Cartesian closed Dialectica categories

In 1930, A. Heyting first specified a formal system for part of intuitionistic mathematics. Although his rules were presumably motivated by the “intended interpretation” or meaning of the logical symbols, over the years a number of other possible interpretations have been discovered for which the rules are also valid. In particular, one might mention the realizablity interpretation of Kleene, the (Dialectica) interpretation of Gödel, and various semantic interpretations, such as Kripke models. (Each of these has several variants or close relatives.) Each such interpretation can be regarded as defining precisely a certain “notion of constructivity”, the study of which may illuminate the still rather vague notions which underlie the intended interpretation; or, if one doubts that there is a single interpretation “intended” by all constructivist mathematicians, the study of precisely defined interpretations may help to delineate and distinguish the possibilities.In the last few years, Heyting's systems have been vastly extended, in order to encompass the large and growing body of constructive mathematics. Several kinds of new systems have been put forward and studied. The present author has extended the various realizability interpretations to several of these systems [B1], [B2] and drawn a number of interesting applications. The mathematical content of the present paper is an interpretation in the style of Godel's Dialectica interpretation, but applicable to the new systems put forward by Feferman [Fl]. The original motivation for this work was to obtain certain metamathematical applications: roughly speaking, Markov's rule and its variants.

We give a Natural Deduction formulation of an adaptation of Gödel’s functional (Dialectica) interpretation to the extraction of (more) efficient programs from (classical) proofs. We adapt Jørgensen’s formulation of pure Dialectica translation by eliminating his “Contraction Lemma” and allowing free variables in the extracted terms (which is more suitable in a Natural Deduction setting). We also adapt Berger’s uniform existential and universal quantifiers to the Dialectica-extraction context. The use of such quantifiers without computational meaning permits the identification and isolation of contraction formulas which would otherwise be redundantly included in the pure-Dialectica extracted terms. In the end we sketch the possible combination of our refinement of Gödel’s Dialectica interpretation with its adaptation to the extraction of bounds due to Kohlenbach into a light monotone functional interpretation.KeywordsProgram extraction from (classical) proofsComplexity of extracted programsBerger’s uniform quantifiersGödel’s Functional interpretationProof-Carrying CodeProof Mining

Wilfried Buchholz. Notation systems for infinitary derivations. Archive for mathematical logic, vol. 30 no. 5–6 (1991), pp. 277–296. - Wilfried Buchholz. Explaining Gentzen's consistency proof within infinitary proof theory. Computational logic and proof theory, 5th Kurt Gödel colloquium, KGC '97, Vienna, Austria, August 25–29, 1997, Proceedings, edited by Georg Gottlob, Alexander Leitsch, and Daniele Mundici, Lecture notes in computer science, vol. 1289, Springer, Berlin, Heidelberg, New York, etc., 1997, pp. 4–17. - Sergei Tupailo. Finitary reductions for local predicativity, I: recursively regular ordinals. Logic Colloquium '98, Proceedings of the annual European summer meeting of the Association for Symbolic Logic, held in Prague, Czech Republic, August 9–15, 1998, edited by Samuel R. Buss, Petr Háajek, and Pavel Pudlák, Lecture notes in logic, no. 13, Association for Symbolic Logic, Urbana, and A K Peters, Natick, Mass., etc., 2000, pp. 465–499. - Volume 8 Issue 3

Predicative functionals and an interpretation of [formula omitted

Chapter 17 Computation models and function algebras

Computability is one of the fundamental notions of mathematics and computer science, trying to capture the effective content of mathematics and its applications. Computability Theory explores the frontiers and limits of effectiveness and algorithmic methods. It has its origins in Godel's Incompleteness Theorems and the formalization of computability by Turing and others, which later led to the emergence of computer science as we know it today. Computability Theory is strongly connected to other areas of mathematics and theoretical computer science. The core of this theory is the analysis of relative computability and the induced degrees of unsolvability; its applications are mainly to Kolmogorov complexity and randomness as well as mathematical logic, analysis and algebra. Current research in computability theory stresses these applications and focuses on algorithmic randomness, computable analysis, computable model theory, and reverse mathematics (proof theory). Recent advances in these research directions have revealed some deep interactions not only among these areas but also with the core parts of computability theory. The goal of this Dagstuhl Seminar is to bring together researchers from all parts of computability theory and related areas in order to discuss advances in the individual areas and the interactions among those.

