1. 89041.152831
    Résumé : Cet article cherche à montrer comment la pratique mathématique, particulièrement celle admettant des représentations visuelles, peut conduire à de nouveau résultats mathématiques. L’argumentation est basée sur l’étude du cas d’un domaine des mathématiques relativement récent et prometteur: la théorie géométrique des groupes. L’article discute comment la représentation des groupes par les graphes de Cayley rendit possible la découverte de nouvelles propriétés géométriques de groupes. Abstract: The paper aims to show how mathematical practice, in particular with visual representations can lead to new mathematical results. The argument is based on a case study from a relatively recent and promising mathematical subject—geometric group theory. The paper discusses how the representation of groups by Cayley graphs made possible to discover new geometric properties of groups.
    Found 1 day ago on PhilSci Archive
  2. 93876.152912
    The slogan ‘Evidence of evidence is evidence’ may sound plausible, but what it means is far from clear. It has often been applied to connect evidence in the current situation to evidence in another situation. The relevant link between situations may be diachronic (White 2006: 538): is present evidence of past or future evidence of something present evidence of that thing? Alternatively, the link may be interpersonal (Feldman 2007: 208): is evidence for me of evidence for you of something evidence for me of that thing? Such interperspectival links have been discussed because they can destabilize inter-perspectival disagreements. In their own right they have become the topic of a lively recent debate (Fitelson 2012, Feldman 2014, Roche 2014, Tal and Comesaña 2014).
    Found 1 day, 2 hours ago on Timothy Williamson's site
  3. 369704.152943
    I give an account of proof terms for derivations in a sequent calculus for classical propositional logic. The term for a derivation δ of a sequent Σ  ∆ encodes how the premises Σ and conclusions ∆ are related in δ. This encoding is many–to–one in the sense that different derivations can have the same proof term, since different derivations may be different ways of representing the same underlying connection between premises and conclusions. However, not all proof terms for a sequent Σ  ∆ are the same. There may be different ways to connect those premises and conclusions.
    Found 4 days, 6 hours ago on Greg Restall's site
  4. 372918.152961
    Computational complexity theory is a branch of computer science that is dedicated to classifying computational problems in terms of their difficulty. Unlike computability theory, whose object is to determine what we can compute in principle, the object of complexity theory is to inform us with regards to our practical limits. It thus serves as a natural conceptual bridge between the study of mathematics and the study of technology, in the sense that computational complexity theory
    Found 4 days, 7 hours ago on PhilSci Archive
  5. 374136.152976
    It is not news that we often make discoveries or find reasons for a mathematical proposition by thinking alone. But does any of this thinking count as conducting a thought experiment? The answer to that question is “yes”, but without refinement the question is uninteresting. Suppose you want to know whether the equation [ 8x + 12y = 6 ] has a solution in the integers. You might mentally substitute some integer values for the variables and calculate. In that case you would be mentally trying something out, experimenting with particular integer values, in order to test the hypothesis that the equation has no solution in the integers. Not getting a solution first time, you might repeat the thought experiment with different integer inputs.
    Found 4 days, 7 hours ago on PhilSci Archive
  6. 374336.152991
    Hintikka taught us that S5 was the wrong epistemic logic because of the unwarranted powers of negative introspection afforded by the (5) schema, ♦p → ♦p. (Tim Williamson later targeted the (4) schema, and with it S4, but that is another story.) The punchline here is that the problem is really the (B) schema, also known as the Brouwershe schema: (B) p → ♦p. In fact, you should think of the (5) schema within S5 as the best hand to play in a classical modal system with (K) when you are dealt the (B) schema. That is the conclusion of (Wheeler 2015), which is about the logic of information rather than logic for lunatics. The behavior of (B), when interpreted as an epistemic modal rather than as a provability operator, is so bizarre, so unreasonable qua epistemic modal, that epistemic logicians should stop referring to (B) as the Brouwershe schema to avoid sullying Brouwer’s good name. Instead, I recommend hereafter for epistemic logicians to refer to (B) as The Blog Schema.
    Found 4 days, 7 hours ago on PhilSci Archive
  7. 374361.153006
    The Univalent Foundations (UF) offer a new picture of the foundations of mathematics largely independent from set theory. In this paper I will focus on the question of whether Homotopy Type Theory (HoTT) (as a formalization of UF) can be justified intuitively as a theory of shapes in the same way that ZFC (as a formalization of set-theoretic foundations) can be justified intuitively as a theory of collections. I first clarify what I mean by an “intuitive justification” by distinguishing between formal and pre-formal “meaning explanations” in the vein of Martin-Löf. I then explain why Martin-Löf’s original meaning explanation for type theory no longer applies to HoTT. Finally, I outline a pre-formal meaning explanation for HoTT based on spatial notions like “shape”, “path”, “point” etc. which in particular provides an intuitive justification of the axiom of univalence. I conclude by discussing the limitations and prospects of such a project.
