We expose the main ideas, concepts and results about Jaśkowski's discussive logic, and apply that logic to the concept of pragmatic truth and to the Dalla Chiara-di Francia view of the foundations of physics.

We present some recent technical results of us on the incompleteness of classical analysis and then discuss our work on the Arnol'd decision problems for the stability of fixed points of dynamical systems.

We propose a formalization of Meinong's theory of objects with the help of Hilbert's $\epsilon$-symbol and a paraconsistent logical system, with an eye towards its application in an axiomatization of the natural sciences.

Maps and territory suggest problems which have to do with the opening of pathways in some poorly explored domain.

We expose the main ideas, concepts and results about Jakowski's discussive logic, and apply that logic to the concept of pragmatic truth and to the Dalla Chiara-di Francia view of the foundations of physics.

We conjecture that the existence of some fast-growing functions implies in a simple way the existence of some inaccessible cardinals. This note expands some previous work by the second author.

We first show that a theorem by Cartan that generalizes the Frobenius integrability theorem allows us (given certain conditions) to obtain noncurvature solutions for the differential Bianchi conditions and for higher-degree similar relations. We then prove that there is no algorithmic procedure to determine, for a reasonable restricted algebra of functions on spacetime, whether a given connection form satisfies the preceding conditions. A parallel result gives a version of Gödel's first incompleteness theorem within an (axiomatized) theory of gauge fields.

We present some recent technical results of us on the incompleteness of classical analysis and then discuss our work on the Arnol'd decision problems for the stability of fixed points of dynamical systems.

We first state a few previously obtained results that lead to general undecidability and incompleteness theorems in axiomatized theories that range from the theory of finite sets to classical elementary analysis. Out of those results we prove several incompleteness theorems for axiomatic versions of the theory of noncooperative games with Nash equilibria; in particular, we show the existence of finite games whose equilibria cannot be proven to be computable.

We first state a few previously obtained results that lead to general undecidability and incompleteness theorems in axiomatized theories that range from the theory of finite sets to classical elementary analysis. Out of those results we prove several incompleteness theorems for axiomatic versions of the theory of noncooperative games with Nash equilibria; in particular, we show the existence of finite games whose equilibria cannot be proven to be computable.

‘My name is Ephraïm Naana, I have already told you on one of these many occasions when you have come here before closing time to speak about my school. In my village, since you ask me, there are not all the things which are to be found on the market here, there are only those which grow there: mangoes, potatoes, maize, bananas, papayas. And my primary school teacher told me that in other places there might be even fewer things; he (...) said that in the market in Khartoum, in Sudan, the nation is vast but on the stalls there are only onions and white and red beans; and even though they were nicely arranged in piles, they were still only white beans, red beans and onions; and then he said that the world was big and that the hunger we experienced was not the worst of famines. (shrink)

We show that the P 2 0 sentence, and explore some of theconsequences of that fact. This paper summarizes recent workby the author with N. C. A. da Costa on the P

Logic and Philosophy of Logic

Direct download (5 more)  

 

Export citation  

 

Bookmark  

In 1962 Tibor Radó uncovered a monster hiding in the midst of innocent-looking Turing machines.

Esta coletânea é uma homenagem aos 70 anos do professor Emmanuel Carneiro Leão. Reúne artigos de seus alunos, amigos e admiradores que foram atraídos pela abrangência e rigor de seu pensamento. A diversidade de temáticas, abordagens e premissas destes ensaios refletem bem a generosidade do espírito de Emmanuel Carneiro.

We give a overview of the main areas in theoretical physics, with emphasis on their relation to Lagrangian formalism in classical mechanics. This review covers classical mechanics; the road from classical mechanics to Schrödinger’s quantum mechanics; electromagnetism, special and general relativity, and gauge field theory and the Higgs mechanism. We shun mathematical rigor in favor of a straightforward presentation.

Out of a folklore–like fact about the counterexample function to P=NP, a function which grows about as fast as the Busy Beaver function, we review the consequences of our exotic formalization for P=NP and then speculate about possible ways to extend our work.

