Les articles de ce recueil discutent les principes de la pensee mathematique qui ont fait le partage entre les trois fameuses options philosophiques: logicisme, formalisme et intuitionnisme. Leur auteur, Paul Bernays, fut un des plus proches collaborateurs de David Hilbert, qui a si profondement marque de son empreinte les mathematiques du XXe siecle et la philosophie construite a leur sujet. Defenseur de l'infini, de la methode axiomatique, des structures generales, des raisonnements abstraits, de la formalisation logique, Hilbert a investi (...) une grande energie pour ancrer ces audacieuses conquetes sur le terrain balise et controlable du fini, de l'intuitif, du concret, du constructible. Les essais de Bernays sont un precieux guide pour mieux s'orienter dans la pensee logique et philosophique de Hilbert et se debarasser une bonne fois des prejuges hatifs qui jalonnent les opinions ressassees sur le formalisme, le finitisme, le platonisme, etc. Mais ces essais nous decouvrent aussi un penseur libre, parlant de sa propre voix. Sensible a la pluralite des approches possibles, assumant le risque de se rectifier sans cesse, mesure dans sa visee d'une certitude pratique plutot que d'une verite absolue, Bernays defend l'idees que les mathematiques ne sont pas un reservoir de connaissance, mais une activite de l'esprit reagissant a des situations. (shrink)

I. Philosophie scientifique et Syntaxe logique. Nécessité d’une interprétation. — II. Logique et mathématique. La distinction kantienne : « analytique »-« synthétique » est remplacée par une distinction entre « formel » et « objectif ». Touchant ici la mathématique et la logique, on traite surtout du côté objectif : qui, en mathématique, consiste dans l’existence de rapports mathématiques, indépendants de la formulation en proposition, et dans la véri- ficabilité de lois arithmétiques ; en logique, dans le rapport implicite des (...) termes et des principes à certains caractères de la réalité. — III. Arithmétique et géométrique sont distingués selon la considération du discret et du continu. Précision formelle des concepts mathématiques intuitifs. — IV. Pour la problématique des fondements. Réflexions et indications sur la situation présente des recherches. (shrink)

Est‐il possible d'exposer une doctrine sans la comparer et l'opposer à d'autres doctrines, proches ou éloignées ? Et pour faire bien ressortir ce qui lui appartient en propre, ne faut‐il pas établir une utile dispute avec tels ou tels points de vue qui semblent au premier abord ne pas trop s'en écarter? La question que les trois brèves études critiques qui suivent cherchent à. mettre en lumière est des plus importante: c'est l'idée même de la cohérence de la vie intellectuelle (...) qu'elles abordent. Leur but est moins de critiquer autrui nue de s'exposer elles‐mêmes à la critique. (shrink)

ZusammenfassungAn Hand der Betrachtung einiger Hauptzüge in der neueren Entwicklung der Philosophie des logischen Empirismus und nahestehender Richtungen wird dargelegt, wie die Korrektur der zu simplifizierenden Thesen in dem ursprünglichen Programm der Wiener Schule auf eine Auseinandersetzung mit den traditionellen erkenntnistheoretischen Problemen zurückführt. — P.B.RésuméCe travail prend en considération certains développements récents de l'empirisme logique et des points de vues apparentés. Ces développements tendent sinon à un abandon, du moins à une revision de certaines positions par trop simplificatrices, tout d'abord (...) occupées par l'Ecole de Vienne. Or, ce faisant, on se trouve amené à rendre leur sens plein aux questions philosophiques qu'on pensait avoir pu écarter.Newer trends of development in the schools of logical empirism are considered. It is shown how, by the correction of the original too simplifying theses of the Vienna school, one becomes reduced to the study of the traditional problems of the philosophy of science. — P. B. (shrink)

Eighteen articles concerning modern logic by Ackermann, Carnap, Curry, Gödel, Heyting, among others, reprinted from the double issue of the review Dialectica for 1958.--L. S. F.

