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  1. Peter Smith, Basic Reading on Computable Functions.
    This is an annotated reading list on the beginning elements of the theory of computable functions. It is now structured so as to complement the first eight lectures of Thomas Forster’s Part III course in Lent 2011 (see the first four chapters of his evolving handouts).
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  2. Peter Smith, The MRDP Theorem.
    Here is Hilbert is his famous address of 1900: The supply of problems in mathematics is inexhaustible, and as soon as one problem is solved numerous others come forth in its place. Permit me in the following, tentatively as it were, to mention particular definite problems, drawn from various branches of mathematics, from the discussion of which an advancement of science may be expected.
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  3. Peter Smith, Tennenbaum's Theorem.
    We are going to prove a key theorem that tells us just a bit more about the structure of the non-standard countable models of first-order Peano Arithmetic; and then we will very briefly consider whether any broadly philosophical morals can be drawn from the technical result.
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  4. Peter Smith, Back to Basics: Revisiting the Incompleteness Theorems.
    Preface 1 The First Theorem revisited 1.1 Notational preliminaries 1.2 Definitional preliminaries 1.3 A general version of G¨ odel’s First Theorem 1.4 Giving the First Theorem bite 1.5 Generic G¨ odel sentences and arithmetic truth 1.6 Canonical and standard G¨ odel sentences 2 The Second Theorem revisited 2.1 Definitional preliminaries 2.2 Towards G¨ odel’s Second Theorem 2.3 A general version of G¨ odel’s Second Theorem 2.4 Giving the Second Theorem bite 2.5 Comparisons 2.6 Further results about provability predicates 2.7 Back (...)
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  5. Peter Smith, Cuts, Consistency and Axiomatized Theories.
    In the Wednesday Logic Reading Group, where we are working through Sara Negri and Jan von Plato’s Structural Proof Theory – henceforth ‘NvP’ – I today introduced Chapter 6, ‘Structural Proof Analysis of Axiomatic Theories’. In their commendable efforts to be brief, the authors are sometimes a bit brisk about motivation. So I thought it was worth trying to stand back a bit from the details of this action-packed chapter as far as I understood it in the few hours I (...)
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  6. Peter Smith, Curry's Paradox, Lukasiewicz, and Field.
    In approaching Ch. 4 of Saving Truth from Paradox, it might be helpful first to revisit Curry’s original paper, and to revisit Lukasiewicz too, to provide more of the scenesetting that Field doesn’t himself fill in. So in §1 I’ll say something about Curry, in §2 we’ll look at what Lukasiewicz was up to in his original three-valued logic, and in §3 we’ll look at the move from a three-valued to a many-valued Lukasiewicz logic. In §4, I move on to (...)
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  7. Peter Smith, Church's Thesis After 70 Years.
    In the section ‘Further reading’, I listed a book that arrived on my desk just as I was sending IGT off to the press, namely Church’s Thesis after 70 Years edited by Adam Olszewski et al. On the basis of a quick glance, I warned that the twenty two essays in the book did seem to be of ‘variable quality’. But actually, things turn out to be a bit worse than that: the collection really isn’t very good at all! After (...)
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  8. Peter Smith, Corrections to Igt.
    An Introduction to G¨ odel’s Theorems now exists in two versions – the original 2007 printing, and a corrected reprint in 2008. The quick way of telling these apart is to glance at the imprints page (the verso of the title page): the later version notes, halfway down that page, “Reprinted with corrections 2008”. Corrections marked with marginal side-bars as here are those that have been noted after the second printing went to press, and hence still need correction/improvement. If you (...)
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  9. Peter Smith, Developing a Writing Style.
    Like the other major journals, ANALYSIS can accept less than 10% of submissions. So standards are fierce. Many submissions are ruled out of court for being badly argued or for re-inventing the wheel or for being plain boring. But a fair proportion end up on the rejection pile simply because they are badly written. I saw far too much bad prose (to be sure, some of the prose that gets published is not exactly wonderful: I assure you that a lot (...)
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  10. Peter Smith, Expressing and Capturing the Primitive Recursive Functions.
