Conciliationism faces a challenge that has not been satisfactorily addressed. There are clear cases of epistemically significant merely possible disagreement, but there are also clear cases where merely possible disagreement is epistemically irrelevant. Conciliationists have not yet accounted for this asymmetry. In this paper, we propose that the asymmetry can be explained by positing a selection constraint on all cases of peer disagreement—whether actual or merely possible. If a peer’s opinion was not selected in accordance with the proposed constraint, then (...) it lacks epistemic significance. This allows us to distinguish the epistemically significant cases of merely possible disagreement from the insignificant ones. (shrink)

This comparative examination of Nietzsche and the Islamic philosopher al-Kindi emphasizes their mutual commitment to the recovery of classical Greek and Hellenistic thought and the idea of philosophy as a way of life. Affiliating both thinkers with the Stoic lineage in particular, I examine the ways in which they appropriate common themes such as fatalism, self-cultivation via spiritual exercises, and the banishing of sorrow. Focusing primarily on their respective conceptions of self and nature, I argue that the antipodal worldviews of (...) al-Kindi and Nietzsche can be understood as a bifurcation of Stoic philosophy. (shrink)

Logic has pride of place in mathematics and its 20th century offshoot, computer science. Modern symbolic logic was developed, in part, as a way to provide a formal framework for mathematics: Frege, Peano, Whitehead and Russell, as well as Hilbert developed systems of logic to formalize mathematics. These systems were meant to serve either as themselves foundational, or at least as formal analogs of mathematical reasoning amenable to mathematical study, e.g., in Hilbert’s consistency program. Similar efforts continue, but have been (...) expanded by the development of sophisticated methods to study the properties of such systems using proof and model theory. In parallel with this evolution of logical formalisms as tools for articulating mathematical theories (broadly speaking), much progress has been made in the quest for a mechanization of logical inference and the investigation of its theoretical limits, culminating recently in the development of new foundational frameworks for mathematics with sophisticated computer-assisted proof systems. In addition, logical formalisms developed by logicians in mathematical and philosophical contexts have proved immensely useful in describing theories and systems of interest to computer scientists, and to some degree, vice versa. Three examples of the influence of logic in computer science are automated reasoning, computer verification, and type systems for programming languages. (shrink)

I propose a novel metaphysical explanation of identity and distinctness facts called the Modal Proposal. According to the Modal Proposal, for each identity fact – that is, each fact of the form a=b – that fact is metaphysically explained by the fact that it is necessary that the entities involved are indiscernible, and for each distinctness fact –that is, each fact of the form a≠b – that fact is metaphysically explained by the fact that it is possible for the entities (...) involved to be discernible. I argue that the Modal Proposal has greater payoffs at less cost than any of its competitors. It gives simple, uniform, and intuitive explanations of identity and distinctness that conserve longstanding philosophical insights about identity that go back to Leibniz. It does this while making our fundamental base more parsimonious, determining whether controversial cases of identity or distinctness are possible, and expanding our understanding of these central philosophical relations. It does all this while avoiding controversial ontological commitments. (shrink)

Prevailing opinion—defended by Jason Brennan and others—is that voting to change the outcome is irrational, since although the payoffs of tipping an election can be quite large, the probability of doing so is extraordinarily small. This paper argues that prevailing opinion is incorrect. Voting is shown to be rational so long as two conditions are satisfied: First, the average social benefit of electing the better candidate must be at least twice as great as the individual cost of voting, and second, (...) the chance of casting the decisive vote must be at least 1/N, where N stands for the number of citizens. It is argued that both of these conditions are often true in the real world. (shrink)

