A proof by Allan Gibbard (Ifs: Conditionals, beliefs, decision, chance, time. Reidel, Dordrecht, 1981) seems to demonstrate that if indicative conditionals have truth conditions, they cannot be stronger than material implication. Angelika Kratzer's theory that conditionals do not denote two-place operators purports to escape this result [see Kratzer (Chic Linguist Soc 22(2):1–15, 1986, 2012)]. In this note, I raise some trouble for Kratzer’s proposed method of escape and then show that her semantics avoids this consequence of Gibbard’s (...) class='Hi'>proof by denying modus ponens. I also show that the same holds for Anthony Gillies’ semantics (Philos Rev 118(3):325–349, 2009) and argue that this consequence of these theories is not obviously prohibitive—hence, both remain viable theories of indicative conditionals. (shrink)

Philosophers have long suspected that thought and discourse about what we ought to do differ in some fundamental way from statements about what is. But the difference has proved elusive, in part because the two kinds of statement look alike. Focusing on judgments that express decisions--judgments about what is to be done, all things considered--Allan Gibbard offers a compelling argument for reconsidering, and reconfiguring, the distinctions between normative and descriptive discourse--between questions of "ought" and "is." Gibbard considers how our actions, (...) and our realities, emerge from the thousands of questions and decisions we form for ourselves. The result is a book that investigates the very nature of the questions we ask ourselves when we ask how we should live, and that clarifies the concept of "ought" by understanding the patterns of normative concepts involved in beliefs and decisions. An original and elegant work of metaethics, this book brings a new clarity and rigor to the discussion of these tangled issues, and will significantly alter the long-standing debate over "objectivity" and "factuality" in ethics. Table of Contents: I. Preliminaries 1. Introduction: A Possibility Proof 2. Intuitionism as Template: Emending Moore II. The Thing to Do 3. Planning and Ruling Out: The "Frege-Geach" Problem 4. Judgment, Disagreement, Negation 5. Supervenience and Constitution 6. Character and Import III. Normative Concepts 7. Ordinary Oughts: Meaning and Motivation 8. Normative Kinds: Patterns of Engagement 9. What to Say about the Thing to Do: The Expressivistic Turn and What it Gains Us IV. Knowing What to Do 10. Explaining with Plans 11. Knowing What to Do 12. Ideal Response Concepts 13. Deep Vindication and Practical Confidence 14. Impasse and Dissent References Index This is a remarkable book. It takes up a central and much-discussed problem - the difference between normative thought and "descriptive" thought. It develops a compelling response to that problem with ramifications for much else in philosophy. But perhaps most importantly, it brings new clarity and rigor to the discussion of these tangled issues. It will take some time to come to terms with the details of Gibbard's discussion. It is absolutely clear, however, that the book will reconfigure the debate over objectivity and "factuality" in ethics. --Gideon Rosen, Professor of Philosophy, Princeton University Gibbard,/author> writes elegantly, and the theory he develops is innovative, philosophically sophisticated, and challenging. Gibbard defends his theory vigorously and with admirable intellectual honesty. --David Copp, Professor of Philosophy, Bowling Green State University. (shrink)

herent and rational way. Several proposals have been made for information merging in which it is possible to encode the preferences of sources (Benferhat, Dubois, Prade, & Williams, 1999; Benferhat, Dubois, Kaci, & Prade, 2000; Lafage & Lang, 2000; Meyer, 2000, 2001; Andreka, Ryan, & Schobbens, 2001). Information merging has much in common with social choice theory, which aims to define operations reflecting the preferences of a society from the individual preferences of the members of the society. Given this connection, (...) frameworks for information merging should provide satisfactory resolutions of problems raised in social choice theory. We investigate the link between the merging of epistemic states and two important results in social choice theory. We show that Arrow’s well-known impossibility theorem (Arrow, 1963) does not hold in merging frameworks when the preferences of sources are represented in terms of epistemic states. This is achieved by providing a consistent set of properties for merging from which Arrow-like properties can be derived. Similarly, by extending these to a consistent framework which includes properties corresponding to the notion of being strategy-proof, we show that results due to Gibbard and Satterthwaite (Gibbard, 1973; Satterthwaite, 1973, 1975) and other (Benoit, 2000; Barber´a, Dutta, & Sen, 2000) do not hold in merging frameworks. (shrink)