    The last Episode wasn’t about logic or formal theories at all: it was about common-or-garden arithmetic and the informal notion of computability. We noted that addition can be defined in terms of repeated applications of the successor function. Multiplication can be defined in terms of repeated applications of addition. The exponential and factorial functions can be defined, in different ways, in terms of repeated applications of multiplication. There’s already a pattern emerging here! The main task in the last Episode was (...)
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  11. Peter Smith, Entailment, with Nods to Lewy and Smiley.
    Last week, we talked a bit about the Anderson-Belnap logic of entailment, as discussed in Priest’s Introduction to Non-Classical Logic. For a quite different approach to entailment, we’ll look next week at Neil Tennant’s account. Doing things rather out of order, this week I’d like to say something more basic about the problems to which both Anderson and Belnap, on the one hand, and Tennant on the other, are responding. This will give me the chance for a bit of nostalgic (...)
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  12. Peter Smith, Formal Logic.
    ... 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 (...)
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  13. Peter Smith, Field on Truth: How Complex is Too Complex?
    In a reading group, we’ve been working through the first three parts of Field’s Saving Truth from Paradox, by the end of which he has presented his core proposals. At this point, we’ve now rather lost the will to continue – for this is an astonishingly badly written book, which makes ridiculous demands on the patience of even a sympathetic reader. It so happened that it fell to me to introduce the last two chapters in Part III, Ch. 17 in (...)
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  14. Peter Smith, First-Order Peano Arithmetic.
    Theorem 1. If T is a sound formalized theory whose language contains the language of basic arithmetic, then there will be a true sentence GT of basic arithmetic such that T ￿ GT and ￿ ¬GT, so T must be negation incomplete.
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  15. Peter Smith, Getting Published.
    Publish or perish? Well, like it or not (and I for one don't!--for I fear it encourages narrowness and scholasticism), having a track record of pieces accepted for publication is now more or less a sine qua non for getting a foot on the first rung of the profession, as a junior research fellow or temporary lecturer. And when it comes to applying for a permanent lectureship a good track record of publication and clear evidence that you are going to (...)
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  16. Peter Smith, Godel's Theorem: A Proof From the Book?
    Here’s one version G¨ odel’s 1931 First Incompleteness Theorem: If T is a nice, sound theory of arithmetic, then it is incomplete, i.e. there are arithmetical sentences ϕ such that T proves neither ϕ nor ¬ϕ. There are three things here to explain straight away.
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  17. Peter Smith, Induction and Predicativity.
    I am interested in the philosophical prospects of what is called ‘predicativism given the natural numbers’. And today, in particular, I want to critically discuss one argument that has been offered to suggest that this kind of predicativism can’t have a stable philosophical motivation. Actually you don’t really need to know about predicativism to find some stand-alone interest in the theme I will be discussing. But still, it’s worth putting things into context. So I’m going to start by spending a (...)
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  18. Peter Smith, Incompleteness and Undecidability.
    In Episode 1, we introduced the very idea of a negation-incomplete formalized theory T . We noted that if we aim to construct a theory of basic arithmetic, we’ll ideally like the theory to be able to prove all the truths expressible in the language of basic arithmetic, and hence to be negation complete. But Gödel’s First Incompleteness Theorem says, very roughly, that a nice theory T containing enough arithmetic will always be negation incomplete. Now, the Theorem comes in two (...)
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  19. Peter Smith, Induction, More or Less.
    The first main topic of this paper is a weak second-order theory that sits between firstorder Peano Arithmetic PA1 and axiomatized second-order Peano Arithmetic PA2 – namely, that much-investigated theory known in the trade as ACA0. What I’m going to argue is that ACA0, in its standard form, lacks a cogent conceptual motivation. Now, that claim – when the wraps are off – will turn out to be rather less exciting than it sounds. It isn’t that all the work that (...)
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  20. Peter Smith, Incompleteness – the Very Idea.
    Why these notes? After all, I’ve written An Introduction to Gödel’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 (...)
     
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  21. Peter Smith, Introducing the Second Theorem.
    This episode introduces the Second Incompleteness Theorem, says something about what it takes to prove it, and why it matters. Just two very quick reminders before we start. We said..
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  22. Peter Smith, Introducing Wilfrid Hodges, a Shorter Model Theory.
