related results. The formal explanations are interwoven with discussions of the wider significance of the two Theorems. This book will be accessible to philosophy students with a limited formal background. It is equally suitable for mathematics students taking a first course in mathematical logic. (shrink)

Vagueness is currently the subject of vigorous debate in the philosophy of logic and language. Vague terms -- such as 'tall', 'red', 'bald', and 'tadpole' -- have borderline cases ; and they lack well-defined extensions. The phenomenon of vagueness poses a fundamental challenge to classical logic and semantics, which assumes that propositions are either true or false and that extensions are determinate.This anthology collects for the first time the most important papers in the field. After a substantial introduction that surveys (...) the field, the essays form four groups, starting with some historically notable pieces. The 1970s saw an explosion of interest in vagueness, and the second group of essays reprints classic papers from this period. The following group of papers represent the best recent work on the logic and semantics of vagueness. The essays in the final group are contributions to the continuing debate about vague objects and vague identity. (shrink)

Theories of Theories of Mind brings together contributions by a distinguished international team of philosophers, psychologists, and primatologists, who between them address such questions as: what is it to understand the thoughts, feelings, and intentions of other people? How does such an understanding develop in the normal child? Why, unusually, does it fail to develop? And is any such mentalistic understanding shared by members of other species? The volume's four parts together offer a state of the art survey of the (...) major topics in the theory-theory/simulationism debate within philosophy of mind, developmental psychology, the aetiology of autism and primatology. The volume will be of great interest to researchers and students in all areas interested in the 'theory of mind' debate. (shrink)

Timothy Smiley’s wonderful paper ‘Rejection’ (1996) is still perhaps not as well known or well understood as it should be. This note first gives a quick presentation of themes from that paper, though done in our own way, and then considers a putative line of objection – recently advanced by Julien Murzi and Ole Hjortland (2009) – to one of Smiley’s key claims. Along the way, we consider the prospects for an intuitionistic approach to some of the issues discussed in (...) Smiley’s paper. (shrink)

Chaotic dynamics has been hailed as the third great scientific revolution in physics this century, comparable to relativity and quantum mechanics. In this book, Peter Smith takes a cool, critical look at such claims. He cuts through the hype and rhetoric by explaining some of the basic mathematical ideas in a clear and accessible way, and by carefully discussing the methodological issues which arise. In particular, he explores the new kinds of explanation of empirical phenomena which modern dynamics can deliver. (...) Explaining Chaos will be compulsory reading for philosophers of science and for anyone who has wondered about the conceptual foundations of chaos theory. (shrink)

Arguably, there is no substantial, general answer to the question of what makes for the approximate truth of theories. But in one class of cases, the issue seems simply resolved. A wide class of applied dynamical theories can be treated as two-component theories—one component specifying a certain kind of abstract geometrical structure, the other giving empirical application to this structure by claiming that it replicates, subject to arbitrary scaling for units etc., the geometric structure to be found in some real-world (...) dynamical phenomenon. In such a case, a theory is approximately true just if the one geometric structure approximately replicates the other (and if problems remain here, they are problems in geometry, of specifying suitable metric approximation relations, not conceptual problems). This article amplifies and defends this simple approach to approximate truth for dynamical theories. (shrink)

Formal logic provides us with a powerful set of techniques for criticizing some arguments and showing others to be valid. These techniques are relevant to all of us with an interest in being skilful and accurate reasoners. In this highly accessible book, Peter Smith presents a guide to the fundamental aims and basic elements of formal logic. He introduces the reader to the languages of propositional and predicate logic, and then develops formal systems for evaluating arguments translated into these languages, (...) concentrating on the easily comprehensible 'tree' method. His discussion is richly illustrated with worked examples and exercises. A distinctive feature is that, alongside the formal work, there is illuminating philosophical commentary. This book will make an ideal text for a first logic course, and will provide a firm basis for further work in formal and philosophical logic. (shrink)

Children’s ability to pretend, and the apparent lack of pretence in children with autism, have become important issues in current research on ‘theory of mind’, on the assumption that pretend play may be an early indicator of metarepresentational abilities.