There is a well-known moral quandary concerning how to account for the rightness or wrongness of acts that clearly contribute to some morally significant outcome – but which each seem too small, individually, to make any meaningful difference. One consequentialist-friendly response to this problem is to deny that there could ever be a case of this type. This paper pursues this general strategy, but in an unusual way. Existing arguments for the consequentialist-friendly position are sorites-style arguments. Such arguments imagine varying (...) a subject’s predicament bit by bit until it is clear that a relevant difference has been achieved. The arguments offered in this paper are structurally different, and do not rely on any sorites series. For this reason, they are not vulnerable to objections that have been leveled against the sorites-style arguments. (shrink)

이 책자는 형식 논리의 일종인 양상논리에 입문하고 싶으신 분들을 위한 짧은 교재입니다. “양상논리 맛보기” 라는 말마따나, 이 책자는 양상논리에 관심은 있지만 아직 본격적으로 공부를 시작하진 않은 분들께서 ‘맛보기’를 하기에 적합한 안내 책자입니다. 아무쪼록 이 책자가 양상논리를 공부해나가시는데 유용한 첫 발판이 될 수 있기를 바랍니다. / This booklet is a Korean adaptation and translation of Part VIII of forall x: Calgary (Fall 2021 edition), which is intended to be introductory material for modal logic. The original text is based on Robert Trueman's (...) A Modal Logic Primer, which is revised and expanded by Richard Zach and Aaron Thomas-Bolduc in forall x: Calgary. (forall x: Calgary is based on forall x: Cambridge by Tim Button, which is in turn based on forall x by P. D. Magnus, and also includes material from forall x: Lorain County Remix by Cathal Woods and J. Robert Loftis.) This booklet is licensed under a Creative Commons Attribution 4.0 license. (shrink)

Should we believe our controversial philosophical views? Recently, several authors have argued from broadly conciliationist premises that we should not. If they are right, we philosophers face a dilemma: If we believe our views, we are irrational. If we do not, we are not sincere in holding them. This paper offers a way out, proposing an attitude we can rationally take toward our views that can support sincerity of the appropriate sort. We should arrive at our views via a certain (...) sort of ‘insulated’ reasoning – that is, reasoning that involves setting aside certain higher-order worries, such as those provided by disagreement – when we investigate philosophical questions. (shrink)

What should a person do when, through no fault of her own, she ends up believing a false moral theory? Some suggest that she should act against what the false theory recommends; others argue that she should follow her rationally held moral beliefs. While the former view better accords with intuitions about cases, the latter one seems to enjoy a critical advantage: It seems better able to render moral requirements ‘followable’ or ‘action-guiding.’ But this tempting thought proves difficult to justify. (...) Indeed, whether it can be justified turns out to depend importantly on the rational status of epistemic akrasia. Furthermore, it can be argued, from premises all parties to the moral ignorance debate should accept, that rational epistemic akrasia is possible. If the argument proves successful, it follows that a person should sometimes act against her rationally held moral convictions. (shrink)

This paper is about how to aggregate outside opinion. If two experts are on one side of an issue, while three experts are on the other side, what should a non-expert believe? Certainly, the non-expert should take into account more than just the numbers. But which other factors are relevant, and why? According to the view developed here, one important factor is whether the experts should have been expected, in advance, to reach the same conclusion. When the agreement of two (...) (or of twenty) thinkers can be predicted with certainty in advance, their shared belief is worth only as much as one of their beliefs would be worth alone. This expectational model of belief dependence can be applied whether we think in terms of credences or in terms of all-or-nothing beliefs. (shrink)

This article re-exams the old tension between the philosopher and the city. Reading Ibn Bājja’s Governance of the Solitary and Nietzsche’s Thus Spoke Zarathustra against the background of Plato’s Republic, I argue that they both embrace several key aspects of Platonic political philosophy: the assumption that philosophical natures can grow spontaneously in sick cities, the ideal of the philosopher legislator and the correlative project of founding a virtuous new regime. Yet in preparation for this final task, each prescribes a regimen (...) of solitude for philosophers, so that they might preserve their own health and autonomy. While this spiritual exercise at first appears merely temporary and provisional—aimed at the cultivation of a philosopher ruler and the eventual establishment of a healthy political regime —I argue that both Ibn Bājja and Zarathustra ultimately abandon their Platonic ambitions and opt instead for the apolitical contemplative life. (shrink)

A discussion of the different ways in which the Islamicate philosophical tradition has been characterized and categorized in Anglo-European scholarship.