The article discusses the problem of teaching computer science, mathematics and logic in modern education, in particular, in a secondary general education school as inherited from the problems of the existing relationships between the sciences of computer science and mathematics, as well as generated by the prevailing forms and practice of teaching in the system of general education. The methodological aspects of teaching computer science, mathematics and logic are considered inherent in them and applied to them in a differentiated form: - a meaningful approach inherent in computer science and applied in mathematics and logic; - a formal approach inherent in mathematics and used in computer science and logic; - a socio-cultural approach characteristic of the triad ”computer science-mathematics-logic”. Against the background of the existing correspondence of the selected approaches (or the measure of this correspondence) to each local and independent element of the triad, it is assumed that there is an identification of the regular relations between the subject and the subject of education, the principles of organizing the processes of implementing methodological knowledge based on these regularities. For our study, it is important that the methodology and methodology of each differentiated system of the triad should contain everything that is ”necessary and sufficient” for the functioning of the educational system of the corresponding level. In this aspect, the article notes that mathematics studies form, computer science - form and content. The study of objects only in form leads to formalism in the information sphere (which is unacceptable), therefore, possible integration must be combined with the necessary differentiation. Since the processes of the development of knowledge of science, its methodology and teachingmethods are interrelated and interdependent, it is unconditionally interesting to reveal the potential (possibilities) of intersubject and metasubject connections and relations of the triad under consideration as close, but not homogeneous educational systems that have their own subject. specificity of the content, methods and forms of training. In this perspective, possible ways of solving the problems that have arisen are proposed in the context of changes in the school curricula of the content of information and mathematics education under the influence of the requirements of the new federal state educational standards and the structure of subject teaching in computer science and mathematics, ensuring their efficiency and effectiveness.

Metapredicativity is a new general term in proof theory which describes the analysis and study of formal systems whose proof-theoretic strength is beyond the Feferman-Schutte ordinal Γ0 but which are nevertheless amenable to purely predicative methods. Typical examples of formal systems which are apt for scaling the initial part of metapredicativity are the transfinitely iterated fixed point theories IDα whose detailed proof-theoretic analysis is given by Jager, Kahle, Setzer and Strahm in [18]. In this paper we assume familiarity with [18]. For natural extensions of Friedman’s ATR that can be measured against transfinitely iterated fixed point theories the reader is referred to Jager and Strahm [20]. In the mid seventies, Feferman [3, 4] introduced systems of explicit mathematics in order to provide an alternative foundation of constructive mathematics. More precisely, it was the origin of Feferman’s program to give a logical account of Bishop-style constructive mathematics. Right from the beginning, systems of explicit mathematics turned out to be of general interest for proof theory, mainly in connection with the proof-theoretic analysis of subsystems of first and second order arithmetic and set theory, cf. e.g. Jager [15] and Jager and Pohlers [19]. More recently, systems of explicit mathematics have been used to develop a general logical framework for functional programming and type theory, where it is possible to derive correctness and termination properties of functional programs. Important references in this connection are Feferman [6, 7, 9] and Jager [17]. Universes are a frequently studied concept in constructive mathematics at least since the work of Martin-Lof, cf. e.g. Martin-Lof [23] or Palmgren [27] for

Ramified recurrence and computational complexity III: Higher type recurrence and elementary complexity

The purpose of this paper is to study identity issues from different points of view. The analysis is based on the economic science tools. Economic identity introduces a framework that develops the emerging literature on personal identity by using a form of production functions in order to depict investment decisions in social identities. In particular, it focuses on the interpretation of the production functions. The proposal uses the structures of the social realm that are external to the individual, but which can affect his/her social identity. The concept of personal identity in economics has received a lot of attention in recent years. The use of personal identity requires expanding the psychology foundation behind the concept of identity in economics in order to account for the psychological dimension of what is required when trying to maintain a personal identity. This paper tends to examine the usefulness of considering developmental psychology in addition to the traditional social psychological foundation of identity economics within a framework of institutional economy and economic identity based on so-called insiders – scientific category.