    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.
     
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  23. Peter Smith, Kleene's Proof of G¨Odel's Theorem.
    There is a familiar derivation of G¨ odel’s Theorem from the proof by diagonalization of the unsolvability of the Halting Problem. That proof, though, still involves a kind of self-referential trick, as we in effect construct a sentence that says ‘the algorithm searching for a proof of me doesn’t halt’. It is worth showing, then, that some core results in the theory of partial recursive functions directly entail G¨ odel’s First Incompleteness Theorem without any further self-referential trick.
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  24. Peter Smith, Notes on How to Tackle the Essay Paper.
    In each of Parts 1A, IB and II of the Philosophy Tripos, there is an Essay paper in which you are asked to write for three hours on a single topic. In these notes I offer some suggestions about how to tackle this paper, and try to answer some Frequently Asked Questions. The notes are based (in the second half, very closely indeed) on notes written by Jane Heal -- I'm very grateful to her for allowing me to snaffle some (...)
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  25. Peter Smith, Godel Without (Too Many) Tears.
    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 (...)
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  26. Peter Smith, Primitive Recursive Functions.
    In our preamble, it might be helpful this time to give a story about where we are going, rather than (as in previous episodes) review again where we’ve been. So, at the risk of spoiling the excitement, here’s what’s going to happen in this and the following three Episodes.
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  27. Peter Smith, Reading Notes on Logic Options –.
    LO : John L. Bell, David DeVidi and Graham Solomon, Logical Options, Broadview Press, 2001. ILF : Peter Smith, Introduction to Formal Logic, CUP 2003. LFP : Ted Sider, Logic for Philosophy, OUP forthcoming: draft available at http://tedsider.org/books/lfp/lfp.pdf.
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  28. Peter Smith, Squeezing Church's Thesis Again.
    In the very last chapter of my Introduction to Gödel Theorems, I rashly claimed that there is a sense in which we can informally prove Church’s Thesis. This sort of claim isn’t novel to me: but it certainly is still very much the minority line. So maybe it is worth rehearsing some of the arguments again. Even if I don’t substantially add to the arguments in the book, it might help to approach things in a different order, with some different (...)
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  29. Peter Smith, There Are Sea-Serpents, Jim, but Not as We Know Them.
    At the last meeting, Tim Crane gave a talk in which he made play with a distinction between ‘believing in’ and ‘believing that’. And he claimed that this distinction could be put to serious philosophical work of interest to serious metaphysicians. My hunch at the time was that this distinction in fact can’t bear any real weight. But I can’t now reconstruct Tim’s own arguments sufficiently to give a fair evaluation of them. However, Tim did say that the distinction he (...)
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  30. Peter Smith, Topoi, Chaps 5,.
    Why should we care about topoi so defined? Indeed, why should it be said that the idea of a topos is a sort of categorial generalization of the idea of a universe of sets? (By ‘categorial’ I mean a treatment where we give characteristics of the relevant objects and the mappings between them in terms of their relations to other objects and mappings belonging to the same category.).
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  31. Peter Smith, The Diagonalization Lemma, Rosser and Tarski.
    We’ve now proved our key version of the First Theorem, Theorem 42. If T is the right kind of ω-consistent theory including enough arithmetic, then there will be an arithmetic sentence GT such that T ￿ GT and T ￿ ¬GT. Moreover, GT is constructed so that it is true if and only if unprovable-in T (so it is true). Now recall that, for a p.r. axiomatized theory T , Prf T(m, n) is the relation which holds just if m (...)
     
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  32. Peter Smith, The First Incompleteness Theorem.
    • How to construct a ‘canonical’ Gödel sentence • If PA is sound, it is negation imcomplete • Generalizing that result to sound p.r. axiomatized theories whose language extends LA • ω-incompleteness, ω-inconsistency • If PA is ω-consistent, it is negation imcomplete • Generalizing that result to ω-consistent p.r. axiomatized theories which extend Q..
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  33. Peter Smith, The Galois Connection Between Syntax and Semantics.
    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..
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  34. Peter Smith, Two Weak Arithmetics.