A philosophical examination of 'new wave' as a musical genre, focusing on its developmental history and relation to punk as well as its unique ethos and aesthetic. Forthcoming 2021.

Although arguments for and against competing theories of vagueness often appeal to claims about the use of vague predicates by ordinary speakers, such claims are rarely tested. An exception is Bonini et al. (1999), who report empirical results on the use of vague predicates by Italian speakers, and take the results to count in favor of epistemicism. Yet several methodological difficulties mar their experiments; we outline these problems and devise revised experiments that do not show the same results. We then (...) describe three additional empirical studies that investigate further claims in the literature on vagueness: the hypothesis that speakers confuse ‘P’ with ‘definitely P’, the relative persuasiveness of different formulations of the inductive premise of the Sorites, and the interaction of vague predicates with three different forms of negation. (shrink)

In The Boundary Stones of Thought, Rumfitt defends classical logic against challenges from intuitionistic mathematics and vagueness, using a semantics of pre-topologies on possibilities, and a topological semantics on predicates, respectively. These semantics are suggestive but the characterizations of negation face difficulties that may undermine their usefulness in Rumfitt’s project.

David Hilbert's finitistic standpoint is a conception of elementary number theory designed to answer the intuitionist doubts regarding the security and certainty of mathematics. Hilbert was unfortunately not exact in delineating what that viewpoint was, and Hilbert himself changed his usage of the term through the 1920s and 30s. The purpose of this paper is to outline what the main problems are in understanding Hilbert and Bernays on this issue, based on some publications by them which have so far received (...) little attention, and on a number of philosophical reconstructions of the viewpoint (in particular, by Hand, Kitcher, and Tait). (shrink)

The use of the symbol ∨for disjunction in formal logic is ubiquitous. Where did it come from? The paper details the evolution of the symbol ∨ in its historical and logical context. Some sources say that disjunction in its use as connecting propositions or formulas was introduced by Peano; others suggest that it originated as an abbreviation of the Latin word for “or,” vel. We show that the origin of the symbol ∨ for disjunction can be traced to Whitehead and (...) Russell’s pre-Principia work in formal logic. Because of Principia’s influence, its notation was widely adopted by philosophers working in logic (the logical empiricists in the 1920s and 1930s, especially Carnap and early Quine). Hilbert’s adoption of ∨ in his Grundzüge der theoretischen Logik guaranteed its widespread use by mathematical logicians. The origins of other logical symbols are also discussed. (shrink)

I argue that a powers-based metaphysics radically reconfigures the existing free will problematic. This is different from claiming that such an approach solves the ill-conceived problems that emerge from Humean-Kantian default commitments.

Some of the most important developments of symbolic logic took place in the 1920s. Foremost among them are the distinction between syntax and semantics and the formulation of questions of completeness and decidability of logical systems. David Hilbert and his students played a very important part in these developments. Their contributions can be traced to unpublished lecture notes and other manuscripts by Hilbert and Bernays dating to the period 1917-1923. The aim of this paper is to describe these results, focussing (...) primarily on propositional logic, and to put them in their historical context. It is argued that truth-value semantics, syntactic ("Post-") and semantic completeness, decidability, and other results were first obtained by Hilbert and Bernays in 1918, and that Bernays's role in their discovery and the subsequent development of mathematical logic is much greater than has so far been acknowledged. (shrink)