The proposition that a decreasing sequence of ordinals necessarily terminates has been given a new, and perhaps unexpected, importance by the rôle which it plays in Gentzen's proof of the freedom from contradiction of the “reine Zahlentheorie.” Gödel's construction of non-demonstrable propositions and the establishment of the impossibility of a proof of freedom from contradiction, within the framework of a certain type of formal system, showed that a proof of freedom from contradiction could be found only by transcending the axioms (...) and proof processes of that formal system. Gentzen's proof succeeds by utilisingtransfinite inductionto prove that certain sequences of reduction processes, enumerated by ordinals less than ε (the first ordinal to satisfy ε = ὡ) are finite. Were it possible to provethe restricted ordinal theorem, that a descending sequence of ordinals, less thanε, is finite, in Gentzen's “reine Zahlentheorie,” then it would be possible to determine a contradiction in that number system. In his paper, Gentzen proves the theorem of transfinite induction, which he requires, by an intuitive argument. There is also a method of reducing transfinite induction, for ordinals less than ε, to a number-theoretic principle given by Hilbert and Bernays, and a similar method by Ackermann. None of these proofs of transfinite induction isfinitist. (shrink)

Full proofs of the Gödel incompleteness theorems are highly intricate affairs. Much of the intricacy lies in the details of setting up and checking the properties of a coding system representing the syntax of an object language (typically, that of arithmetic) within that same language. These details are seldom illuminating and tend to obscure the core of the argument. For this reason a number of efforts have been made to present the essentials of the proofs of Gödel’s theorems without getting (...) mired in syntactic or computational details. One of the most important of these efforts was made by Löb [8] in connection with his analysis of sentences asserting their own provability. Löb formulated three conditions (now known as the Hilbert-Bernays-Löb derivability conditions), on the provability predicate in a formal system which are jointly sufficient to yield the Gödel’s second incompleteness theorem for it. A key role in Löb’s analysis is played by (a special case of) what later became known as the diagonalization or fixed point property of formal systems, a property which had already, in essence, been exploited by Gödel in his original proofs of the incompleteness theorems. The fixed point property plays a central role in Lawvere’s [7] category-theoretic account of incompleteness phenomena (see also [10]). Incompleteness theorems have also been subjected to intensive investigation within the framework of modal logic (see, e.g.[4], [5]). In this formulation the modal operator takes up the role previously played by the provability predicate, and the derivability conditions on the latter are translated into algebraic conditions (the so-called GL, i.e., Gödel–Löb, conditions) on the former. My purpose here is to present a framework for incompleteness phenomena, fully compatible with intuitionistic or constructive principles, in which the idea of a coding system is retained, only in a 2 simple, but very general form, a form wholly free of syntactical notions. As codes we shall take the elements of an arbitrary given nonempty set, possibly, but not necessarily, the set of natural numbers.. (shrink)

In his response to my article, “Historicism and Science: Thoughts on Quine” , Paul Bernays distinguishes between two aspects of the article: my criticism of Quine's theory of knowledge and my contention that the epistemological foundations of modern science lead to historicism. Bernays substantially accepts my criticism of Quine, particularly insofar as I reject Quine's behaviorism and “physicalism”. He opposes, however, my claim that historicism is implicit in the presuppositions of modern science. I argued that the historicism inherent (...) in modern science is due to the essentially changeable character of modern science. Quine's theory of knowledge tacitly takes account of this changeable character a) through his view of sense experience as being constituted by posits, b) through his recognition that individual scientific statements are dependent upon a broad linguistic‐conceptual framework that is itself subject to change, and c) through his pragmatic conception of reason and the concomitant emphasis upon the criterion of simplicity. (shrink)

Modern public relations practices have been dominated by appeals to impulses, desires, and images that affect publics defined predominantly in demographic terms. This paper argues that abandoning basic rhetorical assumptions about the ability of ordinary people to engage in practical reason has serious ethical implications for the marketplace as well as for society in general. The study applies recent rhetorical scholarship on issues of public discourse and rhetorical culture to public relations practices, considering how rhetoric can contribute to more effective (...) and ethical public discourse in our dominant modes of marketplace communication. (shrink)

L'histoire des fondements des mathématiques du xxe siècle montre qu'il nous faut réviser la signification des notions philosophiques traditionnelles comme « évidence », « existence », « expérience » ou « rationalité ». On expose comment le logicien Paul Bernays, familier des conceptions de la philosophie de Jacob Friedrich Fries et de Léonard Nelson, donne aux résultats techniques une interprétation philosophique dont il s'inspire de plus en plus - à partir du milieu du siècle - de la « philosophie (...) ouverte » de Ferdinand Gonseth. Bien avant Thomas Kuhn, Bernays conçoit la révision envisagée non pas simplement comme une question de vérité ou de fausseté, mais comme un problème exigeant l'introduction d'un nouveau système conceptuel. Cependant, l'affaiblissement du concept de vérité est pour lui une conséquence épistémologique provenant de la théorie de la connaissance adoptée et non une conséquence méthodologique issue d'un changement de paradigme et donc d'une incommensurabilité. (shrink)

Let NBG be von Neumann-Bernays-Gödel set theory without the axiom of choice and let NBGA be the modification which allows atoms. In this paper we consider some of the well-known class or global forms of the wellordering theorem, the axiom of choice, and maximal principles which are known to be equivalent in NBG and show they are not equivalent in NBGA.

