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)

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.

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)

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)

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)

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)

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)

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)

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)

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)

After MacIntyre contains original essays by leading moral and political philosophers who assess both the merits and limitations of Alasdair MacIntyre's work. Among the themes explored here are MacIntyre's historical arguments about the sources of the failure of modernity; the validity and relevance of his attempt to reinstate the ideas of Aristotle and Aquinas as central to any satisfactory moral understanding; the effectiveness of his critique of modern liberalism; and the adequacy of key concepts, such as tradition and practice, in (...) his analyses and arguments. In addition, MacIntyre responds in a substantial essay to the issues raised by the other contributors. (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)

For several decades, there has been a debate in the philosophical literature concerning whether those of us who live in developed countries are morally required to give some of our money to aid agencies. Many contributors to this debate have apparently taken it that one may simply assume that the effects of the work such agencies do are overwhelmingly positive. If one turns to the literature on such agencies that has emerged in recent decades, however, one finds a number of (...) concerns about such agencies and the work they do that put that assumption in serious doubt. This situation raises a number of pressing questions, many of which have received little or no attention from philosophers. After articulating a number of those questions, I focus mainly on what I call the ‘Epistemic Question (for potential contributors to aid agencies)’: How can those of us who are not experts in international aid arrive at an estimate concerning the effects of the work aid agencies do that we have at least some good reason to believe accurate? (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)

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.

Translation (French to English) of Jean-Luc Nancy's "L'Annoncée.".

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)

The Promise of Friendship investigates what makes friendship possible and good for human beings. In dialogue with authors ranging from Aristotle and Montaigne to Proust, Levinas, and Derrida, Sarah Horton argues that friendship is suited to our finitude—that is, to the limits within which human beings live—and proposes a novel understanding of friendship as translation: friends translate the world for each other so that each one experiences the world not as the other does but in light of the friend's (...) always-unknowable experience. The very distance between friends that makes it impossible for them to know each other wholly also makes it possible for them to be transformed by friendship. Friendship, then, is possible and good for those who love precisely that they can never wholly know the friend. Friendship is a profound, mutual self-giving that highlights the irreplaceability of each person, fundamentally shapes the self, and is one of the greatest joys of human existence. (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)

Some moral principles require agents to do more than their fair share of a common task, if others won’t do their fair share – each agent’s fair share being what they would be required to do if all contributed as they should. This seems to provide a strong basis for objecting to such principles. For it seems unfair to require agents who have already done their fair share to do more, just because other agents won’t do their fair share. The (...) philosopher who has written most about this issue, however, Liam Murphy, argues that it is not unfair to do so, at least in the standard sense of that term. In this paper, I give Murphy’s reasons for saying this, explain why I think he’s wrong, and then say a little about why this issue might be important. (shrink)

The Authority Account provides a new explanation why commonsense morality contains prudential options—options that permit agents to perform actions that promote their own wellbeing more than the action they have most reason to do, from the moral point of view. At the core of that explanation are two claims. The first is that moral requirements are traditionally widely taken to have an authoritative status; that is, to be rules that morality imposes by right. The second is that in order for (...) moral requirements to have such a status, morality must contain prudential options. If both of these claims are true, then they will create a pressure to think of morality as containing prudential options. And according to the Authority Account, the fact that commonsense morality contains such options is the result of this pressure. (shrink)

One of the ways in which transnational medical agencies (TMAs) such as Medicins Sans Frontieres aim to increase the access of the global poor to health services is by supplying medical aid to people who need it in developing countries. The moral imperative supporting such work is clear enough, but a variety of factors can make such work difficult. One of those factors is the wrongdoing of other agents and agencies. For as a result of such wrongdoing, the attempt to (...) supply medical aid can sometimes lead to significant negative effects. What should TMAs do in such situations? On one view, TMAs should take account of any negative effects arising from the wrongdoing of others in just the same way in which they take account of negative effects arising more directly from their own actions, or from natural forces. To many people, this view seems wrong. In this paper, I articulate and discuss several different reasons why one might think this. In doing so, I hope to contribute to a debate about the more general question of how TMAs should respond to the wrongdoing of others. (shrink)

Global ethics is no ordinary subject. It includes some of the most urgent and momentous issues the world faces, such as extreme poverty and climate change. Given this, any adequate review of that subject should, I suggest, ask some questions about the relation between what those working in that subject do and the real-world phenomena that are the object of their study. The main question I focus on in this essay is this: should academics and others working in the field (...) of global ethics take new measures aimed at having more real-world positive impact on the phenomena they study? Should they take new measures, that is, aimed at bringing about more improvements in those phenomena, improvements such as reductions in extreme poverty and in emissions of greenhouse gases? I defend a positive answer to this question against some objections, and also discuss some of the kinds of measure we might take in an attempt to have more positive impact. (shrink)