This article considers the significance of the Blessed Isles in Nietzsche’s Thus Spoke Zarathustra. They are the isolated locale to which Zarathustra and his fellow creators retreat in the Second Part of the book. I trace Zarathustra’s Blessed Isles back to the ancient Greek paradisiacal afterlife of the makarōn nēsoi and frame them against Nietzsche’s Platonic conception of philosophers as “commanders and legislators,” but I argue that they represent something more like a modern Epicurean Garden. Ultimately, I suggest that Zarathustra’s (...) Epicurean impulse toward withdrawal (whether into a sequestered friendship community or mountain solitude) undermines his Platonic attempts at great politics. (shrink)

Angell's logic of analytic containment AC has been shown to be characterized by a 9-valued matrix NC by Ferguson, and by a 16-valued matrix by Fine. We show that the former is the image of a surjective homomorphism from the latter, i.e., an epimorphic image. The epimorphism was found with the help of MUltlog, which also provides a tableau calculus for NC extended by quantifiers that generalize conjunction and disjunction.

On some accounts of vagueness, predicates like “is a heap” are tolerant. That is, their correct application tolerates sufficiently small changes in the objects to which they are applied. Of course, such views face the sorites paradox, and various solutions have been proposed. One proposed solution involves banning repeated appeals to tolerance, while affirming tolerance in any individual case. In effect, this solution rejects the reasoning of the sorites argument. This paper discusses a thorny problem afflicting this approach to vagueness. (...) In particular, it is shown that, on the foregoing view, whether an object is a heap will sometimes depend on factors extrinsic to that object, such as whether its components came from other heaps. More generally, the paper raises the issue of how to count heaps in a tolerance-friendly framework. (shrink)

The proof theory of many-valued systems has not been investigated to an extent comparable to the work done on axiomatizatbility of many-valued logics. Proof theory requires appropriate formalisms, such as sequent calculus, natural deduction, and tableaux for classical (and intuitionistic) logic. One particular method for systematically obtaining calculi for all finite-valued logics was invented independently by several researchers, with slight variations in design and presentation. The main aim of this report is to develop the proof theory of finite-valued first order (...) logics in a general way, and to present some of the more important results in this area. In Systems covered are the resolution calculus, sequent calculus, tableaux, and natural deduction. This report is actually a template, from which all results can be specialized to particular logics. (shrink)

Hilbert’s program was an ambitious and wide-ranging project in the philosophy and foundations of mathematics. In order to “dispose of the foundational questions in mathematics once and for all,” Hilbert proposed a two-pronged approach in 1921: first, classical mathematics should be formalized in axiomatic systems; second, using only restricted, “finitary” means, one should give proofs of the consistency of these axiomatic systems. Although Gödel’s incompleteness theorems show that the program as originally conceived cannot be carried out, it had many partial (...) successes, and generated important advances in logical theory and metatheory, both at the time and since. The article discusses the historical background and development of Hilbert’s program, its philosophical underpinnings and consequences, and its subsequent development and influences since the 1930s. (shrink)

This paper initiates a dialogue between classical Islamic philosophy and late modern European thought, by focusing on two peripheral, ‘heretical’ figures within these traditions: Abū Bakr Muḥammad ibn Zakariyāʾ al-Rāzī and Friedrich Nietzsche. What affiliates these thinkers across the cultural and historical chasm that separates them is their mutual fascination with, and profound indebtedness to, ancient Greek and Hellenistic philosophy. Given the specific themes, concerns and doctrines that they appropriate from this common source, I argue that al-Rāzī and Nietzsche should (...) be read as Epicureans and seen in relation to their predecessor as part of the subterranean tradition of philosophical naturalism. However, while each figure appropriates the philosophy of Epicurus, they ultimately transform it in important ways: al-Rāzī offers us a qualified “Platonic” Epicureanism, while Nietzsche forges a radicalized “Dionysian” Epicureanism. The point of such a comparative examination is twofold. First, analyzing the divergent paths of these “wayward” Epicureans sheds new light on the historical trajectory of naturalism as a philosophical stance (as well as its current prospects and limitations). Second, by initiating a dialogue between the classical Islamic and modern European philosophical traditions, we can better understand the antagonisms and shared concerns between these two horizons, as well as the ambiguities and self-questioning that occur within them. (shrink)