The Hilbert–Bernays Theorem establishes that for any satisfiable first-order quantificational schema S, one can write out linguistic expressions that are guaranteed to yield a true sentence of elementary arithmetic when they are substituted for the predicate letters in S. The theorem implies that if L is a consistent, fully interpreted language rich enough to express elementary arithmetic, then a schema S is valid if and only if every sentence of L that can be obtained by substituting predicates of L (...) for predicate letters in S is true. The theorem therefore licenses us to define validity substitutionally in languages rich enough to express arithmetic. The heart of the theorem is an arithmetization of Gödel's completeness proof for first-order predicate logic. Hilbert and Bernays were the first to prove that there is such an arithmetization. Kleene established a strengthened version of it, and Kreisel, Mostowski, and Putnam refined Kleene's result. Despite the later refinements, Kleene's presentation of th.. (shrink)

The Löwenheim-Hilbert-Bernays theorem states that, for an arithmetical first-order language L, if S is a satisfiable schema, then substitution of open sentences of L for the predicate letters of S...

Rather than "transitional," the American interwar years constituted a contiguous and seminal era during which the social, religious, and aesthetic consequences of a changed environment, modernity, became powerful forces in shaping the patterns in recent popular culture. Increased literacy and affluence, media technologies, and changes in work and leisure encouraged a mass marketplace of ideas. Popular intellectuals, namely D. W. Griffith, Bruce Barton, John B. Watson, Edward Bernays, Harry Emerson Fosdick, Edward L. Bernays, George Creel, Pearl Buck, John (...) Steinbeck, and Gilbert Selders helped Americans reimagine entire areas of existence for the new context. They illuminate important cultural shifts regarding the nature of self in a rationalized, urban world, the legitimacy of religion in the shadow of skeptical empiricism, and the qualities and social role of art in the age of science, alienation, and political apathy. ;Many of our persistent forms of popular expression can be understood as cultural exchanges over the problem of modernity, evidence of the obtuse ways in which modernism has affected the general culture. Griffith's mawkish melodrama is, upon serious scrutiny, deeply informed by modern fears of personal anomie, naturalism, and the aloof power of the new state. New concepts of "personality" and the "masses," based on behavioristic assumptions, envisioned both self and society as malleable and helped mitigate feelings of alienation with promises of empowerment and self-transformation. With Fosdick and Barton, mainstream Protestantism absorbed "positive thinking" and therapeutic conceptions of religion's role in modern life, and in the process redefined the nature of God, self, and the Church, Buck and Steinbeck reshaped melodramatic formulae into response to modern tensions with a blend of brutal naturalism and sentimental romanticism, a vision of an embattled "human spirit." In these thinkers, we can see Americans confronting the century's most compelling issues, disorder, naturalism, scientism, and relativism. Each phenomenon was a way of admitting modern change while imagining techniques for manipulating it. The relentless pursuit in popular culture for social connectedness, mastery of environment, spiritual assurance, managerial order, and personal power can best be understood as multifarious compensatory strategies for our ambivalence with modernity. (shrink)

Dans un procès qui eut lieu à la fin du XVIIe siècle en Livonie (aujourd’hui Latvia) un homme surnommé « le vieux Thiess » avoua être un loup garou. Les anomalies qui marquent ses aveux ont été l’objet d’interprétations différentes, qui soulèvent le problème des ambitions, et des limites, de la comparaison. L’essai explore, à travers le cas du « vieux Thiess », la possibilité d’utiliser a) la notion d’anomalies conjonctives, inspirée par la critique textuelle, b) la notion de « (...) ressemblances de famille » proposée par Francis Galton, en tant qu’instruments de comparaison dans le domaine de l’histoire des religions. (shrink)