In 1931 the mathematical logician Kurt Godel published a revolutionary paper that challenged certain basic assumptions underpinning mathematics and logic. A colleague of Albert Einstein, his theorem proved that mathematics was partly based on propositions not provable within the mathematical system and had radical implications that have echoed throughout many fields. A gripping combination of science and accessibility, Godel’s Proof by Nagel and Newman is for both mathematicians and the idly curious, offering those with a taste for logic and (...) philosophy the chance to satisfy their intellectual curiosity. (shrink)

1. Fairness is about the just distribution of society's resources, and in ML, the main resource being distributed is model performance, e.g. the translation quality produced by machine translation...

According to the Principle of Conditional Non-Contradiction (CNC), conditionals of the form “If p, q” and “If p, not q” cannot both be true, unless p is inconsistent. This principle is widely regarded as an adequacy constraint on any semantics that attributes truth conditions to conditionals. Gibbard has presented an example of a pair of conditionals that, in the context he describes, appear to violate CNC. He concluded from this that conditionals lack truth conditions. We argue that this conclusion is (...) rash by proposing a new diagnosis of what is going on in Gibbard’s argument. We also provide empirical evidence in support of our proposal. (shrink)

Gibbard''s theory of rationality is evolutionary in terms of its result as well as its underpinning argument. The result is that judgments about what is rational are analyzed as being similar to judgments of morality — in view of what Darwin suggests concerning the latter. According to the Darwinian theory, moral judgments are based on sentiments which evolve to promote the survival and welfare of human societies. On Gibbard''s theory, rationality judgments should be similarly regarded as expressing emotional attachments to (...) behavioral norms which originate and function to coordinate social interaction. Consequently, Gibbard''s theory of rationality might be used to illuminate Darwin''s theory of morality, and vice versa. Additionally, as argued in the present essay, both can be further elaborated, and defended, by developing related themes in philosophical ethics: viz., connected with Hume and 20th-century emotivists. The main problem is that this general Darwinian approach faces widespread opposition nowadays, not only in ethics but in philosophy of science. The purpose of this essay is to analyze Gibbard''s theory, critically and constructively, with emphasis on the pertinent commonalities in Darwin, Hume and the emotivists, while also critically addressing their common enemies. The pervasive methodological orientation is to relate this analysis to (philosophy of) science in general, and biological science in particular. (shrink)

_'Nagel and Newman accomplish the wondrous task of clarifying the argumentative outline of Kurt Godel's celebrated logic bomb.'_ _– The Guardian_ In 1931 the mathematical logician Kurt Godel published a revolutionary paper that challenged certain basic assumptions underpinning mathematics and logic. A colleague of physicist Albert Einstein, his theorem proved that mathematics was partly based on propositions not provable within the mathematical system. The importance of Godel's Proof rests upon its radical implications and has echoed throughout many fields, from (...) maths to science to philosophy, computer design, artificial intelligence, even religion and psychology. While others such as Douglas Hofstadter and Roger Penrose have published bestsellers based on Godel’s theorem, this is the first book to present a readable explanation to both scholars and non-specialists alike. A gripping combination of science and accessibility, _Godel’s Proof_ by Nagel and Newman is for both mathematicians and the idly curious, offering those with a taste for logic and philosophy the chance to satisfy their intellectual curiosity. _Kurt Godel_ Born in Brunn, he was a colleague of physicist Albert Einstein and professor at the Institute for Advanced Study in Princeton, N.J. (shrink)