Some bilateralists have suggested that some of our negative answers to yes-or-no questions are cases of rejection. Mark Textor (2011. Is ‘no’ a force-indicator? No! Analysis 71: 448–56) has recently argued that this suggestion falls prey to a version of the Frege-Geach problem. This note reviews Textor's objection and shows why it fails. We conclude with some brief remarks concerning where we think that future attacks on bilateralism should be directed.

The course of my experience is quite consistent with the hypothesis that it is being produced by a mad scientist who is feeding into my sensory receptors entirely delusive stimuli. Indeed, I could at this very moment be nothing more than a brain floating in a vat of nutrients, my nerve ends linked up to some infernal apparatus by means of which my unknown deceiver induces in me utterly erroneous beliefs about the world.So begins a familiar line of thought which (...) dramatizes an equally familiar sceptical problem about the relation between our experiences and the world. However, Hilary Putnam has recently offered a marvellously ambitious argument which is intended to kill stone-dead the philosophical fantasy that we might be deluded brains in a vat. (shrink)

This is a straightforward, elementary textbook for beginning students of philosophy. The general aim is to provide a clear introduction to the main issues arising in the philosophy of mind. Part I discusses the Cartesian dualist view which many find initially appealing, and contains a careful examination of arguments for and against. Part II introduces the broadly functionalist type of physicalism which has Aristotelian roots. This approach is developed to yield accounts of perception, action, belief and desire, and the emerging (...) theory of the mind is compared at each stage with rival historical and contemporary views. In Part III the functionalist approach is further explored in giving analyses of sensation, thought and freedom of will. The discussions throughout are exceptionally clear, and the writing uncomplicated, to make available to the students a wealth of detailed argument in the philosophy of mind. (shrink)

This Gentle Introduction is very much still work in progress. Roughly aimed at those who want something a bit more discursive, slower-moving, than Awodey's or Leinster's excellent books. -/- The current [Jan 2018] version is 291pp.

In the opening chapter of ‘the Shorter Hodges’, we get a lot of fixing of terminology and notation, and some fairly natural definitions of ideas like that of isomorphism between structures. There are no really tricky ideas which need further exploration, nor any nasty proofs that could do with more elaboration. So I don’t pretend to have anything very thrilling by way of introductory comments. But let me make some more general philosophical comments.

This book examines the philosophical foundations of the realist view of the progress of science as cumulative. It is a view that has recently been faced with a number of powerful attacks in which successive scientific theories are seen, not as extending their scope and honing their explanations, but as incommensurable. There is, it is held, in principle no way of establishing that they are about the same things. From the voluminous literature on the topic, Dr Smith has selected relevantly (...) and incisively and his exposition of the contending arguments is vigourous and clear, without undue technicality. As an explication and defence of realism it will interest all those concerned with this basic question in the philosophy of science and the philosophy of language. (shrink)

odel’s Theorems (CUP, heavily corrected fourth printing 2009: henceforth IGT ). Surely that’s more than enough to be going on with? Ah, but there’s the snag. It is more than enough. In the writing, as is the way with these things, the book grew far beyond the scope of the lecture notes from which it started. And while I hope the result is still pretty accessible to someone prepared to put in the time and effort, there is – to be (...) frank – a lot more in the book than is really needed by philosophers meeting the incompleteness theorems for the first time, or indeed by mathematicians wanting a brisk introduction. You might reasonably want to get your heads around only those technical basics which are actually necessary for understanding how the theorems are proved and for appreciating philosophical discussions about incompleteness. So you need a cut-down version of the book – an introduction to the Introduction! Well, isn’t that what lectures are for? Indeed. But there’s another snag. I haven’t got many lectures to play with. So either (A) I crack on at a very fast pace (hard-core mathmo style), cover those basics, but perhaps leave too many people puzzled and alarmed. Or (B) I do relaxed talk’n’chalk, highlighting the really Big Ideas, making sure everyone is grasping them as we go along, but inevitably omit important stuff and leave quite a gap between what happens in the lectures and what happens in the book. What to do? I’m going for plan (B). But then I ought to do something to fill that gap between lectures and book. Hence these notes. The idea, then, is to give relaxed lectures, highlighting Big Ideas, not worrying too much about depth or fine-detail (nor even about getting through all of the day’s intended menu of topics). These notes then expand things just enough, and give pointers to relevant chunks of IGT. Though I hope these notes will be to a fair extent be stand-alone, and tell a brief but coherent story read by themselves: so occasionally I’ll copy a paragraph or two from the book, rather than just refer to them.. (shrink)