We discuss Kunen’s algorithmic implementation of a proof for the Paris–Harrington theorem, and the author’s and da Costa’s proposed “exotic” formulation for the P = NP hypothesis. Out of those two examples we ponder the relation between mathematics within an axiomatic framework, and intuitive or informal mathematics.

I submit that epistemic progress in key areas of contemporary academic philosophy has been compromised by politically correct ideology. First, guided by an evolutionary account of ideology, results from social and cognitive psychology and formal philosophical methods, I expose evidence for political bias in contemporary Western academia and sketch a formalization for the contents of beliefs from the PC worldview taken to be of core importance, the theory of social oppression and the thesis of anthropological mental egalitarianism. Then, aided by (...) discussions from contemporary epistemology on epistemic values, I model the problem of epistemic appraisal using the frameworks of multi-objective optimization theory and multi-criteria decision analysis and apply it to politically correct philosophy. I conclude that philosophy guided by politically correct values is bound to produce constructs that are less truth-conducive and that spurious values which are ideologically motivated should be abandoned. Objections to my framework stemming from contextual empiricism, the feminine voice in ethics and political philosophy are considered. I conclude by prescribing the epistemic value of empirical adequacy, the contextual value of political diversity and the moral virtue of moral courage to reverse unwarranted trends in academic philosophy due to PC ideology. (shrink)

Preference orderings assigned by coherent lower and upper conditional previsions are defined and they are considered to define maximal random variables and Bayes random variables. Sufficient conditions are given such that a random variable is maximal if and only if it is a Bayes random variable. In a metric space preference orderings represented by coherent lower and upper conditional previsions defined by Hausdorff inner and outer measures are given.

Partindo de dois modelos clássicos que fundamentaram a emergência do conceito do que entendemos hoje enquanto cidade, a polis e a civitas, este texto busca compreender e questionar a cidade contemporânea em sua condição metropolitana, marcada por políticas de imobilidade e sufocamento que demandam uma intensa produção de subjetividades. Trazendo pensadores dos campos da filosofia e da política como Giorgio Agamben, Massimo Cacciari e Vinícius NicastroHonesko, entre outros, nos propomos a pensar outras chaves de relações para o que denominamos cidade, (...) que sejam capazes de produzir modos de habitá-la para além da metrópole. Uma cidade outra, espaço de composição e transposição de mundos que não se encerre em si mesma; uma cidade ainda por vir.Palavras-chave: Cidade. Metrópole. Mundo. Política. Habitar. (shrink)

This volume presents essays by pioneering thinkers including Tyler Burge, Gregory Chaitin, Daniel Dennett, Barry Mazur, Nicholas Humphrey, John Searle and Ian Stewart. Together they illuminate the Map/Territory Distinction that underlies at the foundation of the scientific method, thought and the very reality itself. It is imperative to distinguish Map from the Territory while analyzing any subject but we often mistake map for the territory. Meaning for the Reference. Computational tool for what it computes. Representations are handy and tempting that (...) we often end up committing the category error of over-marrying the representation with what is represented, so much so that the distinction between the former and the latter is lost. This error that has its roots in the pedagogy often generates a plethora of paradoxes/confusions which hinder the proper understanding of the subject. What are wave functions? Fields? Forces? Numbers? Sets? Classes? Operators? Functions? Alphabets and Sentences? Are they a part of our map? Or do they actually belong to the territory? Researcher, like a cartographer, clothes the reality by stitching multitudes of maps that simultaneously co-exist. A simple apple, for example, can be analyzed from several viewpoints beginning with evolution and biology, all the way down its microscopic quantum mechanical components. Is there a reality out there apart from these maps? How do these various maps interact/intermingle with each other to produce a coherent reality that we interact with? Or do they not? Does our brain uses its own internal maps to facilitate “physicist/mathematician” in us to construct the maps about the external territories in turn? If so, what is the nature of these internal maps? Are there meta-maps? Evolution definitely fences our perception and thereby our ability to construct maps, revealing to us only those aspects beneficial for our survival. But the question is, to what extent? Is there a way out of the metaphorical Platonic cave erected around us by the nature? While “Map is not the territory” as Alfred Korzybski remarked, join us in this journey to know more, while we inquire on the nature and the reality of the maps which try to map the reality out there. The book also includes a foreword by Sir Roger Penrose and an afterword by Dagfinn Follesdal. (shrink)