U radu se nudi opis konteksta unutar kojeg je formuliran poznati dokaz za nužnost identiteta. Iznosi se formalni prikaz ovog dokaza kako ga je formulirao poznati filozof i logičar Saul Kripke. Također se razmatra gledište filozofa Allana Gibbarda koji nasuprot Kripkeu brani tvrdnju da neki iskazi identiteta mogu biti kontingentni. Osnovni cilj rada je upoznati domaćeg čitatelja s formalnim aspektom rasprave o nužnosti identiteta te dati kratki pregled konteksta unutar kojeg su formulirani argumenti za nužnost identiteta. In the paper, we (...) offer an overview of the context within which the well-known proof of the necessity of identity is formulated. A formal account of this proof is presented as formulated by the renowned philosopher and logician Saul Kripke. Furthermore, we discuss the account of the philosopher Allan Gibbard, who defends the claim that some statements of identity can be contingent. The main goal of the paper is to acquaint the readers who understand the Croatian language with the formal aspect of the discussion on the necessity of identity and to give a brief overview of the context within which the arguments for the necessity of identity are formulated. (shrink)

This chapter contains sections titled: Generic Argument for Traditional Utilitarianism Mill's Proof of Utilitarianism (Straightforward Interpretation) Mill's Proof of Utilitarianism (One Alternative Interpretation) Mill's Proof of Utilitarianism (Another Alternative Interpretation).

This paper examines key aspects of Allan Gibbard's psychological account of moral activity. Inspired by evolutionary theory, Gibbard paints a naturalistic picture of morality mainly based on two specific types of emotion: guilt and anger. His sentimentalist and expressivist analysis is also based on a particular conception of rationality. I begin by introducing Gibbard's theory before testing some key assumptions underlying his system against recent empirical data and theories. The results cast doubt on some crucial aspects of Gibbard's philosophical theory, (...) namely his reduction of morality to anger and guilt, and his theory of “normative governance.” Gibbard's particular version of expressivism may be undermined by these doubts. (shrink)

According to the dispositional theory of meaning and content, what a speaker means by an expression is determined by her dispositions to use it. The literature contains two well-known objections against this theory: the problem of finitude and the problem of error. In his bookMeaning and Normativity, Allan Gibbard propounds a novel defence against these objections. In this paper, I argue that Gibbard’s suggestions fail to save the dispositional theory. Moreover, I argue that Gibbard’s deflationary view about facts (...) prevents him from saying anything about the property of meaning that would hold any interest for a naturalist.Selon la théorie dispositionnelle du sens et du contenu, ce qu’un locuteur veut dire par une expression est déterminé par ses dispositions à l’utiliser. La littérature contient deux objections bien connues contre cette théorie: le problème de la finitude et le problème de l’erreur. Dans son livreMeaning and Normativity, Allan Gibbard propose une nouvelle défense de la théorie dispositionnelle du sens et du contenu. Dans cet article, je soutiens que les suggestions de Gibbard ne parviennent pas à sauver la théorie dispositionnelle. De plus, je soutiens que le point de vue déflationniste de Gibbard sur les faits l’empêche de dire quoi que ce soit à propos de la propriété du sens qui aurait un intérêt quelconque pour un naturaliste. (shrink)

In these three Tanner lectures, distinguished ethical theorist Allan Gibbard explores the nature of normative thought and the bases of ethics. In the first lecture he explores the role of intuitions in moral thinking and offers a way of thinking about the intuitive method of moral inquiry that both places this activity within the natural world and makes sense of it as an indispensable part of our lives as planners. In the second and third lectures he takes up the kind (...) of substantive ethical inquiry he has described in the first lecture, asking how we might live together on terms that none of us could reasonably reject. Since working at cross purposes loses fruits that might stem from cooperation, he argues, any consistent ethos that meets this test would be, in a crucial way, utilitarian. It would reconcile our individual aims to establish, in Kant's phrase, a "kingdom of ends." The volume also contains an introduction by Barry Stroud, the volume editor, critiques by Michael Bratman (Stanford University), John Broome (Oxford University), and F. M. Kamm (Harvard University), and Gibbard's responses. (shrink)