A structure which generalizes formulas by including substitution terms is used to represent proofs in classical logic. These structures, called expansion trees, can be most easily understood as describing a tautologous substitution instance of a theorem. They also provide a computationally useful representation of classical proofs as first-class values. As values they are compact and can easily be manipulated and transformed. For example, we present an explicit transformations between expansion tree proofs and cut-free sequential proofs. A theorem prover which represents proofs using expansion trees can use this transformation to present its proofs in more human-readable form. Also a very simple computation on expansion trees can transform them into Craig-style linear reasoning and into interpolants when they exist. We have chosen a sublogic of the Simple Theory of Types for our classical logic because it elegantly represents substitutions at all finite types through the use of the typed λ-calculus. Since all the proof-theoretic results we shall study depend heavily on properties of substitutions, using this logic has allowed us to strengthen and extend prior results: we are able to prove a strengthen form of the firstorder interpolation theorem as well as provide a correct description of Skolem functions and the Herbrand Universe. The latter are not straightforward generalization of their first-order definitions.

Mathematical logic and computer science have driven the design of a growing number of logics and related formalisms such as set theories and type theories. In response to this population explosion, logical frameworks have been developed as formal meta-languages in which to represent, structure, relate and reason about logics.Research on logical frameworks has diverged into separate communities, often with conflicting backgrounds and philosophies. In particular, two of the most important logical frameworks are the framework of institutions, from the area of model theory based on category theory, and the Edinburgh Logical Framework LF, from the area of proof theory based on dependent type theory. Even though their ultimate motivations overlap – for example in applications to software verification – they have fundamentally different perspectives on logic.In the current paper, we design a logical framework that integrates the frameworks of institutions and LF in a way that combines their complementary advantages while retaining the elegance of each of them. In particular, our framework takes a balanced approach between model theory and proof theory, and permits the representation of logics in a way that comprises all major ingredients of a logic: syntax, models, satisfaction, judgments and proofs. This provides a theoretical basis for the systematic study of logics in a comprehensive logical framework. Our framework has been applied to obtain a large library of structured and machine-verified encodings of logics and logic translations.

This handbook provides a systematic survey of the central areas of philosophical logic. It is divided into four volumes, each devoted to a major sub-field within the disciplines. The articles should be of value to general philosophers, linguists, logicians, mathematicians and computer scientists. Volume 1, Elements of Classical Logic, deals with the background to what has come to be considered the standard formulation of predicate logic - both as far as its semantics and proof theory are concerned. The central chapter on predicate logic is followed by chapters outlining various alternative, but essentially equivalent ways of constructing the semantics for first-order logic as well as its proof theory. In addition, this volume contains a discussion of higher-order extensions of first-order logic and a compendium of the algorithmic and decision-theoretic prerequisites in the study of logical systems. Volume 2, Extensions of Classical Logic, surveys the most significant extensions of predicate logic and their applications to various philosophical fields of inquiry. The 12 chapters in this volume together provide a succinct introduction to a variety of intensional frameworks, a discussion of the most well-known logical systems, as well as an overview of major applications and of the open problems in the respective fields. Volume 3, Alternatives to Classical Logic, consists of a series of surveys of some of the alternatives to the basic assumptions of classical logic. These include many-valued logic, partial logic, free logic, relevance and entailment logics, dialogue logic, quantum logic and intuitionism. Volume 4, Topics in the Philosophy of Language, presents a panorama of the applications of logical tools and methods in the formal analysis of natural language. Since a number of developments in philosophical logic were originally stimulated by concern arising in the semantic analysis of natural language discourse, the chapters in this volume provide some criteria of evaluation of the applications of work in philosophical logic. In revealing both the adequacies and inadequacies of logical investigations in the semantic structures of natural discourse, these chapters also point the way to future developments in philosophical logic in general and thus close again the circle of inquiry relating logic and language.