The period from 1900 to 1935 was particularly fruitful and important for the development of logic and logical metatheory. This survey is organized along eight "itineraries" concentrating on historically and conceptually linked strands in this development. Itinerary I deals with the evolution of conceptions of axiomatics. Itinerary II centers on the logical work of Bertrand Russell. Itinerary III presents the development of set theory from Zermelo onward. Itinerary IV discusses the contributions of the algebra of logic tradition, in particular, Löwenheim (...) and Skolem. Itinerary V surveys the work in logic connected to the Hilbert school, and itinerary V deals specifically with consistency proofs and metamathematics, including the incompleteness theorems. Itinerary VII traces the development of intuitionistic and many-valued logics. Itinerary VIII surveys the development of semantical notions from the early work on axiomatics up to Tarski's work on truth. (shrink)

Roger White (2015) sketches an ingenious new solution to the problem of induction. He argues from the principle of indifference for the conclusion that the world is more likely to be induction- friendly than induction-unfriendly. But there is reason to be skeptical about the proposed indifference-based vindication of induction. It can be shown that, in the crucial test cases White concentrates on, the assumption of indifference renders induction no more accurate than random guessing. After discussing this result, the paper explains (...) why the indifference-based argument seemed so compelling, despite ultimately being unsound. (shrink)

This paper examines Nietzsche’s conflicted relation to Epicurus, an important naturalistic predecessor in the ‘art of living’ tradition. I focus in particular on the Epicurean credo “live unnoticed” (lathe biōsas), which advocated an inconspicuous life of quiet philosophical reflection, self-cultivation and friendship, avoiding the public radar and eschewing the larger ambitions and perturbations of political life. Perhaps unsurprisingly, the idea looms largest and is most warmly received in Nietzsche’s middle period writings, where one finds a repeated concern with prudence, withdrawal (...) and concealment, and where the primary emphasis is on private pluralistic experiments in therapeutic self-cultivation among small groups of free spirits. The idea of the Epicurean Garden appeals greatly to Nietzsche at this time as well, and I suggest that Zarathustra’s Blessed Isles are best understood as a friendship community along these lines. However, Nietzsche’s growing impatience with human imperfections and the siren song of great politics eventually lead him away from Epicurus and back to Plato. Beginning with Zarathustra, the paradigm of the modest, hidden helpful philosopher-therapist is replaced by the more ambitious philosopher-legislator who takes upon himself the task of determining the future of humanity. Nonetheless, I argue that we can profit more from the modest, practical insights of Nietzsche’s Epicurean art of living. (full version). (shrink)

The last twenty-five years or so have seen the emergence of exciting comparative work on Nietzsche and various philosophical traditions beyond the bounds of Europe. So far, however, the emphasis has been primarily on the cultures of India, China and Japan, with an almost exclusive focus on Buddhist, Hindu, Daoist, and Confucian traditions. Surprisingly, little work has been done on Nietzsche and the Islamic tradition. In this paper, I sketch out Nietzsche’s understanding of Islam, the ways in which he uses (...) it as a resource for his critique of Christianity and European modernity and the criticisms he has to make of it as one of the great monotheistic world religions. I then argue for the need to engage Nietzsche with specific Islamic falāsifa of the classical period (9-12th c.) rather than Islam itself, as some recent scholars have attempted to do. Although Nietzsche himself seems to have had no direct familiarity with any of the falāsifa, there are, I argue, many entry points for productive comparison and dialogue. This is at least in part because they share a significant common heritage: both were careful students of classical Greek and Hellenistic thought, and both put the insights they encountered there to work in bold new ways. Indeed, they appropriated, transformed and reanimated Greek ideas in new contexts and towards new ends that their progenitors would scarcely have recognized, and that were often radically challenging to their contemporaries. I focus here on a few select themes—the idea of philosophy as a “way of life,” the ideal of “becoming like God so far as it is possible” and the Platonic figure of the philosopher-ruler, all of which get taken up and re-imagined in radically novel and sometimes antipodal ways. (shrink)