In 1931 the mathematical logician Kurt Godel published a revolutionary paper that challenged certain basic assumptions underpinning mathematics and logic. A colleague of Albert Einstein, his theorem proved that mathematics was partly based on propositions not provable within the mathematical system and had radical implications that have echoed throughout many fields. A gripping combination of science and accessibility, _Godel’s Proof_ by Nagel and Newman is for both mathematicians and the idly curious, offering those with a taste for logic and philosophy (...) the chance to satisfy their intellectual curiosity. (shrink)

In Prior Analytics 1.15, Aristotle attempts to give a proof through the impossible of Barbara, Celarent, Darii, and Ferio with an assertoric first premiss, a contingent second premiss, and a possible conclusion. These proofs have been controversial since antiquity. I shall show that they are valid, and that Aristotle is able to explain them by relying on two meta-syllogistic lemmas on the nature of possibility interpreted as syntactic consistency. It will turn out that Aristotle's proofs are not of the (...) intended schemata. I shall investigate some of the results that the impact of this reconstruction on the modal syllogistic has: the relationship between Aristotle's syllogistic and the logics of relevance; the value of Aristotle's requirement that universal affirmative propositions be taken ‘absolutely’; the destruction of many Aristotelian proofs; the recovery of certain principles of modal opposition from a charge of inconsistency. (shrink)

_'Nagel and Newman accomplish the wondrous task of clarifying the argumentative outline of Kurt Godel's celebrated logic bomb.'_ _– The Guardian_ In 1931 the mathematical logician Kurt Godel published a revolutionary paper that challenged certain basic assumptions underpinning mathematics and logic. A colleague of physicist Albert Einstein, his theorem proved that mathematics was partly based on propositions not provable within the mathematical system. The importance of Godel's Proof rests upon its radical implications and has echoed throughout many fields, from (...) maths to science to philosophy, computer design, artificial intelligence, even religion and psychology. While others such as Douglas Hofstadter and Roger Penrose have published bestsellers based on Godel’s theorem, this is the first book to present a readable explanation to both scholars and non-specialists alike. A gripping combination of science and accessibility, _Godel’s Proof_ by Nagel and Newman is for both mathematicians and the idly curious, offering those with a taste for logic and philosophy the chance to satisfy their intellectual curiosity. _Kurt Godel_ Born in Brunn, he was a colleague of physicist Albert Einstein and professor at the Institute for Advanced Study in Princeton, N.J. (shrink)

Arrow’s impossibility result stems chiefly from a combination of two requirements: independence and fixity. Independence says that the social choice is independent of individual preferences involving unavailable alternatives. Fixity says that the social choice is fixed by a social preference relation that is independent of what is available. Arrow found that requiring, further, that this relation be transitive yields impossibility. Here it is shown that allowing intransitive social indifference still permits only a vastly unsatisfactory system, a liberum veto oligarchy. Arrow’s (...) argument for independence, though, undermines any rationale for fixity. (shrink)

Is there anything morally special about free exchange? In asking this, I am asking not only about extreme, so-called “libertarian” views, on which free exchange is sacrosanct, but about more widespread, moderate views, on which there is at least something morally special about free exchange. On these more compromising views, other moral considerations may override the moral importance of free exchange, but even when rights of free exchange are restricted for good reason, something morally important is lost. For some, free (...) exchange may preserve liberty, in some morally significant sense, or realize some such moral value as “to each his own.” Alternatively, a system of free exchange may have a special moral status by virtue of the kinds of pragmatic arguments that economists give, arguments that free exchange produces good social results. Whether free exchange has any such virtues as these is the broad question I address in this paper. I offer what I have to say somewhat in the spirit of an overview. Philosophical scrutiny and economic analysis combine, it seems to me, to delineate fairly clearly what is, and what is not, morally special about free exchange. (shrink)