... and a reading knowledge of formal logical symbolism is essential too. (Philosophers often use bits of logical symbolism to clarify their arguments.) Because the artificial and simply formal languages of logic give us highly illuminating objects of comparison when we come thinking about how natural languages work. (Relevant to topics in ‘philosophical logic’ and the philosophy of language.) But mainly because it us the point of entry into the study of one of the major intellectual achievements by philosophers of (...) the 20th – i.e. the development of mathematical logic. (Not least in Cambridge: Russell and Whitehead, Ramsey, Turing.). (shrink)

The course of my experience is quite consistent with the hypothesis that it is being produced by a mad scientist who is feeding into my sensory receptors entirely delusive stimuli. Indeed, I could at this very moment be nothing more than a brain floating in a vat of nutrients, my nerve ends linked up to some infernal apparatus by means of which my unknown deceiver induces in me utterly erroneous beliefs about the world.So begins a familiar line of thought which (...) dramatizes an equally familiar sceptical problem about the relation between our experiences and the world. However, Hilary Putnam has recently offered a marvellously ambitious argument which is intended to kill stone-dead the philosophical fantasy that we might be deluded brains in a vat. (shrink)

Preface 1 Partially ordered sets 1.1 Posets introduced 1.2 Partial orders and strict orders 1.3 Maps between posets 1.4 Compounding maps 1.5 Order similarity 1.6 Inclusion posets as typical..

Needless to say, Charles Parsons’s long awaited book1 is a must-read for anyone with an interest in the philosophy of mathematics. But as Parsons himself says, this has been a very long time in the writing. Its chapters extensively “draw on”, “incorporate material from”, “overlap considerably with”, or “are expanded versions of” papers published over the last twenty-five or so years. What we are reading is thus a multi-layered text with different passages added at different times. And this makes for (...) a rather bumpy read. There is another route Parsons could have taken: he could have reprinted the relevant papers with postscripts, and then top-and-tailed the collection with a preface and added concluding reflections. It must sound very ungrateful, but I rather suspect that that might have worked better. Much of the book is about arithmetic. But Parsons has woven into the discussion claims about mathematics more generally and about set theory in particular. We might well have a basic worry about this structure: for do defensible claims about the ontology and epistemology of arithmetic have to be generalizable to apply to more infinitary mathematics? For example, suppose you are attracted to a Hellman-like modal structuralist account of arithmetic: then should you think it a problem if you suspect that such an account can’t readily be extended to cope e.g. with set theory (since you boggle at the idea of possible worlds free of abstracta but with enough structure to somehow model ZFC)? Parsons himself seems to waver over such questions. So for present purposes, I’ll focus just on arithmetic, and in what follows I’ll revisit two of the most familiar Parsonian themes, his views on structuralism as an account of the ontology of arithmetic, and his exploration of the role of intuition in grounding arithmetical knowledge. (shrink)

This is a straightforward, elementary textbook for beginning students of philosophy. The general aim is to provide a clear introduction to the main issues arising in the philosophy of mind. Part I discusses the Cartesian dualist view which many find initially appealing, and contains a careful examination of arguments for and against. Part II introduces the broadly functionalist type of physicalism which has Aristotelian roots. This approach is developed to yield accounts of perception, action, belief and desire, and the emerging (...) theory of the mind is compared at each stage with rival historical and contemporary views. In Part III the functionalist approach is further explored in giving analyses of sensation, thought and freedom of will. The discussions throughout are exceptionally clear, and the writing uncomplicated, to make available to the students a wealth of detailed argument in the philosophy of mind. (shrink)