Charles Chihara's new book develops and defends a structural view of the nature of mathematics, and uses it to explain a number of striking features of mathematics that have puzzled philosophers for centuries. The view is used to show that, in order to understand how mathematical systems are applied in science and everyday life, it is not necessary to assume that its theorems either presuppose mathematical objects or are even true. Chihara builds upon his previous work, in which (...) he presented a new system of mathematics, the constructibility theory, which did not make reference to, or presuppose, mathematical objects. Now he develops the project further by analysing mathematical systems currently used by scientists to show how such systems are compatible with this nominalistic outlook. He advances several new ways of undermining the heavily discussed indispensability argument for the existence of mathematical objects made famous by Willard Quine and Hilary Putnam. And Chihara presents a rationale for the nominalistic outlook that is quite different from those generally put forward, which he maintains have led to serious misunderstandings.A Structural Account of Mathematics will be required reading for anyone working in this field. (shrink)

This paper addresses John Burgess's answer to the ‘Benacerraf Problem’: How could we come justifiably to believe anything implying that there are numbers, given that it does not make sense to ascribe location or causal powers to numbers? Burgess responds that we should look at how mathematicians come to accept: There are prime numbers greater than 1010 That, according to Burgess, is how one can come justifiably to believe something implying that there are numbers. This paper investigates what lies behind (...) Burgess's answer and ends up as a rebuttal to Burgess's reasoning. (shrink)

Charles Chihara gives a thorough critical exposition of modal realism, the philosophical doctrine that there exist many possible worlds of which the actual world--the universe in which we live--is just one. The striking success of possible-worlds semantics in modal logic has made this ontological doctrine attractive. Modal realists maintain that philosophers must accept the existence of possible worlds if they wish to have the benefit of using possible-worlds semantics to assess modal arguments and explain modal principles. Chihara challenges (...) this claim, and argues instead for modality without worlds; he offers a new account of the role of interpretations or structures of the formal languages of logic. (shrink)

Charles Chihara gives a thorough critical exposition of modal realism, the philosophical doctrine that there exist many possible worlds of which the actual world--the universe in which we live--is just one. The striking success of possible-worlds semantics in modal logic has made thisontological doctrine attractive. Modal realists maintain that philosophers must accept the existence of possible worlds if they wish to have the benefit of using possible-worlds semantics to assess modal arguments and explain modal principles. Chihara challenges this (...) claim, and argues instead formodality without worlds; he offers a new account of the role of interpretations or structures of the formal languages of logic. (shrink)

This paper discusses and rejects some objections raised by Chihara to the book Scientific Reasoning: the Bayesian Approach, by Howson and Urbach. Some of Chihara's objections are of independent interest because they reflect widespread misconceptions. One in particular, that the Bayesian theory presupposes logical omniscience, is widely regarded as being fatal to the entire Bayesian enterprise, It is argued here that this is no more true than the parallel charge that the theory of deductive logic is fatally comprised (...) because it presupposes logical omniscience. Neither theory presupposes logical omniscience. (shrink)

This paper discusses and rejects some objections raised by Chihara to the book Scientific Reasoning: the Bayesian Approach, by Howson and Urbach. Some of Chihara's objections are of independent interest because they reflect widespread misconceptions. One in particular, that the Bayesian theory presupposes logical omniscience, is widely regarded as being fatal to the entire Bayesian enterprise, It is argued here that this is no more true than the parallel charge that the theory of deductive logic is fatally comprised (...) because it presupposes logical omniscience. Neither theory presupposes logical omniscience. (shrink)

The Worlds of Possibility (1998)에서 찰스 치하라는 S5 양상 양화 논리에 대한 현실주의적 모형 이론 의미론을 제시한다. 치하라의 의미론은 단지 가능하기만한 대상이 존재한다는 가정을 수반하지 않기에 현실주의적이다. 본 논문은 치하라의 의미론의 핵심 개념인 참과 만족 개념에 대해 자세한 분석, 해석, 재정식화를 제시하여, 치하라의 의미론에 대해 제기될 수 있는 여러 문제들을 해결하고자 한다. 치하라의 의미론의 ‘참’ 술어가 단항이 아닌 이항 술어로 이해되어야 하고, 만족 개념은 의미 장치가 있는 준거점과 문장의 진리값이 평가되는 평가점에 상대화된 삼항 관계로 이해되어야 한다고 논증한다.