_'Nagel and Newman accomplish the wondrous task of clarifying the argumentative outline of Kurt Godel's celebrated logic bomb.'_ _– The Guardian_ In 1931 the mathematical logician Kurt Godel published a revolutionary paper that challenged certain basic assumptions underpinning mathematics and logic. A colleague of physicist Albert Einstein, his theorem proved that mathematics was partly based on propositions not provable within the mathematical system. The importance of Godel's Proof rests upon its radical implications and has echoed throughout many fields, from (...) maths to science to philosophy, computer design, artificial intelligence, even religion and psychology. While others such as Douglas Hofstadter and Roger Penrose have published bestsellers based on Godel’s theorem, this is the first book to present a readable explanation to both scholars and non-specialists alike. A gripping combination of science and accessibility, _Godel’s Proof_ by Nagel and Newman is for both mathematicians and the idly curious, offering those with a taste for logic and philosophy the chance to satisfy their intellectual curiosity. _Kurt Godel_ Born in Brunn, he was a colleague of physicist Albert Einstein and professor at the Institute for Advanced Study in Princeton, N.J. (shrink)

Cantor’s proof that the powerset of the set of all natural numbers is uncountable yields a version of Richard’s paradox when restricted to the full definable universe, that is, to the universe containing all objects that can be defined not just in one formal language but by means of the full expressive power of natural language: this universe seems to be countable on one account and uncountable on another. We argue that the claim that definitional contexts impose restrictions on (...) the scope of quantifiers reveals a natural way out. (shrink)

What are ethical judgments about? And what is their relation to practice? How can ethical judgment aspire to objectivity? The past two decades have witnessed a resurgence of interest in metaethics, placing questions such as these about the nature and status of ethical judgment at the very center of contemporary moral philosophy. Moral Discourse and Practice: Some Philosophical Approaches is a unique anthology which collects important recent work, much of which is not easily available elsewhere, on core metaethical issues. Naturalist (...) moral realism, once devastated by the charge of "naturalistic fallacy," has been reinvigorated, as have versions of moral realism that insist on the discontinuity between ethics and science. Irrealist, expressivist programs have also developed with great subtlety, encouraging the thought that a noncognivist account may actually be able to explain ethical judgments' aspirations to objectivity. Neo-Kantian constructivist theories have flourished as well, offering hope that morality can be grounded in a plausible conception of reasonable conduct. Together, the positions advanced in the essays collected here address these recent developments, constituting a rich array of approaches to contemporary moral philosophy's most fundamental debates. An extensive introduction by Darwall, Gibbard, and Railton is also included, making this volume the most comprehensive and up-to-date work of its kind. Moral Discourse is ideally suited for use in courses in contemporary ethics, ethical theory, and metaethics. (shrink)

This paper argues that there is only one proof for God’s existence in Avicenna, and only one way for establishing the proof within his metaphysical system. This metaphysical proof is essentially derived from a priori notions, among which the notion of existence has the central role. Avicenna’s proof is structured in such a way that all its concepts are either derived from the meaning of ‘existence’ or are connected with this meaning. In this sense Avicenna’s (...) class='Hi'>proof sets out a scenario of discursive a priori knowledge established purely on considerations of fundamental ontological meanings such as ‘existence’, ‘existent’, ‘thing’, ‘necessary’, ‘possible’, ‘impossible’, ‘one’ and ‘cause’. (shrink)

I argue that if we allow that Moore’s Method, which involves taking an ordinary knowledge claim to support a substantive metaphysical conclusion, can be used to support Moore’s proof an external world, then we should accept that Moore’s Method can be used to support a variety of incompatible metaphysical conclusions. I shall refer to this as “the problem of many proofs”. The problem of many proofs, I claim, stems from the theory-ladenness of perception. I shall argue further that this (...) plethora of proofs for incompatible positions leads to a darker form of skepticism, one which maintains that each of the dogmatic views is probably false. We will conclude by considering various ways a Moorean might respond to these difficulties. (shrink)

This chapter contains sections titled: Hegel's Discussion of the Proofs On “Proof” and “Existence” The Proofs, Religious Elevation, and the Communion of Spirit The Multiplicity of Proofs and the One God The Cosmological Proof The Teleological Proof The Ontological Proof The Dialectic of the Proofs and the Speculative Reversal Hegel's Proofs Today.