The main objective of this work is to understand the daily practices of user management in post-reformist devices in the Brazilian mental health field. Through Foucault's genealogical work on government practices, understood as forms of conducting the behavior of others, it is possible to open a possible field for the study of the practices of psi knowledge, considering them as forms of management that act by through the free and natural acts of individuals. More specifically, our goal is to examine (...) day-to-day practices on some reformist devices such as CAPS through user files. Basically, two driving models have been found in the new devices: 1) Cases with frank intervention of production of commitments and agreements with users, seeking the entry of this to varied activities and political leadership ; 2) Cases with little response from the user, in which the forms of conduction are implemented by policies of inclusion of family members and close reference characters in the management of the patient's life. (shrink)

We analyze N. C. A. da Costa and F. A. Doria’s “exotic formalization” of the conjecture P = NP [3–7]. For any standard axiomatic PA extension T and any number-theoretic sentence ${\varphi }$ , we let ${\varphi ^{\star} := \varphi \vee \lnot \mathsf{Con}\left( \mathsf{T}\right)}$ and prove the following “exotic” inferences 1–3. 1. ${\mathsf{T}+\varphi ^{\star}}$ is consistent, if so is T, 2. ${\mathsf{T}+\varphi}$ is consistent, provided that ${\mathsf{T}+\varphi ^{\star}}$ is ω-consistent, 3. ${\mathsf{T}+\varphi}$ is consistent, provided that T is consistent and (...) has the same provably total recursive functions as ${\mathsf{T}+\left( \varphi \leftrightarrow \varphi ^{\star }\right) }$ . Furthermore we show that 1–3 continue to hold for ${\varphi ^{\star} := \varphi _{S} :=\varphi \vee \lnot S}$ , where ${S=\forall x\exists yR\left( x,y\right)}$ is any ${\Pi _{2}^{0}}$ sentence satisfying: 4. ${\left( \forall n\in \omega \right) \left( \mathsf{T}\vdash S_{x}\left[ \underline{n}\right] \right) }$ , 5. ${\mathsf{Con}\left( \mathsf{T}\right) \Rightarrow \mathsf{T}\nvdash S}$ . We observe that if ${\varphi :=\left[ \mathsf{P}=\mathsf{NP}\right] }$ and ${S:= \left[\digamma total\right] }$ , where ${\digamma=\digamma _{\mathsf{T}}}$ is da Costa-Doria “exotic” function with respect to T, then 4, 5 are satisfied for most familiar (presumably) consistent T in question, while ${\varphi _{S}}$ becomes equivalent to da Costa-Doria “exotic formalization” ${\left[ \mathsf{P}=\mathsf{NP}\right]^{\digamma}}$ . Moreover, the corresponding “exotic” inferences 1–3 generalize analogous da Costa-Doria results. Hence these “exotic” inferences are universal for all number-theoretic sentences and not characteristic to the conjecture P = NP. Nor do they infer relative consistency of P = NP (see Conclusion 15 in the text). (shrink)

L'histoire des femmes depuis la Seconde Guerre mondiale est bien plus développée en Italie qu'en France. Le dernier ouvrage d'Anna Rossi-Doria nous en fournit une preuve supplémentaire. S'appuyant sur ses recherches personnelles, mais aussi sur une solide bibliographie, l'auteure s'interroge sur les multiples facettes du droit de vote féminin. Le décret qui institue le droit de vote féminin présente bien des similarités avec l'ordonnance française. Comme en France, celui-ci ne résulte..