The indispensability argument, which claims that science requires beliefs in mathematical entities, gives a strong motivation for mathematical realism. However, mathematical realism bears Benacerrafian ontological and epistemological problems. Although recent accounts of mathematical realism have attempted to cope with these problems, it seems that, at least, a satisfactory account of epistemology of mathematics has not been presented. For instance, Maddy's realism with perceivable sets and Resnik's and Shapiro's structuralism have their own epistemological problems. This fact has been a reason to (...) rebut the indispensability argument and adopt mathematical nominalism. Since mathematical nominalism purports to be committed only to concretia, it seems that mathematical nominalism is epistemically friendlier than mathematical realism. However, when it comes to modal mathematical nominalism, this claim is not trivial. There is a reason for doubting the modal primitives that it invokes. In this thesis, this doubt is investigated through Chihara's Constructibility Theory. Chihara's Constructibility Theory purports not to be committed to abstracta by replacing existential assertions of the standard mathematics with ones of constructibility. However, the epistemological status of the primitives in Chihara's system can be doubted. Chihara might try to argue that the problem would dissolve by using possible world semantics as a didactic device to capture the primitive notions. Nonetheless, his analysis of possible world semantic is not plausible, when considered as a part of the project of nominalizing mathematics in terms of the Constructibility Theory. (shrink)

Chihara seeks to avoid commitment to mathematical objects by replacing traditional assertions of the existence of mathematical objects with assertions about possibilities of constructing certain open-sentence tokens. I argue that Chihara's project can be defended against several important objections, but that it is no less epistemologically problematic than its platonistic competitors.

Charles Chihara has presented a problem he claims Bayesian confirmation theory cannot handle. Chihara gives examples in which he claims the change in belief cannot be construced as conditionalizing on new evidence. These are situations in which the agent suddenly thinks of new possibilities. I propose a solution that incorporates the important ideas of Bayesian theory. In particular, I present a principle which shows that the change of belief in Chihara's example is due to simple conditionalization.

Chihara introduces this book as a response to critics of his last book, which gave an account of mathematical objects in terms of possible constructions of open sentences. Several reviewers charged him with exchanging an ontology of platonistic mathematical objects for an equally extravagant ontology of possible entities. In this book Chihara replies with an extended account how one can use modal logic, and even the notions of possible worlds semantics, without accepting merely possible worlds or objects. A (...) final chapter extends his anti-platonist arguments about philosophy of mathematics, directed primarily against Penelope Maddy and the “Indispensibility” argument associated with Quine and Putnam. (shrink)

I first discuss Chihara's claim that the presence of Liar-paradoxical sentences presents no problem for our understanding of natural languages, and argue that this cannot be held as easily as he suggests. I then consider the theories advanced by Martin, van Fraassen, Kripke and Burge which attempt to meet some of the problems involved. I argue that the claim in the first two theories that Liar sentences are ill-formed cannot be maintained, and that Burge's theory is methodologically unsound and (...) seriously incomplete. Kripke's theory, though it gives implausible truth-values to some sentences, is more satisfactory. Finally, I discuss Gupta's revision theory, and provide my own alternative which has the following advantages. First, it facilitates our understanding of Kripke's and Gupta's theories, and of their relationship. Second, it provides a third theory in which we can understand how the presence of the Liar does not radically infect a language. (shrink)

Professor Charles S. Chihara has criticized the views on the subject of inconsistency which Wittgenstein put forward in his recently published 1939 lectures. Chihara notes that these views are not peculiar to the 1939 lectures, and in fact they are to be found in all Wittgenstein's later writings on mathematics . So these ideas about inconsistency appear not to be just a momentary aberration on Wittgenstein's part. One would therefore expect that he had some good reasons for holding (...) them. But Chihara justly complains that the kind of strong argumentation one would hope for is not forthcoming in these lectures. Instead Wittgenstein's usual procedure is to try to defend his views by producing series of rather unconvincing examples. (shrink)

While taking Charles Chihara's metaphysics course as a graduate student at U.C. Berkeley, I wrote an advice columnist to ask about the puzzle at the center of the course. Marilyn Vos Savant writes a weekly column for Parade Magazine , which is included in the Sunday editions of many newspapers. She claims to be listed in the Guinness Book of World Records for "highest IQ".

