Martin-Löf on the Validity of Inference

An inference is valid if it guarantees the transferability of knowledge from the premisses to the conclusion. If knowledge is here understood as demonstrative knowledge, and demonstration is explained as a chain of valid inferences, we are caught in an explanatory circle. In recent lectures, Per Martin-Löf has sought to avoid the circle by specifying the notion of knowledge appealed to in the explanation of the validity of inference as knowledge of a kind weaker than demonstrative knowledge. The resulting explanation is the main topic of this article

Carnap’s Turn to the Thing Language

Les contributions de Carnap au Congrès de 1935 marquent un triple changement dans sa philosophie: son tournant sémantique; ce qui sera appelé plus tard « la libéralisation de l’empirisme»; et son adoption du « langage des choses» comme base du langage de la science. C’est ce troisième changement qui est examiné ici. On s’interroge en particulier sur les motifs qui ont poussé Carnap à adopter le langage des choses comme langage protocolaire de la science unifiée et sur les vertus de ce langage, comparé aux autres types de langage protocolaire.Rudolf Carnap’s contributions to the Paris 1935 Congress for scientific philosophy signal three important changes in his philosophy: his semantic turn; what would later be called the “liberalization of empiricism”; and his adoption of the so-called thing language as a basis for the language of science. This paper examines this third change. In particular, it considers Carnap’s motivation for adopting the thing language as the protocol language of unified science and the virtues of the thing language in comparison with other types of protocol language

Modality and the structure of assertion

A solid foundation of modal logic requires a clear conception of the notion of modality. Modern modal logic treats modality as a propositional operator. I shall present an alternative according to which modality applies primarily to illocutionary force, that is, to the force, or mood, of a speech act. By a first step of internalization, modality applied at this level is pushed to the level of speech-act content. By a second step of internalization, we reach a propositional operator validating the modal logic S4. After a brief discussion of problematic modality and possibility, the article concludes with an extension of the account that identifies modality with illocutionary force. Throughout, close attention is paid to the intended interpretation of the formalism. All of the rules stipulated will be justified on the basis of this interpretation

Carnap’s Turn to the Thing Language

Les contributions de Carnap au Congrès de 1935 marquent un triple changement dans sa philosophie: son tournant sémantique; ce qui sera appelé plus tard « la libéralisation de l’empirisme»; et son adoption du « langage des choses» comme base du langage de la science. C’est ce troisième changement qui est examiné ici. On s’interroge en particulier sur les motifs qui ont poussé Carnap à adopter le langage des choses comme langage protocolaire de la science unifiée et sur les vertus de ce langage, comparé aux autres types de langage protocolaire.Rudolf Carnap’s contributions to the Paris 1935 Congress for scientific philosophy signal three important changes in his philosophy: his semantic turn; what would later be called the “liberalization of empiricism”; and his adoption of the so-called thing language as a basis for the language of science. This paper examines this third change. In particular, it considers Carnap’s motivation for adopting the thing language as the protocol language of unified science and the virtues of the thing language in comparison with other types of protocol language

Carnap and Husserl

The first part of this entry details what is known about the personal encounters between Rudolf Carnap and Edmund Husserl. The second part looks at all the places in Carnap’s works where Husserl is cited