The class forcing theorem, which asserts that every class forcing notion ${\mathbb {P}}$ admits a forcing relation $\Vdash _{\mathbb {P}}$, that is, a relation satisfying the forcing relation recursion—it follows that statements true in the corresponding forcing extensions are forced and forced statements are true—is equivalent over Gödel–Bernays set theory $\text {GBC}$ to the principle of elementary transfinite recursion $\text {ETR}_{\text {Ord}}$ for class recursions of length $\text {Ord}$. It is also equivalent to the existence of truth predicates for (...) the infinitary languages $\mathcal {L}_{\text {Ord},\omega }$, allowing any class parameter A; to the existence of truth predicates for the language $\mathcal {L}_{\text {Ord},\text {Ord}}$ ; to the existence of $\text {Ord}$ -iterated truth predicates for first-order set theory $\mathcal {L}_{\omega,\omega }$ ; to the assertion that every separative class partial order ${\mathbb {P}}$ has a set-complete class Boolean completion; to a class-join separation principle; and to the principle of determinacy for clopen class games of rank at most $\text {Ord}+1$. Unlike set forcing, if every class forcing notion ${\mathbb {P}}$ has a forcing relation merely for atomic formulas, then every such ${\mathbb {P}}$ has a uniform forcing relation applicable simultaneously to all formulas. Our results situate the class forcing theorem in the rich hierarchy of theories between $\text {GBC}$ and Kelley–Morse set theory $\text {KM}$. (shrink)

Professor P. Bernays has kindly drawn my attention to the fact that the assignment of Godel numbers to formulas described on page 295 of volume I of this Journal, is not correct. Under this assignment, different formulas may obtain the same Godel number and therefore the argument given on page 296 under (i) is not cogent. The correct assignment would give, for the formula considered as an example, the Godel number 28. 320. 514. 782 The correction to be made (...) on page 294, L 20 is obvious. (shrink)

Professor P. Bernays has kindly drawn my attention to the tact that the assignment of Gödel numbers to formulas described on page 295 of volume I of this Journal, is not correct. Under this assignment, different formulas may obtain the same Gödel number and therefore the argument given on page 296 under (i) is not cogent. The correct assignment would give, for the formula considered as an example, the Gödel number 23. 329. 514. 732 The correction to be made (...) on page 294, l. 20 is obvious. (shrink)

We say that a first order formula Φ distinguishes a structure M over a vocabulary L from another structure M' over the same vocabulary if Φ is true on M but false on M'. A formula Φ defines an L-structure M if Φ distinguishes M from any other non-isomorphic L-structure M'. A formula Φ identifies an n-element L-structure M if Φ distinguishes M from any other non-isomorphic n-element L-structure M'. We prove that every n-element structure M is identifiable by a (...) formula with quantifier rank less than (1- 1/2k)n+k²-k+4 and at most one quantifier alternation, where k is the maximum relation arity of M. Moreover, if the automorphism group of M contains no transposition of two elements, the same result holds for definability rather than identification. The Bernays-Schönfinkel class consists of prenex formulas in which the existential quantifiers all precede the universal quantifiers. We prove that every n-element structure M is identifiable by a formula in the Bernays-Schönfinkel class with less than (1-1/2k²+2)n+k quantifiers. If in this class of identifying formulas we restrict the number of universal quantifiers to k, then less than n-√ n+k²+k quantifiers suffice to identify M and, as long as we keep the number of universal quantifiers bounded by a constant, at total n-O(√ n) quantifiers are necessary. (shrink)

RésuméL'auteur analyse la conception de la méthode dialectique chez Gaston Bachelard et Ferdinand Gonseth qui est à L'origine de la «philosophie ouverte». Lorsqu'il s'est agi de donner un nom à la revue qu'ils allaient fonder avec Paul Bernays, le choix s'est porté sur celui de Dialectica, en accord avec L'orientation qu'ils comptaient donner à leurs publications.SummaryThe author describes Gaston Bachelard's and Ferdinand Gonseth's dialectical method which is at the origin of their so‐called open philosophy. When faced with the task (...) of giving a name to a planned review, edited together with Paul Bernays, they settled their choice on Dialectica in order to announce the orientation of the future publications.ZusammenfassungEs werden Gaston Bachelards und Ferdinand Gonseths Auffassungen der dialektischen Methode dargelegt, die zur Entwicklung der «offenen Philosophie» geführt hat. Die Tatsache, dass sich die beiden Philosophen, die gemeinsam mit Paul Bernays Dialectica gegründet haben, über die Vorzüge dieser Methode einig waren, hat für die Wahl des Namens der Zeitschrift den Ausschlag gegeben. (shrink)