In this paper I focus on the figure of the ape in Nietzsche's texts, and how it fits into his putative naturalism. I examine the lowly status and ignoble qualities that he associates with this animal and argue that they betray a residual anthropocentricism profoundly at odds with Nietzsche's dehumanized conception of nature. Accordingly, I suggest a reading of Nietzsche's ape remarks that brings them more into accord with his non-teleological and non-hierarchical conception of species. Ultimately, I argue that a (...) robust, thorough going Nietzschean naturalism would require the affirmation of the ape, including everything it comes to represent for Nietzsche. (shrink)

A vast historical, cultural and philosophical chasm separates the thought of the 10th century Islamic philosopher al-Farabi and Friedrich Nietzsche, the progenitor of postmodernity. However, despite their significant differences, they share one important commitment: an attempt to resuscitate and reappropriate the project of Platonic political philosophy, particularly through their conceptions of the “true philosopher” as prophet, leader, and lawgiver. This paper examines al-Farabi and Nietzsche’s respective conceptions of the philosopher as commander and legislator against the background of their Platonic source, (...) and reflects upon the human cost each thinker is willing to abide in order to establish a regime that would be conducive to the perfectibility and flourishing of the human being. (shrink)

This paper examines the significance of Epicureanism for Nietzsche’s critique of Christian monotheism and his subsequent attempt to reanimate a kind of this-worldly, affirmative religiosity of immanence. After a brief overview of the pivotal role that Epicurus’ thought plays in the death of God, I focus on Epicurus’ own residual conception of the gods and the ways in which Nietzsche strategically retrieves it and puts it use in his writing. Nietzsche juxtaposes the distant, serene, indifferent Epicurean gods with the omniscient, (...) intrusive, jealous and needy God of middle-Eastern monotheisms. One might say that they constitute a ‘halfway house’ of sorts between Christianity and Nietzsche’s new Dionysian religion of the earth. But they also figure prominently in his own conflicted desire to intervene in the aleatory course of natural history and legislate the the future of humanity: Nietzsche finds he cannot simply look down with “the mocking and aloof eyes” of an Epicurean god upon the degeneration and diminution of the human being under Christian ascetic regimes of cultivation. Ultimately, Nietzsche fails to achieve the divine temperament of the Epicurean gods (looking down from afar on human ignorance, desire and suffering with a noble pathos of distance). Forthcoming 2022. (shrink)

This paper examines Nietzsche’s conflicted relation to Epicurus, an important naturalistic predecessor in the ‘art of living’ tradition. I focus in particular on the Epicurean credo “live unnoticed” (lathe biōsas), which advocated an inconspicuous life of quiet philosophical reflection, self-cultivation and friendship, avoiding the public radar and eschewing the larger ambitions and perturbations of political life. Perhaps unsurprisingly, the idea looms largest and is most warmly received in Nietzsche’s middle period writings, where one finds a repeated concern with prudence, withdrawal (...) and concealment, and where the primary emphasis is on private pluralistic experiments in therapeutic self-cultivation among small groups of free spirits. The idea of the Epicurean Garden appeals greatly to Nietzsche at this time as well, and I suggest that Zarathustra’s Blessed Isles are best understood as a friendship community along these lines. However, Nietzsche’s growing impatience with human imperfections and the siren song of great politics eventually lead him away from Epicurus and back to Plato. Beginning with Zarathustra, the paradigm of the modest, hidden helpful philosopher-therapist is replaced by the more ambitious philosopher-legislator who takes upon himself the task of determining the future of humanity. Nonetheless, I argue that we can profit more from the modest, practical insights of Nietzsche’s Epicurean art of living. (short version). (shrink)

I examine Zarathustra's increasing ambivalence about his role as philosopher-prophet-legislator, connecting the speech "On Passing By" (Z III.7) with his doctrine of amor fati (GS 276) as a pivotal moment in his gradual ascent up the ladder of love/affirmation and consequent overcoming of great politics. Forthcoming 2022.