This paper offers an answer to Glymour's ‘old evidence’ problem for Bayesian confirmation theory, and assesses some of the objections, in particular those recently aired by Chihara, that have been brought against that answer. The paper argues that these objections are easily dissolved, and goes on to show how the answer it proposes yields an intuitively satisfactory analysis of a problem recently discussed by Maher. Garber's, Niiniluoto's and others’ quite different answer to Glymour's problem is considered and rejected, and (...) the paper concludes with some brief reflections on the prediction/accommodation issue. (shrink)

While many are impressed with the utility of possible worlds in linguistics and philosophy, few can accept the modal realism of David Lewis, who regards possible worlds as sui generis entities of a kind with the concrete world we inhabit.1 Not all uses of possible worlds require exotic ontology. Consider, for instance, the use of Kripke models to establish formal results in modal logic. These models contain sets often regarded for heuristic reasons as sets of “possible worlds”. But the “worlds” (...) in these sets can be anything at all; they can be numbers, or people, or sh. The set of worlds, together with the accessibility relation and the rest of the model, is used as a purely formal structure.2 One can even go beyond establishing results about formal systems and apply Kripke models to English, as Charles Chihara has recently argued.3 Chihara shows, for instance, how to use Kripke models (plus primitive modal notions) to give an account of validity for English modal sentences. In other cases worlds are not really needed at all. It is often vivid to give a counterexample thus: “There is a possible world in which P. Since your theory implies that in all worlds, not-P, your theory is wrong.” But the counterexample could just as easily be given using modal operators: “Possibly, P. Since your theory implies that it is necessary that not-P, your theory is wrong.”. (shrink)

Il dibattito sul ruolo e le implicazioni del teorema di Gödel per l'intelligenza artificiale ha recentemente ricevuto nuovo impeto grazie a due importanti volumi pubblicati da Roger Penrose, The Emperor's New Mind [1989] e Shadows of the Mind [1994]. Naturalmente, Penrose non è il primo né l'ultimo a usare il teorema di Gödel allo scopo di trarne conseguenze per i fondamenti dell'intelligenza artificiale. Tuttavia il recente dibattito suscitato dai due libri di Penrose è significativo sia per ampiezza sia per profondità. (...) In queste pagine si vuole dare una rassegna di tale dibattito, cominciando dai suoi precursori negli anni '60 (fra cui Lucas, Putnam, e Chihara), per passare poi alle complesse argomentazioni proposte da Penrose e le reazioni di una serie di commentatori (ad esempio Dennett, Feferman, McDermott, Davis). (shrink)

Haack, S. Is truth flat or bumpy?--Chihara, C. S. Ramsey 's theory of types.--Loar, B. Ramsey 's theory of belief and truth.--Skorupski, J. Ramsey on Belief.--Hookway, C. Inference, partial belief, and psychological laws.--Skyrms, B. Higher order degrees of belief.--Mellor, D. H. Consciousness and degrees of belief.--Blackburn, S. Opinions and chances.--Grandy, R. E. Ramsey, reliability, and knowledge.--Cohen, L. J. The problem of natural laws.--Giedymin, J. Hamilton's method in geometrical optics and Ramsey 's view of theories.

For quite some time the indispensability arguments of Quine and Putnam were considered a formidable obstacle to anyone who would reject the existence of mathematical objects. Various attempts to respond to the indispensability arguments were developed, most notably by Chihara and Field. Field tried to defend mathematical fictionalism, according to which the existential assertions of mathematics are false, by showing that the mathematics used in applications is in fact dispensable. Chihara suggested, on the other hand, that mathema­tics makes (...) true existential assertions, but that these can be interpreted so as to remove the commitment to abstract objects. More recently, there have been various attempts to show that the indispensability arguments contain assumptions that are conceptually misguided in ways having little to do with mathematical content. All of this work is of considerable interest, and the result has been a gathering consensus that the indispensability arguments, as put forth by Quine and Putnam, do not provide convincing reason to accept mathematical realism. The focus here will be on the ways of responding to the indispensability arguments, and in particular on the obstacles to fictionalism that remain after the versions of Quine and Putnam are undercut. (shrink)

In the recent years we can observe a sort of renaissance of the philosophy of mathematics. More and more papers and books are published. A few years ago a new journal (Philosophia Mathematica) devoted exclusively to the philosophy of mathematics started appearing. In the contemporary discussions - especially in the context of the question of the applicability of mathematics to the description of the physical world - the issue of the existence and the ontological status of mathematical objects plays a (...) particular role. Many new conceptions have appeared - both realistic and antirealistic. In the paper one of the antirealistic conceptions formulated in the recent years is presented - namely a conception by C. Chihara [1990]. The author rejects both Gödel's and Quine's arguments for the realistic standpoint. First, these standpoints will be briefly summarised, since they play a significant role in the contemporary philosophy of mathematics - and are particularly important in the context of the discussion upon Chihara's conception. Next, Chihara's arguments will be analysed in detail. As a result it will turn out that these arguments are not conclusive. Chihara's system relies on certain unjustified assumptions. Moreover, the philosophical difficulties that it encounters are not sufficiently discussed. (shrink)