Most of the early proofs of the decidability or completeness of certain mathematical theories were based on the method of eliminations of quantifiers. Various more recent results on completeness were obtained independently of such procedures. However, it is shown in the present paper that, conversely, the completeness of a mathematical theory will in certain circumstances entail the existence of an elimination method. The proof involves the application of the extended first ε‐theorem of Hilbert‐Bernays.ZusammenfassungDie meisten früheren Beweise der Vollständigkeit oder (...) Entscheidbarkeit gewisser mathematischer Theorien benutzten die sogenannte Quantoren‐eliminierungsmethode. Dagegen sind einige neuere Ergebnisse über Voll‐ständigkeit unabhängig von diesem Verfahren. Es ist deshalb von Interesse festzustellen, dass umgekehrt die Vollständigkeit einer mathematischen Theorie unter Umständen die Existenz einer Eliminierungsmethode fürdiese Theorie nach sich zieht. Der Beweis benutzt das erweiterte erste ε‐Theorem von Hilbert‐Bernays.SommaireBeaucoup de démonstrations concernant la complétude de certaines théories mathématiques, ou concernant l'existence d'un procédé de décision pour une telle théorie, sont basées sur des méthodes d'élimination de quantificateurs. Or, il y a des résultats plus récents sur la complétude de certaines théories qui ne dépendent pas de telles méthodes d'élimination. Toutefois on démontre ici que la complétude d'une théorie mathématique entraîne sous certaines conditions l'existence d'une méthode d'élimination. Au cours de la démonstration on se sert d'un des théorèmes‐ε de Hilbert‐Bernays. (shrink)

Conceptions of autonomy in western philosophy and ethics have often centred on self-governance and self-determination. However, a growing bulk of literature also questions such conceptions, including the understanding of the autonomous self as a self-governing independent individual that chooses, acts, and lives in accordance with her or his own values, norms, or sense of self. This article contributes to the critical interrogation of selfhood, autonomy, and autonomous decision making by combining a feminist focus on relational dimensions of selfhood and autonomy (...) with phenomenological philosophy of the embodied self as being-in-the-world. It offers a philosophical investigation of different dimensions of bodily relational autonomy by turning to phenomenological accounts of the lived body as self-reflexive. When so doing, we hope to contribute to bridging the gap that sometimes exists between discussions of autonomy in analytic moral philosophy and of freedom and facticity in phenomenological philosophy. We see this gap as unfortunate, and hold that a nuanced understanding of autonomy and autonomous decision making can be reached if these strands of philosophy are brought into dialogue. (shrink)

RésuméSelon Descartes, la force démonstrative d'une argumentation peut résulter ou bien de l'application des règles universelles de la logique ou bien d'une intuition particulière. Cette doctrine permet d'éviter le reproche de circularité qui est souvent fait à l'égard de certaines argumentations épistémologiques. D'autre part, elle implique le rejet de la méthode du contre‐exemple. L'acception de certaines argumentations en vertu d'une intuition particulière ne crée pas une situation définitive; elle constitue une phase dans la genèse de ce que Bernays a (...) caractérisé comme une évidence acquise.ZusammenfassungNach Descartes kann die Beweiskraft einer Argumentation entweder von der Anwendung der allgemeingültigen Regeln der Logik oder von einer besonderen Anschauung herrühren. Diese Ansicht erlaubt uns, dem Einwand der Zirkelhaftigkeit zu entgehen, welcher öfters gegen gewisse erkenntnistheoretische Argumentationen erhoben wird. Sie erfordert anderseits den Verzicht auf die Methode der Gegenbeispiele. Die Hinnahme gewisser Argumentationen auf Grund einer besonderen Anschauung schafft niemals eine bleibende Situation; sie bildet eine Phase im Zustandekommen dessen, was Bernays als eine erworbene Evidenz charakterisiert hat.According to Descartes, the demonstrative power of an argumentation may result either from the application of the universal rules of logic or from a particular intuition. This doctrine permits us to avoid the reproach of circularity which is often raised as an objection to certain argumentations in epistemology. On the other hand, it implies the rejection of the method of the counter‐example. The acceptance of certain argumentations based on a particular intuition never creates a permanent situation; it rather constitutes a stage in the genesis of what Bernays has characterized as an évidence acquise. (shrink)

Let Q be an equivalence relation whose equivalence classes, denoted Q[x], may be proper classes. A function L defined on Field(Q) is a labelling for Q if and only if for all x,L(x) is a set and L is a labelling by subsets for Q if and only if BG denotes Bernays-Gödel class-set theory with neither the axiom of foundation, AF, nor the class axiom of choice, E. The following are relatively consistent with BG. (1) E is true but (...) there is an equivalence relation with no labelling.(2) E is true and every equivalence relation has a labelling, but there is an equivalence relation with no labelling by subsets. (shrink)

The main subject of this very original and engaging book is the history of the idea of formal proof. Von Plato shows how that idea emerged, from the Greeks to the crucial last few decades of the ni...