One relatively straightforward way in which academics could have more impact on global poverty is by doing more to help people make wise decisions about issues relevant to such poverty. Academics could do this by conducting appropriate kinds of research on those issues and sharing what they have learned with the relevant decision makers in accessible ways. But aren’t academics already doing this? In the case of many of those issues, I think the appropriate answer would be that they could (...) do so much better. As an illustration, I examine the academic input into one decision about an issue concerning global poverty in some detail in this paper. I argue that that input has been seriously deficient, and suggest some ways in which it might be improved. Building on this discussion, I then formulate two questions that can be applied to any such decision, answers to which would indicate the quality of the input academics are currently providing. In cases where that input is deficient, and the decision in question an important one, I suggest that academics consider organising themselves in ways that will improve that input. I finish by briefly discussing how Academics Stand Against Poverty might help them do so. (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.

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.

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)

Over the last few decades, psychologists have amassed a great deal of evidence that our thinking is strongly influenced by a number of biases. This research appears to have important implications for moral methodology. It seems likely that these biases affect our thinking about moral issues, and a fuller awareness of them might help us to find ways to counteract their influence, and so to improve our moral thinking. And yet there is little or no reference to such biases in (...) the philosophical literature on many pressing, substantive moral questions. In this paper, I make a start on repairing this omission in relation to one such question, the 'Aid Question', which concerns how much, if anything, we are morally required to give to aid agencies. I begin by sketching a number of biases that seem particularly likely to affect our thinking about that question. I then go on to review the psychological research on 'debiasing' - that is, on attempts to counteract the influence of such biases. And finally I discuss and illustrate certain strategies for counteracting the influence of the biases in question on our thinking about the Aid Question. (shrink)

In light of Jacques Derrida’s writings on death and mourning, it may seem that the Christian teaching that the dead will be raised is a betrayal of others, a failure to take up one’s responsibility to testify to those who have died. In conversation with Emmanuel Falque’s work on finitude, Martin Heidegger’s reading of 1 Thessalonians, and Søren Kierkegaard’s reading of Abraham, I respond in two movements to this objection to faith that God will raise the dead. First, I propose (...) that even for the Christian, the death of the other remains a loss, since the Christian must surrender the other to God. It is, however, this very surrender of the other to God that seems to be an abdication of responsibility. Second, therefore, I argue that faith in the resurrection decenters the self and challenges our understanding of responsibility even more than does Derrida’s own analysis. Faith, I conclude, means giving up the desire to cling to one’s own responsibility. (shrink)

It has become increasingly common recently to construe human natureas setting some pretty stringent limits to moral endeavour. Many consequentialists, in particular, take considerations concerning human nature to defeat certain demanding norms that would otherwise follow from their theory. One argument is that certain commitments ground psychological incapacitiesthat prevent us from doing what would maximize the good. Another is that we would be likely to suffer some kind of psychological demoralization if we tried to become significantly more selfless. I argue (...) that influential versions of both of these arguments underestimate our deliberative resources, and also fail to examine the kind of moral sources that may be able to sustain rigorous moral endeavour. Pessimism about our capacities for such endeavour results from the neglect of these factors, rather than from uncovering any significant limitations in human nature. (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)

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)

In this paper, I argue that if we understand Levinas’s Desire of the Other as gift, we can understand it as joyful—that is, as celebratory. After presenting Levinas’s conception of Desire, I consider his claim, found in Otherwise than Being, that the self is a hostage to the Other, and I contend that, paradoxical as it may seem, being a hostage to the Other is actually liberating. Then, drawing on insights Richard Kearney offers in Reimagining the Sacred, I argue for (...) understanding Desire as a gift that is the condition of possibility for joy. If I offer hospitality to the Other, I thereby accept the gift that makes joy possible, and this joy is not egoistic but is the proper response to the gift. Finally, I ask whether Desire can be joyful in practice, given that the pure gift is an originary condition and never a historical one, and I conclude that imperfect joy remains possible. Moreover, this imperfect joy is better than any solitary enjoyment I might experience in the total absence of the Other. (shrink)