The Philosophy of Mathematics Today gives a panorama of the best current work in this lively field, through twenty essays specially written for this collection by leading figures. The topics include indeterminacy, logical consequence, mathematical methodology, abstraction, and both Hilbert's and Frege's foundational programmes. The collection will be an important source for research in the philosophy of mathematics for years to come. Contributors Paul Benacerraf, George Boolos, John P. Burgess, Charles S. Chihara, Michael Detlefsen, Michael Dummett, Hartry Field, Kit (...) Fine, Bob Hale, Richard G. Heck, Jnr., Geoffrey Hellman, Penelope Maddy, Karl-Georg Niebergall, Charles D. Parsons, Michael D. Resnik, Matthias Schirn, Stewart Shapiro, Peter Simons, W.W. Tait, Crispin Wright. (shrink)

In this paper I examine arguments by Benacerraf and by Chihara against Gödel’s platonistic philosophy of mathematics.Neste artigo examino argumentos de Benacerraf e de Chihara contra a filosofia platônica da matemática de Gödel.

In this paper I examine arguments by Benacerraf and by Chihara against Gödel’s platonistic philosophy of mathematics.

In this paper I examine arguments by Benacerraf and by Chihara against Gödel’s platonistic philosophy of mathematics.

Richard Cartwright's impact on other philosophers has been as much a product of his own personal contact with students and colleagues as the result of his written work. The essays in this book demonstrate the deep influence he has had, not only by his thinking but equally by his style and manner and, above all, by his clarity and purity of intention. All of the essays are concerned with the questions of logic, language, and metaphysics that have been at the (...) core of Cartwright's own work. The chapters and their authors are: The Consistency of Frege's Foundations of Arithmetic, George Boolos. Russell's "No-Classes" Theory of Classes, Leonard Linsky. Quine's Indeterminacy, Charles S. Chihara. Justifying Symbolizations, Harold Levin. Substitutivity, Scott Soames. Moore-Paradox, Sincerity Conditions, and Epistemic Qualification, Charles E. Caton. Matching Illocutionary Act Types, William P. Alston. Scattered Objects, Roderick M. Chisholm. Parts and Places, Helen Morris Cartwright. Ruminations on an Account of Personal Identity, Judith Jarvis Thomson. G_del's Ontological Proof, Jordan Howard Sobel. "The Concept of the Subject Contains the Concept of the Predicate," David Wiggins. Judith Jarvis Thomson is Professor of Philosophy at MIT. Philosophical Essays, the first collection of Richard Cartwright's work, is being published simultaneously by The MIT Press. (shrink)

Review of John Burgess' and Gideon Rosen's A Subject with no Object.

Possible worlds present a formidable challenge for the lover of desert landscapes. One cannot ignore their usefulness; they provide, as David Lewis puts it, “a philosophers’ paradise”.1 But to enter paradise possibilia must be fit into a believable ontology. Some follow Lewis and accept worlds at face value, but most prefer some other choice from the current menu. Part of Chihara’s book is a critical discussion of some of these menu options: Lewis’s modal realism, Alvin Plantinga’s abstract modal realism, (...) Graeme Forbes’s anti-realism and Gideon Rosen’s modal fictionalism. These discussions are very detailed and conversant with the literature. The discussions of Forbes and of paradox within Plantinga’s system are particularly enlightening. The rest of the book is devoted to Chihara’s positive project: developing an account of the status of model theory for non-modal logic, and then applying it to the modal case. The prize is an understanding of possible worlds semantics that requires no commitment to possible worlds at all What does the relativized notion of truth in an interpretation studied in model theory have to do with plain old truth? Chihara’s answer involves “connecting theorems” that relate truth-in to truth. A “natural-language proto-interpretation of the sentential calculus” is a function that assigns meanings of declarative sentences of English to sentence letters. Where I is an NLPI of SC and φ is a sentence letter, Chihara uses ‘[φ/I ]’ to refer to “φ with the meaning it has been assigned by I ” ; where φ is not atomic, he says. (shrink)

On the face of it, statements like, whose truth we are readily prepared to allow, carry an "ontological commitment," in Quine's jargon, to abstract entities: Some shapes are uninstantiated. Can a nominalistic paraphrase of be provided? I take Charles Chihara to be urging a positive answer in his exciting book, with in particular meeting his precise prescription: It is possible to construct a shape predicate, in some language or other, that fails to be satisfied by anything. Not that we (...) are supposed to have in hand an effective procedure for producing any such predicate token. Rather, we are to envisage all the different shape predicate tokens that are to be found in different possible worlds. Some of them will have a null extension in those worlds. But that is not what asks of us. Think first of some shape predicate type of which there are no tokens in the actual world. Now consider a possible world in which there are tokens of that type. Finally, we are to think of those tokens as having a zero extension in our world. (shrink)