Methods available for the axiomatization of arbitrary finite-valued logics can be applied to obtain sound and complete intelim rules for all truth-functional connectives of classical logic including the Sheffer stroke and Peirce’s arrow. The restriction to a single conclusion in standard systems of natural deduction requires the introduction of additional rules to make the resulting systems complete; these rules are nevertheless still simple and correspond straightforwardly to the classical absurdity rule. Omitting these rules results in systems for intuitionistic versions of (...) the connectives in question. (shrink)

In this essay I examine the tension between Nietzsche's doctrine of amor fati and his political project of Zuchtung. As philosophical naturalist, Nietzsche espouses a love of fate and a respect for necessity and reality. However, as philosophical legislator, he apparently denies the fatality of the human being in his attempts to cultivate or perfect it. I argue that Nietzsche's Zuchtung differs importantly from "idealistic" varieties of legislation in that it both requires and aims at the affirmation of fate. On (...) my reading, Nietzsche's nomothetic project neither contradicts nor undermines his naturalistic project: rather it represents its apotheosis and culmination. (shrink)

In the 1920s, David Hilbert proposed a research program with the aim of providing mathematics with a secure foundation. This was to be accomplished by first formalizing logic and mathematics in their entirety, and then showing---using only so-called finitistic principles---that these formalizations are free of contradictions. ;In the area of logic, the Hilbert school accomplished major advances both in introducing new systems of logic, and in developing central metalogical notions, such as completeness and decidability. The analysis of unpublished material presented (...) in Chapter 2 shows that a completeness proof for propositional logic was found by Hilbert and his assistant Paul Bernays already in 1917--18, and that Bernays's contribution was much greater than is commonly acknowledged. Aside from logic, the main technical contribution of Hilbert's Program are the development of formal mathematical theories and proof-theoretical investigations thereof, in particular, consistency proofs. In this respect Wilhelm Ackermann's 1924 dissertation is a milestone both in the development of the Program and in proof theory in general. Ackermann gives a consistency proof for a second-order version of primitive recursive arithmetic which, surprisingly, explicitly uses a finitistic version of transfinite induction up to www . He also gave a faulty consistency proof for a system of second-order arithmetic based on Hilbert's &egr;-substitution method. Detailed analyses of both proofs in Chapter 3 shed light on the development of finitism and proof theory in the 1920s as practiced in Hilbert's school. ;In a series of papers, Charles Parsons has attempted to map out a notion of mathematical intuition which he also brings to bear on Hilbert's finitism. According to him, mathematical intuition fails to be able to underwrite the kind of intuitive knowledge Hilbert thought was attainable by the finitist. It is argued in Chapter 4 that the extent of finitistic knowledge which intuition can provide is broader than Parsons supposes. According to another influential analysis of finitism due to W. W. Tait, finitist reasoning coincides with primitive recursive reasoning. The acceptance of non-primitive recursive methods in Ackermann's dissertation presented in Chapter 3, together with additional textual evidence presented in Chapter 4, shows that this identification is untenable as far as Hilbert's conception of finitism is concerned. Tait's conception, however, differs from Hilbert's in important respects, yet it is also open to criticisms leading to the conclusion that finitism encompasses more than just primitive recursive reasoning. (shrink)

Priest has provided a simple tableau calculus for Chellas's conditional logic Ck. We provide rules which, when added to Priest's system, result in tableau calculi for Chellas's CK and Lewis's VC. Completeness of these tableaux, however, relies on the cut rule.