    Our last big theorem – Theorem 6 – tells us that if a theory meets certain conditions, then it must be negation incomplete. And we made some initial arm-waving remarks to the effect that it seems plausible that we should want theories which meet those conditions. Later, we announced that there actually is a consistent weak arithmetic with a first-order logic which meets the conditions (in which case, stronger arithmetics will also meet the conditions); but we didn’t say anything about (...)
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  35. Peter Smith, Laws of Nature.
    Where to begin? I’ll take three books from my shelves. First, now nearly forty years old, a little book of television lectures by the great physicist Richard Feynman, The Character of Physical Law. He talks about the laws of motion, the inverse square law of gravitation, conservation laws, symmetry principles and the various ways these all hang together. Feynman obviously takes it that it is a prime aim of science to discover such laws. But what are laws? He writes – (...)
     
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  36. Peter Smith, What Are Turing Jumps?
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  37. Peter Smith, Wittgenstein on Mathematics and Games.
    Unlike his other major typescripts, the Big Typescript is divided into titled chapters, themselves divided into titled sections. But within a section we still get a collection of remarks typically without connecting tissue and lacking any transparently significant ordering or helpful signposting. So we still encounter the usual difficulties in trying to think our way through into what Wittgenstein might be wanting to say. Some enthusiasts like to try to persuade us that the aphoristic style is really of the essence. (...)
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  38. Peter Smith, Special Relativity.
    An inertial frame is one in which a freely falling particle obeys Newton’s first law (i.e., continues in a state of uniform motion). Classically, we have the following: Galilean Principle of Relativity: The laws of dynamics are invariant between all inertial frames. In other words, all inertial observers (at rest in an inertial frame) will get experimentally verify the same dynamical laws.
     
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  39. J. C. Belland & P. L. Smith (forthcoming). Reclaiming Virtue: Philosophy in the Field of Educational Technology. Journal of Thought.
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  40. P. Carruthers & P. K. Smith (forthcoming). Theories of Theories Of. Mind.
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  41. Mara Olekalns, Christopher J. Horan & Philip L. Smith (forthcoming). Maybe It's Right, Maybe It's Wrong: Structural and Social Determinants of Deception in Negotiation. Journal of Business Ethics.
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  42. J. Perry & P. Smith (forthcoming). Levels of Valuational Discourse in Education. Philosophy of Education.
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  43. P. M. Smith & C. A. Mayer (forthcoming). La premiere Epigramme francaise: Clement Marot, Jean Bouchet et Michel d'Amboise. Definition, sources, anteriorité. Bibliothèque d'Humanisme Et Renaissance.
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  44. P. T. Smith & F. Jotterand (forthcoming). Toward a Common Grace Christian Bioethics: A Reformed Protestant Engagement with H. Tristram Engelhardt, Jr. Christian Bioethics.
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  45. Patrick Taylor Smith (forthcoming). The Intergenerational Storm: Dilemma or Domination. Philosophy and Public Issues - Filosofia E Questioni Pubbliche.
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  46. Peter Smith (forthcoming). The Paradoxes of Multiculturalism. Journal of Aesthetic Education.
     
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  47. Preserved Smith (forthcoming). A New Light on the Relations of Peter and Paul. Hibbert Journal.
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  48. P. Toldo, A. Cartier, Ca Mayer & Pm Smith (forthcoming). Philibert de Vienne. Bibliothèque d'Humanisme Et Renaissance.
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  49. G. Msiska, P. Smith & T. Fawcett (2014). Emotive Responses to Ethical Challenges in Caring: A Malawian Perspective. Nursing Ethics 21 (1):97-107.
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  50. Patrick Taylor Smith (2014). Laura Valentini: Justice in a Globalized World: A Normative Framework. Ethical Theory and Moral Practice 17 (3):587-588.
    Laura Valentini’s Justice in a Globalized World presents, with admirable clarity, a new, hybrid conception of global justice that builds on insights from both cosmopolitans and statists, especially their relational variants. Relational cosmopolitans generally argue that substantial economic cooperation and interdependence (i.e., the relevant economic relations) trigger robust obligations of distributive justice. They then argue that, as a matter of fact, these relations obtain globally in virtue of intensifying global trade, capital flows, and labor migration. Thus, relational cosmopolitans conclude that (...)
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