Immanent Reasoning or Equality in Action A Dialogical Study

PREFACEProf. Göran Sundholm of Leiden University inspired the group of Logic at Lille and Valparaíso to start a fundamental review of the dialogical conception of logic by linking it to constructive type logic. One of Sundholm's insights was that inference can be seen as involving an implicit interlocutor. This led to several investigations aimed at exploring the consequences of joining winning strategies to the proof-theoretical conception of meaning. The leading idea is, roughly, that while introduction rules lay down the conditions under which a winning strategy for the Proponent may be built, the elimination rules lay down those elements of the Opponent's assertions that the Proponent has the right to use for building winning strategy. It is the pragmatic and ethical features of obligations and rights that naturally lead to the dialogical interpretation of natural deduction.During the Visiting Professorship of Prof. Sundholm at Lille in (2012) the group of Lille started delving into the ways of implementing Per Martin-Löf's Constructive Type Theory with the dialogical perspective. In particular, Aarne Ranta's (1988) paper, the first publication on the subject, was read and discussed during Sundholm’s seminar. The discussions strongly suggested that the game-theoretical conception of quantifiers as deploying interdependent moves provide a natural link between CTT and dialogical logic. This idea triggered several publications by the group of Lille in collaboration with Nicolas Clerbout and Juan Redmond at the University of Valparaíso, including the publication of the book (2015) by Clerbout / Rahman that provides a systematic development of this way of linking CTT and the dialogical conception of logic. However the Clerbout / Rahman book was written from the CTT perspective on dialogical logic, rather than the other way round. The present book should provide the perspective from the other side of the dialogue between the Dialogical Framework and Constructive Type Theory. The main idea of our present study is that Sundholm's (1997) notion of epistemic assumption is closely linked to the Copy-Cat Rule or Socratic Rule that distinguishes the dialogical framework from other game-theoretical approaches. One way to read the present book is as a further development of Sundholm’s extension of Austin's (1946, p. 171) remark on acts of assertion to inference. Indeed, Sundholm (2013, p. 17) gave the following forceful formulation: When I say therefore, I give others my authority for asserting the conclusion, given theirs for asserting the premisses.Per Martin-Löf, in recent lectures, have utilized the dialogical perspective on epistemic assumptions to get out of a certain circle that threatens the explanation of the notions of inference and demonstration. A demonstration may be explained as a chain of (immediate) inferences starting from no premisses. That an inference J1 ... Jn—————Jis valid means that the conclusion J can be made evident on the assumption that J1, …, Jn are known. The notion of epistemic assumption thus enters in the explanation of valid inference. We cannot, however, in this explanation understand 'known' in the sense of demonstrated, for then we are explaining the notion of inference in terms of demonstration, whereas demonstration has been explained in terms of inference. Martin-Löf suggests that we here understand 'known' in the sense of asserted, so that epistemic assumptions are judgements others have made, judgements for which others have taken the responsibility; that the inference is valid then means that, given that others have taken responsbility for the premisses, I can take responsibility for the conclusion:The circularity problem is this: if you define a demonstration to be a chain of immediate inferences, then you are defining demonstration in terms of inference. Now we are considering an immediate inference and we are trying to give a proper explanation of that; but, if that begins by saying: Assume that J1, …, Jn have been demonstrated – then you are clearly in trouble, because you are about to explain demonstration in terms of the notion of immediate inference, hence when you are giving an account of the notion of immediate inference, the notion of demonstration is not yet at your disposal. So, to say: Assume that J1, …, Jn have already been demonstrated, makes you accusable of trying to explain things in a circle. The solution to this circularity problem, it seems to me now, comes naturally out of this dialogical analysis. […]The solution is that the premisses here should not be assumed to be known in the qualified sense, that is, to be demonstrated, but we should simply assume that they have been asserted, which is to say that others have taken responsibility for them, and then the question for me is whether I can take responsibility for the conclusion. So, the assumption is merely that they have been asserted, not that they have been demonstrated. That seems to me to be the appropriate definition of epistemic assumption in Sundholm's sense. One of the main tenets of the present study is that the further move of relating the Socratic Rule rule with judgemental equality provides both a simpler and more direct way to implement the constructive type theoretical approach within the dialogical framework. Such a reconsideration of the Socratic Rule, roughly speaking, amounts to the following. 