Hilbert's ε-calculus is based on an extension of the language of predicate logic by a term-forming operator εx. Two fundamental results about the ε-calculus, the first and second epsilon theorem, play a rôle similar to that which the cut-elimination theorem plays in sequent calculus. In particular, Herbrand's Theorem is a consequence of the epsilon theorems. The paper investigates the epsilon theorems and the complexity of the elimination procedure underlying their proof, as well as the length of Herbrand disjunctions of existential (...) theorems obtained by this elimination procedure. (shrink)

Textbook on Gödel’s incompleteness theorems and computability theory, based on the Open Logic Project. Covers recursive function theory, arithmetization of syntax, the first and second incompleteness theorem, models of arithmetic, second-order logic, and the lambda calculus.

This chapter describes Kurt Gödel's paper on the incompleteness theorems. Gödel's incompleteness results are two of the most fundamental and important contributions to logic and the foundations of mathematics. It had been assumed that first-order number theory is complete in the sense that any sentence in the language of number theory would be either provable from the axioms or refutable. Gödel's first incompleteness theorem showed that this assumption was false: it states that there are sentences of number theory that are (...) neither provable nor refutable. The first theorem is general in the sense that it applies to any axiomatic theory, which is ω-consistent, has an effective proof procedure, and is strong enough to represent basic arithmetic. Their importance lies in their generality: although proved specifically for extensions of system, the method Gödel used is applicable in a wide variety of circumstances. Gödel's results had a profound influence on the further development of the foundations of mathematics. It pointed the way to a reconceptualization of the view of axiomatic foundations. (shrink)

Although Kurt Gödel does not figure prominently in the history of computabilty theory, he exerted a significant influence on some of the founders of the field, both through his published work and through personal interaction. In particular, Gödel’s 1931 paper on incompleteness and the methods developed therein were important for the early development of recursive function theory and the lambda calculus at the hands of Church, Kleene, and Rosser. Church and his students studied Gödel 1931, and Gödel taught a seminar (...) at Princeton in 1934. Seen in the historical context, Gödel was an important catalyst for the emergence of computability theory in the mid 1930s. (shrink)

In Winter 2017, the first author piloted a course in formal logic in which we aimed to (a) improve student engagement and mastery of the content, and (b) reduce maths anxiety and its negative effects on student outcomes, by adopting student oriented teaching including peer instruction and classroom flipping techniques. The course implemented a partially flipped approach, and incorporated group-work and peer learning elements, while retaining some of the traditional lecture format. By doing this, a wide variety of student learning (...) preferences could be provided for. (shrink)

A textbook for modal and other intensional logics based on the Open Logic Project. It covers normal modal logics, relational semantics, axiomatic and tableaux proof systems, intuitionistic logic, and counterfactual conditionals.

An introductory textbook on metalogic. It covers naive set theory, first-order logic, sequent calculus and natural deduction, the completeness, compactness, and Löwenheim-Skolem theorems, Turing machines, and the undecidability of the halting problem and of first-order logic. The audience is undergraduate students with some background in formal logic.

A construction principle for natural deduction systems for arbitrary, finitely-many-valued first order logics is exhibited. These systems are systematically obtained from sequent calculi, which in turn can be automatically extracted from the truth tables of the logics under consideration. Soundness and cut-free completeness of these sequent calculi translate into soundness, completeness, and normal-form theorems for natural deduction systems.

Takeuti and Titani have introduced and investigated a logic they called intuitionistic fuzzy logic. This logic is characterized as the first-order Gödel logic based on the truth value set [0,1]. The logic is known to be axiomatizable, but no deduction system amenable to proof-theoretic, and hence, computational treatment, has been known. Such a system is presented here, based on previous work on hypersequent calculi for propositional Gödel logics by Avron. It is shown that the system is sound and complete, and (...) allows cut-elimination. A question by Takano regarding the eliminability of the Takeuti-Titani density rule is answered affirmatively. (shrink)