A cut-free Gentzen sequent calculus for Ewald’s intuitionistic tense logic \ is established. By the proof-theoretic method, we prove that, for every set of strictly positive implications S, the classical tense logic \ is embedded into its intuitionistic analogue \ via Kolmogorov, Gödel–Genzten and Kuroda translations respectively. A sufficient and necessary condition for Glivenko type theorem in tense logics is established.

Let KP- be the theory resulting from Kripke-Platek set theory by restricting Foundation to Set Foundation. Let G: V → V (V:= universe of sets) be a ▵0-definable set function, i.e. there is a ▵0-formula φ(x, y) such that φ(x, G(x)) is true for all sets x, and $V \models \forall x \exists!y\varphi (x, y)$ . In this paper we shall verify (by elementary proof-theoretic methods) that the collection of set functions primitive recursive in G coincides with (...) the collection of those functions which are Σ1-definable in KP- + Σ1-Foundation + ∀ x ∃!yφ (x, y). Moreover, we show that this is still true if one adds Π1-Foundation or a weak version of ▵0-Dependent Choices to the latter theory. (shrink)

In this paper, we show the equivalence between the provability of a proof system of basic hybrid logic and that of translated formulas of the classical predicate logic with equality and explicit substitution by a purely proof–theoretic method. Then we show the equivalence of two groups of proof systems of hybrid logic: the group of labelled deduction systems and the group of modal logic-based systems.

A proof-theoretical analysis of elementary theories of order relations is effected through the formulation of order axioms as mathematical rules added to contraction-free sequent calculus. Among the results obtained are proof-theoretical formulations of conservativity theorems corresponding to Szpilrajn’s theorem on the extension of a partial order into a linear one. Decidability of the theories of partial and linear order for quantifier-free sequents is shown by giving terminating methods of proof-search.

This thesis is concerned with extending the underlying logical approach as well as the breadth of applications of the proof mining program to various (mostly previously untreated) areas of nonlinear analysis and optimization, with a particular focus being placed on topics which involve set-valued operators.For this, we extend the current logical methodology of proof mining by new systems and corresponding so-called logical metatheorems that cover these more involved areas of nonlinear analysis. Most of these systems crucially rely on (...) the use of intensional methods, treating sets with potentially high quantifier complexity in the defining matrix via characteristic functions and axioms that describe only their properties and do not completely characterize the elements of the sets.The applicability of all of these metatheorems is then substantiated by a range of case studies for the respective areas which in particular also highlight the naturalness of the use of intensional methods in the design of the corresponding systems.The first new area covered thereby is the theory of nonlinear semigroups induced by corresponding evolution equations for accretive operators. In that context, we present (besides an initial foray into the area from 2015) essentially the first applications of proof mining to the theory of partial differential equations. Concretely, we provide quantitative versions of four central results on the asymptotic behavior of solutions to such equations.The second new area unlocked in this thesis is that of the continuous dual of a Banach space and its norm (which are also approached via intensional methods). This in particular relies on a proof-theoretically tame treatment of suprema over (certain) bounded sets in this intensional context which is further exploited later on. These systems, which give access to this until now untreated fundamental notion from functional analysis, are then used to provide further substantial extensions to treat various notions from convex analysis like the Fréchet derivative of a convex function, Fenchel conjugates, Bregman distances, and monotone operators on Banach spaces in the sense of Browder.These systems are then utilized to provide applications in the context of Picard- and Halpern-style iterations of so-called Bregman strongly nonexpansive mappings where we provide both new quantitative and qualitative results.Lastly, we discuss the key notion of extensionality of a set-valued operator and its relation to set-theoretic maximality principles in more depth (which was already singled out—to some degree—in previous work). We thereby exhibit an issue arising with treating full extensionality in the context of these intensional approaches to set-valued operators and present useful fragments of the full extensionality statement where these issues are avoided.Corresponding to these fragments, we discuss a range of uniform continuity statements for set-valued operators beyond the usual notion involving the Hausdorff-metric. In particular, in that context, we utilize the previous tame treatment of suprema over bounded sets to also provide the first proof-theoretic treatment of that Hausdorff-metric in the context of systems for proof mining.The applicability of this treatment of the Hausdorff-metric is then in particular substantiated by a last case study where we provide quantitative information for a Mann-type iteration of set-valued mappings which are nonexpansive w.r.t. the Hausdorff-metric.The abstract was taken directly from the thesis.Abstract prepared by Nicholas Pischke, Technische Universität Darmstadt, Darmstadt, GermanyE-mail: [email protected]: https://doi.org/10.26083/tuprints-00026584. (shrink)

By a result of Paris and Friedman, the collection axiom schema for $\Sigma_{n+1}$ formulas, $B\Sigma_{n+1}$ , is $\Pi_{n+2}$ conservative over $I\Sigma_n$ . We give a new proof-theoretic proof of this theorem, which is based on a reduction of $B\Sigma_n$ to a version of collection rule and a subsequent analysis of this rule via Herbrand's theorem. A generalization of this method allows us to improve known results on reflection principles for $B\Sigma_n$ and to answer some technical questions left (...) open by Sieg [23] and Hájek [9]. We also give a new proof of independence of $B\Sigma_{n+1}$ over $I\Sigma_n$ by a direct recursion-theoretic argument and answer an open problem formulated by Gaifman and Dimitracopoulos [8]. (shrink)