David Botting, in the essay, Why modal pluralism does not require normative pluralism, wants to demonstrate that my introduction of multi-modal argumentation (Gilbert 1997) (a) does not require introducing diverse forms of argument appraisal, and (b) all arguments in any of the modes, whether logical, emotional, kisceral, or visceral, can be reduced to logically deductive arguments. I would like to briefly respond making several points. I Surrender To begin I have to admit that Mr. Botting has me dead to rights. All we have to do is accept his initial assumptions, and I am a goner: there is no need to analyze arguments, let alone appraise them using any but the logical mode. I quote: To persuade rationally is to give reasons. in the logical mode (and not in other modes) arguers state reasons in support of their thesis, and argumentation in this mode can be translated into a formal argument only the customary problems of resolving ambiguous and vague formulations. function of argumentation is rational persuasion. Therefore, all legitimate argumentation is in the logical mode and to evaluate it is to assess the strength of the reason's support for the conclusion. Then, because rational persuasion is, definition, considered the analyst to be the goal of argumentation, the argument is appraised against whatever might be countenanced in the logical mode. (Botting, this issue, p. 170) If we accept this statement, viz., that argument is about giving reasons to support a thesis and that those reasons are all reducible to logically deductive arguments, then Botting is correct. But that assumption and the assertion that by assessment of arguments is in the logical mode flies in the face of a great deal of work in Argumentation Theory. Botting's definition amounts to claiming that: 1. All arguments must be deductively reducible. 2. Deductively reduced arguments all use the same form of appraisal. 3. Gilbert's argument modes are deductively reducible, 4. So, Gilbert's modes do not require distinct forms of appraisal. But the difficulty is that my entire theory is a critique of his assumption. Following Brockriede, Wenzel, Willard, Tindale, and a host of others, I argue that persuasion goes beyond the logical mode, and that persuasion is perfectly rational (Brockriede, 1990; Tindale, 1999; Wenzel, 1979; Willard, 1989). I also want to underscore the phrase, with only the customary problems of resolving ambiguous and vague formulations, which I will refer to later. Foundational Error A great deal of Botting's argument relies on his assertion that any discursive argument is in the logical mode. He writes, The kind of argument we normally deal with-the kind that we get from a or spoken text, for example--is in the logical mode and can be appraised against logical norms (p. 169). I am at a loss as to where the idea comes from that I put all arguments that are presented in or spoken texts into the logical category. Throughout the book, I give examples or scripts of arguments in various modes that are completely verbal and completely in written or spoken I am clear in various places that the way in which an argument is presented, verbal or non-verbal, is not necessarily a factor in contributing a mode to it. Moreover, I find it hard to refer to a spoken argument as a text. Does Mr. Botting mean the transcript of the argument? If so, the question is begged once again, insofar as he begins stripping away much of what I insist requires examination, viz., tone, expression, context and a host of rhetorical considerations. In short, the assertion that, Only in the logical mode (and not in other modes) arguers state reasons, cannot be attributed to me. I never said it and do not believe it. only source for this belief I can imagine is the idea that any argument in whatever mode can be shoehorned into a logical model. …

Recently, the second author, Briseid, and Safarik introduced nonstandard Dialectica, a functional interpretation capable of eliminating instances of familiar principles of nonstandard arithmetic—including overspill, underspill, and generalizations to higher types—from proofs. We show that the properties of this interpretation are mirrored by first-order logic in a constructive sheaf model of nonstandard arithmetic due to Moerdijk, later developed by Palmgren, and draw some new connections between nonstandard principles and principles that are rejected by strict constructivism. Furthermore, we introduce a variant of the Diller–Nahm interpretation with two different kinds of quantifiers, similar to Hernest’s light Dialectica interpretation, and show that one can obtain nonstandard Dialectica by weakening the computational content of the existential quantifiers—a process called herbrandization. We also define a constructive sheaf model mirroring this new functional interpretation, and show that the process of herbrandization has a clear meaning in terms of these sheaf models.