    Found 4 days, 7 hours ago on PhilSci Archive
  8. 376557.153021
    One of Tarski’s stated aims was to give an explication of the classical conception of truth—truth as ‘saying it how it is’. Many subsequent commentators have felt that he achieved this aim. Tarski’s core idea of defining truth via satisfaction has now found its way into standard logic textbooks. This paper looks at such textbook definitions of truth in a model for standard first-order languages and argues that they fail from the point of view of explication of the classical notion of truth. The paper furthermore argues that a subtly different definition—also to be found in classic textbooks but much less prevalent than the kind of definition that proceeds via satisfaction—succeeds from this point of view.
    Found 4 days, 8 hours ago on Nick Smith's site
  9. 1818619.153036
    Provability logic is a modal logic that is used to investigate what arithmetical theories can express in a restricted language about their provability predicates. The logic has been inspired by developments in meta-mathematics such as Gödel’s incompleteness theorems of 1931 and Löb’s theorem of 1953. As a modal logic, provability logic has been studied since the early seventies, and has had important applications in the foundations of mathematics. From a philosophical point of view, provability logic is interesting because the concept of provability in a fixed theory of arithmetic has a unique and non-problematic meaning, other than concepts like necessity and knowledge studied in modal and epistemic logic.
    Found 3 weeks ago on Stanford Encyclopedia of Philosophy
  10. 1993765.15305
    Computer scientists, logicians and functional programmers have studied continuations in laboratory settings for years. As a result of that work, continuations are now accepted as an indispensable tool for reasoning about control, order of evaluation, classical versus intuitionistic proof, and more. But all of the applications just mentioned concern artificial languages; what about natural languages, the languages spoken by humans in their daily life? Do natural languages get by without any of the marvelous control operators provided by continuations, or can we find continuations in the wild? This paper argues yes: that an adequate and complete analysis of natural language must recognize and rely on continuations. In support of this claim, I identify four independent linguistic phenomena for which a simple CPS-based description provides an insightful analysis.
    Found 3 weeks, 2 days ago on Chris Barker's site
  11. 2169391.153065
    Although mathematicians often use it, mathematical beauty is a philosophically challenging concept. How can abstract objects be evaluated as beautiful? Is this related to the way we visualise them? Using a case study from graph theory (the highly symmetric Petersen graph), this paper tries to analyse aesthetic preferences in mathematical practice and to distinguish genuine aesthetic from epistemic or practical judgements. It argues that, in making aesthetic judgements, mathematicians may be responding to a combination of perceptual properties of visual representations and mathematical properties of abstract structures; the latter seem to carry greater weight. Mathematical beauty thus primarily involves mathematicians’ sensitivity to aesthetics of the abstract.
    Found 3 weeks, 4 days ago on PhilSci Archive
  12. 2215200.153079
    Scope-taking is one of the most fundamental, one of the most characteristic, and one of the most dramatic features of the syntax and semantics of natural languages. A phrase takes scope over a larger expression that contains it when the larger expression serves as the smaller phrase’s semantic argument.
    Found 3 weeks, 4 days ago on Chris Barker's site
  13. 2292636.153094
    Brasoveanu 2011 argues that certain expressions exhibit what he calls “association with distributivity” (AWD for short): (1) a. Every boy read a different poem. b. The boys read a different poem. The claim is that in (1a), different can be anaphorically linked to the distributivity introduced by the quantificational determiner every, in which case (1a) entails that no poem was read by more than one boy. In contrast, in (1b) the plural the boys does not introduce distributivity, which is why (on the AWD account) (1b) does not have the reading just described. Instead, (1b) only has an external reading on which different is anaphoric to some element outside to the sentence.
    Found 3 weeks, 5 days ago on Chris Barker's site
  14. 2292661.153109
    In this paper we introduce a computational-level model of theory of mind (ToM) based on dynamic epistemic logic (DEL), and we analyze its computational complexity. The model is a special case of DEL model checking. We provide a parameterized complexity analysis, considering several aspects of DEL (e.g., number of agents, size of preconditions, etc.) as parameters. We show that model checking for DEL is PSPACE-hard, also when restricted to single-pointed models and S5 relations, thereby solving an open problem in the literature. Our approach is aimed at formalizing current intractability claims in the cognitive science literature regarding computational models of ToM.