I present a novel interpretation of Frege’s attempt at Grundgesetze I §§29-31 to prove that every expression of his language has a unique reference. I argue that Frege’s proof is based on a contextual account of reference, similar to but more sophisticated than that enshrined in his famous Context Principle. Although Frege’s proof is incorrect, I argue that the account of reference on which it is based is of potential philosophical value, and I analyze the class of cases (...) to which it may successfully be applied. (shrink)

In his book Philosophy of Logic, Putnam (1971) presents a short argument which reads like—and indeed, can be reconstructed as—a formal proof that a nominalistic physics is impossible. The aim of this paper is to examine Putnam’s proof and show that it is not compelling. The precise way in which the proof fails yields insight into the relation that a nominalistic physics should bear to standard physics and into Putnam’s indispensability argument.

I have argued that without an adequate solution to the knower paradox Fitch's Proof is- or at least ought to be-ineffective against verificationism. Of course, in order to follow my suggestion verificationists must maintain that there is currently no adequate solution to the knower paradox, and that the paradox continues to provide prima facie evidence of inconsistent knowledge. By my lights, any glimpse at the literature on paradoxes offers strong support for the first thesis, and any honest, non-dogmatic reflection (...) on the knower paradox provides strong support for the second. Whether verificationists want to go the route I've suggested is not for me todecide. As in the previous section my aim has been that of defending the mere viability of verificationism in the face of what many, many philosophers have taken to be its death-knell, namely Fitch's Proof. But, as the final objection makes clear, showing that verificationism can live in the face of Fitch's Proof is one thing; showing that it should live is another project. If nothing else, I hope that this papershows that verificationists still have a project to pursue; Fitch's Proof, contrary to popular opinion, need not bury verificationism.13. (shrink)

In the last few years there has been a resurgence of interest in Moore’s Proof of the existence of an external world, which is now often rendered as follows:1 (I) Here’s a hand (II) If there is a hand here, there is an external world Therefore (III) There is an external world The contemporary debate has been mostly triggered by Crispin Wright’s influential—conservative —“Facts and certainty” and further fostered by Jim Pryor’s recent—liberal—“What’s wrong with Moore’s argument?”.2 This debate is (...) worth surveying with care because—so I shall contend—it will help us see that, in fact, it allows for an important view that, so far, hasn’t been explicitly considered. The critical survey will be the task of the next two sections, while, in the remaining.. (shrink)

On my own view likewise, if we look for a straight analysis of the concept of a reason, we will find none that works. I mean the concept of a reason to do something, as when I say that the health effects and addictiveness of smoking are reasons not to take it up. I don't mean the psychological concept of an “operative reason”, as Scanlon calls it, a person's reason for doing what she does, as in “Her reason for taking (...) up smoking was that it would freak out her parents;” saying this doesn't commit us to thinking that that's any reason at all to take up smoking. Now for this normative concept of a reason to do something, I agree with Scanlon: we can't do better than he does by way of straight analysis; we can only say, somewhat uninformatively, that to be a reason to do something is to count in favor of doing it. On much the grounds he lays out, for instance, we can't translate talk of reasons into psychological claims about preferences or desires. (shrink)

This paper argues that morality is objective in a specific sense that accords with a broadly expressivist stance in metaethics. The paper also explains that although there is a kind of subjectivity in moral inquiry, the same holds for other kinds of normative inquiry, including epistemic and even scientific inquiry, and moreover that this kind of subjectivity is no threat to morality’s objectivity. The argument for the objectivity of morality draws strong parallels between ethics, epistemology, and science, but does not (...) depend on equally strong parallels between ethics and mathematics. I argue that there is more to learn from a comparison between ethics and mathematics than I used to think, but the difference between the issues that come up in thinking about objectivity in ethics and those that come up in thinking about objectivity in mathematics is substantial, and we cannot carry over results from the philosophy of mathematics to the case of morality. (shrink)