1.A move from P that brings-forward a play-object in order to defend an elementary proposition A can be challenged by O. 2.The answer to such a challenge, involves P bringing forward a definitional equality that expresses the fact that the play-object chosen by P copies the one O has chosen while positing A. For short, the equality expresses at the object-languaje level that the defence of P relies on the authority given to O to be able to produce the play-objects she brings forward. More generally, according to this view, a definitional equality established by P and brought forward while defending the proposition A, expresses the equality between a play-object (introduced by O) and the instruction for building a play-object deployed by O while affirming A. So it can be read as a computation rule that indicates how to compute O's instructions. Let us recall that from the strategic point of view, O's moves correspond to elimination rules of demonstrations. Thus, the dialogical rules that prescribe how to introduce a definitional equality – correspond – at the strategic level – to the definitional equality rules for CTT as applied to the selector-functions involved in the elimination rules. So, what we are doing here is extending the dialogical interpretation of Sundholm's epistemic assumption to the rules that set up the definitional equality of a type.The present work is structured in the following way:•In the Introduction we state the main tenets that guide our book•Chapter II contains a brief overview of Constructive Type Theory penned by Ansten Klev with exercises and their solutions. •Chapter III offers a novel presentation of the dialogical conception of logic. In fact, the first two sections of the chapter provide an overview of standard dialogical logic – including solved exercises, slightly adapted to the content of the present volume. The rest of the sections develop a new dialogical approach to Constructive Type Theory. •Chapters IV and V contain the main body of our study, namely the relation between the Socratic Rule, epistemic assumptions, and equality. The chapter provides the reader with thoroughly worked out examples with comments. •After the Final Remarks the book containsa.An appendix containing a an overview of all the rules for the new formulation of dialogical approach to constructive type theory developed in the book; b.a brief outline of Per Martin-Löf's informal presentation of the demonstration of the axiom of choice.c.two examples of a tree-shaped graph of an extensive strategy. Acknowledgments: Many thanks to Mark van Atten (Paris1), Giuliano Bacigalupo (Zürich), Charles Zacharie Bowao (Univ. Ma Ngouabi de Brazzaville , Congo), Christian Berner (Lille3), Michel Crubellier (Lille3), Pierre Cardascia (Lille3), Oumar Dia (Dakar), Marcel Nguimbi ((Univ. Marien Ngouabi de Brazzaville, Congo), Adjoua Bernadette Dango (Lille3 / Alassane Ouattara de Bouaké, Côte d'Ivoire), Steephen Rossy Eckoubili (Lille3), Johan Georg Granström (Zürich) , Gerhard Heinzmann (Nancy2), Muhammad Iqbal (Lille3), Matthieu Fontaine (Lille3), Radmila Jovanovic (Belgrad), Hanna Karpenko (Lille3), Laurent Keiff (Lille3), Sébastien Magnier (Saint Dennis, Réunion), Clément Lion (Lille3), Gildas Nzokou (Libreville), Fachrur Rozie (Lille3), Helge Rückert (Mannheim), Mohammad Shafiei (Paris1), and Hassan Tahiri (Lisbon), for rich interchanges, suggestions on the main claims of the present study and proof-reading of specific sections of the text. Special thanks to Gildas Nzokou (Libreville), Steephen Rossy Eckoubili (Lille3) and Clément Lion (Lille3), with whom a Pre-Graduate Textbook in French on dialogical logic and CTT is in the process of being worked out – during the visiting-professorship in 2016 of the former. Eckoubili is the author of the sections on exercises for dialogical logic, Nzokou cared of those involving the development of CTT-demonstrations out of winning-strategies. Special thanks also to Ansten Klev (Prague) for helping with technical details and conceptual issues on CTT, particularly so as author of the chapter introducing CTT

Extending Kleene’s O Using Infinite Time Turing Machines, or How With Time She Grew Taller and Fatter.

We define two successive extensions of Kleene’s O using infinite time Turing machines. The first extension, O +, is proved to code a tree of height λ, the supremum of the writable ordinals, while the second extension, O ++, is proved to code a tree of height ζ, the supremum of the eventually writable ordinals. Furthermore, we show that O + is computably isomorphic to h, the lightface halting problem of infinite time Turing machine computability, and that O ++ is computably isomorphic to s, the set of programs that eventually write a real. The last of these results implies, by work of Welch, that O ++ is computably isomorphic to the Σ2 theory of Lζ, and, by work of Burgess, that O ++ is complete with respect to the class of the arithmetically quasi-inductive sets. This leads us to conjecture the existence of a parallel of hyperarithmetic theory at the level of Σ2(Lζ), a theory in which O ++ plays the role of O, the arithmetically quasi-inductive sets play the role of Π1 1, and the eventually writable reals play the role of ∆1 1.