This article examines Kamel Daoud’s treatment of solidarity and the absurd in Meursault, contre-enquête and posits that the question of how to live in solidarity with others is central to the novel, although the word ‘solidarity’ never appears in it. After recalling Camus’s discussion of the absurd in Le Mythe de Sisyphe and of solidarity in L’Homme révolté, the article examines the manner in which Haroun, Daoud’s narrator and the brother of the Arab Meursault killed in L’Étranger, reveals his own (...) failures of solidarity. He justly criticizes Meursault for privileging his confrontation with the absurd over the death of the Arab he did not even name, but Haroun too has killed. Haroun has, however, a greater understanding of solidarity than his fellows: he at least recognizes that murder is significant. He thus joins Meursault as an unworthy prophet who proclaims the absurd while surrounded by people who flee from it—and proclaiming the absurd can be a gesture of solidarity when one speaks for others, as Haroun speaks for his brother Moussa. Daoud’s novel reminds us that there are no final answers telling us how to live in solidarity with others and that we must do so all the same. (shrink)

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)

Drawing on Sartre's account of violence, I argue that not only is bad faith inevitable in practice, but a limited bad faith is necessary for authenticity. Although violating the freedom of others is bad faith, it is impossible to never violate anyone's freedom. Moreover, and more fundamentally, the ontological structure of the foritself entails that the for-itself can only be authentic in the mode of not being authentic. Seeking to altogether avoid bad faith is bad faith, for it is an (...) attempt to constitute oneself as essentially authentic, yet the for-itself has no preexisting essence. By recognizing one's complete responsibility for choosing bad faith, however, one limits one's bad faith. This limited bad faith is in fact necessary to authenticity, which is a project lived out in concrete situations and not a categorical moral law that forbids bad faith. (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)

Any intermediate propositional logic can be extended to a calculus with epsilon- and tau-operators and critical formulas. For classical logic, this results in Hilbert’s $\varepsilon $ -calculus. The first and second $\varepsilon $ -theorems for classical logic establish conservativity of the $\varepsilon $ -calculus over its classical base logic. It is well known that the second $\varepsilon $ -theorem fails for the intuitionistic $\varepsilon $ -calculus, as prenexation is impossible. The paper investigates the effect of adding critical $\varepsilon $ - (...) and $\tau $ -formulas and using the translation of quantifiers into $\varepsilon $ - and $\tau $ -terms to intermediate logics. It is shown that conservativity over the propositional base logic also holds for such intermediate ${\varepsilon \tau }$ -calculi. The “extended” first $\varepsilon $ -theorem holds if the base logic is finite-valued Gödel–Dummett logic, and fails otherwise, but holds for certain provable formulas in infinite-valued Gödel logic. The second $\varepsilon $ -theorem also holds for finite-valued first-order Gödel logics. The methods used to prove the extended first $\varepsilon $ -theorem for infinite-valued Gödel logic suggest applications to theories of arithmetic. (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)

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.

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)

It is shown how the schema of equivalence can be used to obtain short proofs of tautologies A , where the depth of proofs is linear in the number of variables in A .

Proper attention to the theme of corporeality is crucial for understanding Derrida’s analysis of Hegel in “The Pit and the Pyramid.” This article argues that Derrida’s essay compels us to face the impossibility of giving a wholly coherent account of embodiment. The _Aufhebung_ supposedly unites the exteriority of the corporeal with interiority in a higher unity that cancels and preserves them both; Hegel’s own text reveals, however, that meaning is primordially absent from the body that was thought to incarnate it. (...) And it is this absence of ideal meaning that is originary: Differance conditions the body as it conditions speech, rendering the body other than itself such that it is not categorizable as flesh that is the self or as an object that is not the self. I am and am not my body because the dichotomy between interiority and exteriority breaks down even at the level of the body. Indeed, I am and am not my self; the embodied self is disrupted from the start, never self-contained. Thus embodiment always already testifies to the other. (shrink)

Translation (French to English) of "Philosophie et théologie : nouvelles frontières" by Emmanuel Falque.

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)

While the traditional understanding of the look views it in terms of shame and oppression, I read Sartre’s Notebooks for an Ethics with Beauvoir’s Ethics of Ambiguity to argue that the look always gives me the world and inaugurates my freedom. Even the oppressor’s look reveals that I am free and that my existence is conditioned by the existence of other free beings. Because the look gives me the world as the arena within which I act freely, it is a (...) means of grace, and receiving it only in shame is bad faith. Although my existence remains unjustifiable and this grace cannot promise salvation, the look calls me out of shame to the pursuit of my and others’ freedom, and this call is a gift. (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)

My contributions to this book are the translations (French to English) of the Preface to the American Edition, "Opening: Confrontation with Étienne Gilson," and "Afterword: Saint Thomas and the Entrance of God into Philosophy.".