    Found 3 weeks, 5 days ago on Jakub Szymanik's site
  15. 2334943.153123
    Following Szabolcsi 1982, Naumann 2001, and Lascarides & Asher 2003, this paper promotes actions as an essential element in semantic analysis. It proposes that imperatives denote actions, and speculates about embedding a variant of Segerberg’s Dynamic Logic for imperatives within a Linear Logic treatment of imperatives and deontics along the lines of Barker 2010. The key test case will be Ross’ Paradox and its deontic analog, the problem of free choice permission. So what are actions? Actions change the world. This means that actions can be characterized by before-and-after pictures, that is, by a picture of the world before the action is performed, and a picture of the world afterwards. Technically, then, an action will be a relation over worlds, a set whose elements are ordered pairs w, w where w is the world before the action and w is the world after the action in question has been performed.
    Found 3 weeks, 6 days ago on Chris Barker's site
  16. 2335117.153138
    This paper explores an approach to reconstruction that falls into the general category of semantic reconstruction: the syntax and the semantics collaborate in order to account for a number of reconstruction effects, but without any syntactic movement. The analysis builds on Shan and Barker 2006, Barker and Shan 2008, and Barker 2009. Shan and Barker 2006:123 note that at least some reconstruction effects fall out from the interaction of their particular analyses of scope-taking, binding, and wh-interrogatives. In Barker 2009, I discuss that account of reconstruction, developing especially some of the details of the the treatment of questions and higher-order pronoun meanings. These previous discussions, however, considered only a very small range of example types. One of the main goals of the current paper is to see how well the approach scales up to a wider range of reconstruction effects and example types, including quantificational binding, binding of anaphors, idiom licensing, and especially crossover phenomena, in the context of wh-interrogatives, relative clauses, and wh-relatives.
    Found 3 weeks, 6 days ago on Chris Barker's site
  17. 2345436.153152
    Mathematically, quantum mechanics can be regarded as a non-classical probability calculus resting upon a non-classical propositional logic. More specifically, in quantum mechanics each probability-bearing proposition of the form “the value of physical quantity \(A\) lies in the range \(B\)” is represented by a projection operator on a Hilbert space \(\mathbf{H}\). These form a non-Boolean—in particular, non-distributive—orthocomplemented lattice. Quantum-mechanical states correspond exactly to probability measures (suitably defined) on this lattice. What are we to make of this? Some have argued that the empirical success of quantum mechanics calls for a revolution in logic itself.
    Found 3 weeks, 6 days ago on Stanford Encyclopedia of Philosophy
  18. 2684681.153167
    In response to problems raised by Bench-Capon [4], this paper shows how two models of precedential constraint can be broadened to include legal information represented through dimensions, as well as standard factors.
    Found 1 month ago on John Horty's site
  19. 2693800.153189
    This paper is a further consideration of Hemmo and Shenker’s (2012) ideas about the proper conceptual characterization of macrostates in statistical mechanics. We provide two formulations of how macrostates come about as elements of certain partitions of the system’s phase space imposed on by the interaction between the system and an observer, and we show that these two formulations are mathematically equivalent. We also reflect on conceptual issues regarding the relationship of macrostates to distinguishability, thermodynamic regularity, observer dependence, and the general phenomenon of measurement.
    Found 1 month ago on PhilSci Archive
  20. 2694865.153206
    Hawthorne et al. (2015) argue that the Principal Principle implies a version of the Principle of Indifference. We show that what the Authors take to be the Principle of Indifference can be obtained without invoking anything which would seem to be related to the Principal Principle. In the Appendix we also discuss several Conditions proposed in the same paper.
    Found 1 month ago on PhilSci Archive
  21. 2714208.153226
    Welcome to the first Virtual Colloquium of the spring term! Today’s paper is “Anselm, not Alston: The Reference of ‘God’ Revisited” by H.D.P. Burling. Hugh Burling is a PhD student at the University of Cambridge (UK) and a Visiting Graduate Fellow at the Center for Philosophy of Religion at the University of Notre Dame. …
    Found 1 month ago on The Prosblogion
  22. 3171155.153245
    A common view is that Charles Peirce influenced Josiah Royce. This paper demonstrates that Josiah Royce influenced Charles Peirce. A chronology is presented, followed with a brief description of a change in Peirce’s thinking from studying the writings of Royce.
  23. 3519316.153262
    Computer programs are particular kinds of texts. It is therefore natural to ask what is the meaning of a program or, more generally, how can we set up a formal semantical account of a programming language. There are many possible answers to such questions, each motivated by some particular aspect of programs. So, for instance, the fact that programs are to be executed on some kind of computing machine gives rise to operational semantics, whereas the similarities of programming languages with the formal languages of mathematical logic has motivated the denotational approach that interprets programs and their constituents by means of set-theoretical models.