In 1995 N. C. A. da Costa and F. Doria proposed the modaltype elegant axiomatization of Jaśkowski’s discussive logic D2. Yet his ownproblem which was formulated in 1975 in a following way: Is it possible toformulate natural and simple axiomatization for D2, employing classical disjunction and conjunction along with discussive implication and conjunctionas the only primitive connectives? — still seems left open. The matter of factis there are some axiomatizations of D2 proposed, e.g., by T. Furmanowski, J. Kotas and (...) N. C. A. da Costa , G. Achtelik, L. Dubikajtus,E. Dudek and J. Konior , satisfying da Costa’s conditions, but they arerather looking very complicated and unnatural. An attempt is made to solveda Costa’s problem. The new axiomatization of D2 is proposed essentiallybased on da Costa’s-Doria axiomatization from 1995. (shrink)

In this paper the general notion of Bourbaki structures, interpreted in terms of Suppes predicates, will be used to axiomatize a system of meta-rankings in the sense introduced by A. K. Sen. It will be argued that this axiomatization must take place in a Kantian-ruled world in order to provide a link between meta-rankings and individual actions.Dedicated to Prof. Francisco A. Doria on his 50th birthday.

Non-classical views about important issues in logic and its philosophy are a distinctive trait of Shahid Rahman's work. This volume has been designed, on the occasion of his 50th birthday, as a gathering place for unconventional approaches, original ideas and attempts to question well-established standards. Some of the world top philosophers and logicians contributed to a brilliant collection of papers, some of which doubtlessly leave their mark on the work to come in logic and in philosophy of formal sciences. Contributors (...) are: Philippe Balbiani, Diderik Batens, Johan van Benthem, Giacomo Bonanno, Walter A. Carnielli, Newton C. A. Da Costa, Michel Crubellier, Francisco A. Doria, Dov M. Gabbay, Olivier Gasquet, Gerhard Heinzmann, Andreas Herzig, Jaakko Hintikka, Justine Jacot, Reinhard Kahle, Erik C. W. Krabbe, Decio Krause, Franck Lihoreau, Kuno Lorenz, Ilkka Niiniluoto, Graham Priest, Stephen Read, Manuel Rebuschi, Greg Restall, Gabriel Sandu, Gerhard Schurz, Francois Schwarzentruber, Yaroslav Shramko, Goran Sundholm, John Symons, Christian Thiel, Nicolas Troquard, Tero Tulenheimo, Jean Paul Van Bendegem, Daniel Vanderveken, Yde Venema, Heinrich Wansing, Jan Wolenski and John Woods. (shrink)

This accessible book gives a new, detailed and elementary explanation of the Gödel incompleteness theorems and presents the Chaitin results and their relation to the da Costa-Doria results, which are given in full, but with no ...