This paper draws attention to a proof of the necessity of identity given by Arthur Prior. In its simplicity, it is comparable to a proof of Quine's, popularised by Kripke, but it is slightly different. Prior's Polish notation is transcribed into a more familiar idiom. Prior's proof is followed by a proof of the necessity of difference, possibly the first such proof in the literature, which is also repeated here and transcribed. The paper concludes with (...) a brief discussion of Prior's views on identity and difference over time. (shrink)

I conclude that Gibbard fails to solve several of the traditional problems for expressivism. He solves some of these problems, but his solutions to them in effect give up expressivism. Of course, one might respond that it does not really matter whether his theory is expressivist. In some ways, I agree. Gibbard says many fascinating things about morality which have at most indirect connections to his expressivist analysis. I am thinking especially of his later discussions of hyperscepticism, parochialism, and indirect (...) pragmatism. These views could still be developed even if he gave up expressivism. All I have tried to show here is that he does need to give up expressivism unless he can solve the problems that I have raised. (shrink)

In the recent literature on Moore's Proof of an external world, it has emerged that different diagnoses of the argument's failure are prima facie defensible. As a result, there is a sense that the appropriateness of the different verdicts on it may depend on variation in the kinds of context in which the argument is taken to be a move, with different characteristic aims. In this spirit, Martin Davies has recently explored the use of the argument within two different (...) epistemic projects called respectively ‘deciding what to believe’ and ‘settling the question’. Depending on which project is in hand, according to Davies, the diagnoses of its failure—if indeed it fails—will differ. I believe that, by introducing the idea that the effectiveness of a valid argument may be epistemic project-relative, Davies has pointed the way to an important reorientation of the debates about Moore's Proof. But I wish to take issue with much of the detail of his proposals. I argue that Davies's characterization of his two projects is misleading (§1), and his account of their distinction defective (§2). I then canvass some suggestions about how it may be improved upon and about how further relevant kinds of epistemic projects in which Moore's argument may be taken to be a move can be characterized, bringing out how each of these projects impinges differently on the issue of the Proof's failure and of its diagnosis (§§3 and 4). In conclusion (§5) I offer an overview of the resulting terrain. (shrink)

Gentzen's systems of natural deduction and sequent calculus were byproducts in his program of proving the consistency of arithmetic and analysis. It is suggested that the central component in his results on logical calculi was the use of a tree form for derivations. It allows the composition of derivations and the permutation of the order of application of rules, with a full control over the structure of derivations as a result. Recently found documents shed new light on the discovery of (...) these calculi. In particular, Gentzen set up five different forms of natural calculi and gave a detailed proof of normalization for intuitionistic natural deduction. An early handwritten manuscript of his thesis shows that a direct translation from natural deduction to the axiomatic logic of Hilbert and Ackermann was, in addition to the influence of Paul Hertz, the second component in the discovery of sequent calculus. A system intermediate between the sequent calculus LI and axiomatic logic, denoted LIG in unpublished sources, is implicit in Gentzen's published thesis of 1934—35. The calculus has half rules, half “groundsequents,” and does not allow full cut elimination. Nevertheless, a translation from LI to LIG in the published thesis gives a subformula property for a complete class of derivations in LIG. After the thesis, Gentzen continued to work on variants of sequent calculi for ten more years, in the hope to find a consistency proof for arithmetic within an intuitionistic calculus. (shrink)

Restall :279–291, 2014) proposes a new, proof-theoretic, logical pluralism. This is in contrast to the model-theoretic pluralism he and Beall proposed in Beall and Restall :475–493, 2000) and in Beall and Restall. What I will show is that Restall has not described the conditions on being admissible to the proof-theoretic logical pluralism in such a way that relevance logic is one of the admissible logics. Though relevance logic is not hard to add formally, one critical component of Restall’s (...) pluralism is that the relevance logic that gets added must have connectives which mean the same thing as the connectives in the already admitted logic. This is what I will show is not possible. (shrink)

This chapter contains section titled: Alternatives to Mill's Methodology Conclusion.