    Found 1 month, 1 week ago on Stanford Encyclopedia of Philosophy
  24. 3634966.153277
    A small probability space representation of quantum mechanical probabilities is defined as a collection of Kolmogorovian probability spaces, each of which is associated with a context of a maximal set of compatible measurements, that portrays quantum probabilities as Kolmogorovian probabilities of classical events. Bell’s theorem is stated and analyzed in terms of the small probability space formalism.
    Found 1 month, 1 week ago on PhilSci Archive
  25. 4081614.153291
    Logicism is typically defined as the thesis that mathematics reduces to, or is an extension of, logic. Exactly what “reduces” means here is not always made entirely clear. (More definite articulations of logicism are explored in section 5 below.) While something like this thesis had been articulated by others (e.g., Dedekind 1888 and arguably Leibniz 1666), logicism only became a widespread subject of intellectual study when serious attempts began to be made to provide complete deductions of the most important principles of mathematics from purely logical foundations. This became possible only with the development of modern quantifier logic, which went hand in hand with these attempts. Gottlob Frege announced such a project in his 1884 Grundlagen der Arithmetik (translated as Frege 1950), and attempted to carry it out in his 1893–1902 Grundgesetze der Arithmetik (translated as Frege 2013). Frege limited his logicism to arithmetic, however, and it turned out that his logical foundation was inconsistent.
    Found 1 month, 2 weeks ago on Kevin C. Klement's site
  26. 4081631.153306
    Simplistic accounts of its history sometimes portray logic as having stagnated in the West completely from its origins in the works of Aristotle all the way until the 19th Century. This is of course nonsense. The Stoics and Megarians added propositional logic. Medievals brought greater unity and systematicity to Aristotle’s system and improved our understanding of its underpinnings (see e.g., Henry 1972), and important writings on logic were composed by thinkers from Leibniz to Clarke to Arnauld and Nicole. However, it cannot be denied that an unprecedented sea change occurred in the 19th Century, one that has completely transformed our understanding of logic and the methods used in studying it. This revolution can be seen as proceeding in two main stages. The first dates to the mid-19th Century and is owed most signally to the work of George Boole (1815–1864). The second dates to the late 19th Century and the works of Gott-lob Frege (1848–1925). Both were mathematicians primarily, and their work made it possible to bring mathematical and formal approaches to logical research, paving the way for the significant meta-logical results of the 20th Century. Boolean algebra, the heart of Boole’s contributions to logic, has also come to represent a cornerstone of modern computing. Frege had broad philosophical interests, and his writings on the nature of logical form, meaning and truth remain the subject of intense theoretical discussion, especially in the analytic tradition. Frege’s works, and the powerful new logical calculi developed at the end of the 19th Century, influenced many of its most seminal figures, such as Bertrand Russell, Ludwig Wittgenstein and Rudolf Carnap. Indeed, Frege is sometimes heralded as the “father” of analytic philosophy, although he himself would not live to become aware of any such movement.
    Found 1 month, 2 weeks ago on Kevin C. Klement's site
  27. 4243495.153327
    In this paper, I will examine the representative halfer and thirder solutions to the Sleeping Beauty problem. Then by properly applying the event concept in probability theory and examining similarity of the Sleeping Beauty problem to the Monty Hall problem, it is concluded that the representative thirder solution is wrong and the halfers are right, but that the representative halfer solution also contains a wrong logical conclusion.
    Found 1 month, 2 weeks ago on PhilSci Archive
  28. 4243515.153343
    This paper raises a simple continuous spectrum issue in many-worlds interpretation of quantum mechanics, or Everettian interpretation. I will assume that Everettian interpretation refers to many-worlds understanding based on quantum decoherence. The fact that some operators in quantum mechanics have continuous spectrum is used to propose a simple thought experiment based on probability theory. Then the paper concludes it is untenable to think of each possibility that wavefunction Ψ gives probability as actual universe. While the argument that continuous spectrum leads to inconsistency in the cardinality of universes can be made, this paper proposes a different argument not relating to theoretical math that actually has practical problems.
    Found 1 month, 2 weeks ago on PhilSci Archive
  29. 4593299.153359
    In this paper, I examine the relationship between physical quantities and physical states in quantum theories. I argue against the claim made by Arageorgis (1995) that the approach to interpreting quantum theories known as Algebraic Imperialism allows for “too many states”. I prove a result establishing that the Algebraic Imperialist has very general resources that she can employ to change her abstract algebra of quantities in order to rule out unphysical states.
    Found 1 month, 3 weeks ago on PhilSci Archive
  30. 4671270.153374
    Continuing from the previous post, I’ll consider five elementary textbooks aimed at philosophers, all either first published, or with new editions, well after e.g. Edgington’s State of the Art article. …
    Found 1 month, 3 weeks ago on Peter Smith's blog