In lieu of an abstract, here is a brief excerpt of the content:Reviewed by:The Return of Scepticism: From Hobbes and Descartes to BayleSébastien CharlesGianni Paganini, editor. The Return of Scepticism: From Hobbes and Descartes to Bayle. Dordrecht: Kluwer, 2003. Pp. xxviii + 486. Cloth, $180.00.Cette édition des actes du congrès international « The Return of Scepticism », organisé par Gianni Paganini à l'Université du Piémont-Oriental de Vercelli en mai 2000, a pour ambition de faire le point sur l'état de la (...) recherche actuelle en cherchant à cerner au mieux les implications épistémologiques et les répercussions européennes du scepticisme à la période moderne. Placés sous le patronage de Richard H. Popkin, dont un essai ouvre l'ouvrage et pave la voie à de futures recherches sur la question, les textes rassemblés par Gianni Paganini ont pour trait commun d'insister sur la prégnance du scepticisme à l'époque moderne, tout en prenant garde au fait de bien replacer chaque stratégie sceptique dans l'environnement culturel et philosophique qui fut le sien. Divisé en sept sections, l'ouvrage fournit une vue d'ensemble de la question sceptique qui s'étale sur deux siècles, le XVIIe et le XVIIIe.Dans la première section, l'accent est mis sur un aspect délaissé de l'histoire du scepticisme moderne, à savoir le traitement particulier qui lui fut réservé dans la pensée anglaise des XVIIe et XVIIIe siècles. Sont abordés ici les cas de Hobbes, dont le traitement du scepticisme est pensé en relation inversée avec celui de Descartes (G. Paganini), Locke, qui tend à faire sien le principe sceptique de la limitation du savoir (G. A. J. Rogers), et les Platoniciens de Cambridge, qui s'inspirent de la méthode sceptique pour compartimenter le champ du savoir (F. Tomasoni).Dans la deuxième section, c'est la pensée continentale qui est à l'honneur, avec des textes consacrés à Descartes spécifiquement ou à La Mothe le Vayer et Descartes, avec pour objectif commun de questionner l'originalité du travail cartésien en insistant notamment sur le rôle de l'epoché chez Charron comme source possible du cogito (J. R. Maia Neto) et sur une tradition sceptique non fidéiste présente dans les œuvres de La Mothe le Vayer comme de Descartes (J.-P. Cavaillé).La troisième section évoque la prise de conscience des enjeux sceptiques chez les sectateurs et les adversaires de Descartes, qui s'accordent tous à reconnaître l'insuffisance de la réfutation du scepticisme proposée par ce dernier. Ce point fait l'unanimité tant chez les opposants à Descartes—Martin Schoock (A. Del Prete), Pierre Villemandy (C. Borghero), Pierre-Daniel Huet (T. M. Lennon), et les philosophes italiens Fardella et Doria (M. T. Marcialis)—que chez ceux qui se sont inspirés de sa philosophie—Arnauld (G. Belgioioso), Malebranche, Foucher et Desgabets (E. Scribano).La prise en compte des enjeux sceptiques aux plans politique et religieux est intégrée dans la quatrième partie au sein de laquelle sont évoquées des thématiques plurielles comme la présence du probabilisme néo-académique chez Pascal (A. McKenna), l'interprétation sceptique, voire libertine, du pouvoir et de la liberté rattachée à la théorie hobbesienne de l'état de nature chez Samuel Sorbière (L. Bianchi), ou le rapport existant entre scepticisme et tolérance (L. Simonutti).La section suivante délaisse politique et religion pour aborder le cas de la science, et de l'effet du scepticisme au niveau épistémologique. Le cas de Martín Martínez témoigne du fait que le scepticisme peut être conciliable avec une vision empiriste de la science et qu'une physique sceptique reste possible (J. C. Laursen). De même, le rapport des savants italiens comme Redi, Magalotti ou Dati à l'égard du scepticisme est évocateur de la prise de conscience des dangers que le scepticisme pose pour le domaine du démontrable (M. Torrini), ce qui vaut également pour toute théorie de la signification, dont les implications sceptiques sont patentes (G. Hassler), ou pour toute approche historique minée par des considérations pyrrhoniennes (C. Grell... (shrink)

From Newton da Costa’s works, many people in France know only the revival of paraconsistency. We give some reasons in defence of investigations in this part of logic. But above all we recall one of the major contributions of Newton da Costa: his proof, in 1991, in collaboration with Doria, of the gödelian undecidability of motion in mathematical physics, a result which was somewhat foreseen on other grounds by Duhem in 1906.

The debate about constructivism in physics has led to different kinds of questions that can be conventionally framed in two classes. One concerns the mathematics that is considered for the theoretical development of physics. The other is concerned with the experimental parts of physical theories. It is unnecessary to observe that the intersection between our two classes of problems is far from being empty. In this paper we will mainly deal with topics belonging to the second class. However, let us (...) briefly mention some important problems that have been debated in the framework of our first class. For instance, the following: to what extent do the undecidability and incompleteness results of classical mathematics affect fragments of physical theories, in such way as to have a “real physical meaning”? are the mathematical arguments that seem to be essential for physics justifiable in the framework of traditional mathematical constructivism?The first question has recently been investigated by Pitowski, Penrose, da Costa, Doria, Mundici, Svozil and others. As expected by most logicians, one can construct undecidable sentences whose physical meaning seems to be hardly questionable. This happens both in classical and in quantum mechanics. (shrink)