The paper proposes a semantic value for the logical constants (connectives and quantifiers) within the framework of proof-theoretic semantics, basic meaning on the introduction rules of a meaning conferring natural deduction proof system. The semantic value is defined based on Fregecontributions” to sentential meanings as determined by the function-argument structure as induced by a type-logical grammar. In doing so, the paper proposes a novel proof-theoretic interpretation of the semantic types, traditionally interpreted in Henkin models. The (...) compositionality of the resulting attribution of semantic values is discussed. Elsewhere, the same method was used for defining proof-theoretic meaning of subsentential phrases in a fragment of natural language. Doing the same for (the simpler and clearer case of) logic sheds more light on the proposal. (shrink)

We suggest an algebraic approach to proof-theoretic analysis based on the notion of graded provability algebra, that is, Lindenbaum boolean algebra of a theory enriched by additional operators which allow for the structure to capture proof-theoretic information. We use this method to analyze Peano arithmetic and show how an ordinal notation system up to 0 can be recovered from the corresponding algebra in a canonical way. This method also establishes links between proof-theoretic ordinal analysis (...) and the work which has been done in the last two decades on provability logic and reflection principles. Because of its abstract algebraic nature, we hope that it will also be of interest for non-prooftheorists. (shrink)

It is well known that the modal logic S5 can be embedded in the classical predicate logic by interpreting the modal operator in terms of a quantifier. Wajsberg [10] proved this fact in a syntactic way. Mints [7] extended this result to the quantified version of S5; using a purely proof-theoretic method he showed that the quantified S5 corresponds to the classical predicate logic with one-sorted variable. In this paper we extend Mints' result to the basic modal logic (...) S4; we investigate the correspondence between the quantified versions of S4 (with and without the Barcan formula) and the classical predicate logic (with one-sorted variable). We present a purely proof-theoretic proof-transformation method, reducing an LK-proof of an interpreted formula to a modal proof. (shrink)

The paper briefly surveys the sentential proof-theoretic semantics for fragment of English. Then, appealing to a version of Frege’s context-principle (specified to fit type-logical grammar), a method is presented for deriving proof-theoretic meanings for sub-sentential phrases, down to lexical units (words). The sentential meaning is decomposed according to the function-argument structure as determined by the type-logical grammar. In doing so, the paper presents a novel proof-theoretic interpretation of simple type, replacing Montague’s model-theoretic type (...) interpretation (in arbitrary Henkin models). The domains of derivations are collections of derivations in the associated “dedicated” natural-deduction proof-system, and functions therein (with no appeal to models, truth-values and elements of a domain). The compositionality of the semantics is analyzed. (shrink)

Ever since antiquity, attempts have been made at defining knowledge through belief augmented by additional properties such as truth and justification. These characterizations have been challenged by Gettier counterexamples and their variants. A modern proposal, what is known as defeasibility theory, characterizes knowledge through stability under revision of beliefs on the basis of true or arbitrary information. A formal investigation of such a proposal calls for the methods of dynamic epistemic logic: well developed semantic approaches to dynamic epistemic logic (...) have been given through the plausibility models, but a corresponding proof theory is still in its infancy. We shall recast plausibility models in terms of more general neighbourhood models and develop on their basis complete proof systems. An inferential treatment of epistemic and doxastic notions such as safe belief will give a new way to study their relationships; among these, the characterization of knowledge as belief stable under arbitrary revision will be grounded through formal labelled sequent calculus derivations. (shrink)

The k-provability for an axiomatic system A is to determine, given an integer k 1 and a formula in the language of A, whether or not there is a proof of in A containing at most k lines. In this paper we develop a unification-theoretic method for investigating the k-provability problem for Parikh systems, which are first-order axiomatic systems that contain a finite number of axiom schemata and a finite number of rules of inference. We show that the (...) k-provability problem for a Parikh system reduces to a unification problem that is essentially the unification problem for second-order terms. By solving various subproblems of this unification problem , we solve the k- probability problem for a variety of Parikh systems, including several formulations of Peano arithmetic. Our method for investigating the k-provability problem employs algorithms that compute and characterize unifiers. We give some examples of how these algorithms can be used to solve complexity problems other than the k-provability problem. (shrink)

Proof-theoretic methods are developed for subsystems of Johansson’s logic obtained by extending the positive fragment of intuitionistic logic with weak negations. These methods are exploited to establish properties of the logical systems. In particular, cut-free complete sequent calculi are introduced and used to provide a proof of the fact that the systems satisfy the Craig interpolation property. Alternative versions of the calculi are later obtained by means of an appropriate loop-checking history mechanism. Termination of the (...) new calculi is proved, and used to conclude that the considered logical systems are PSPACE-complete. (shrink)

Fuzzy logics are many-valued logics that are well suited to reasoning in the context of vagueness. They provide the basis for the wider field of Fuzzy Logic, encompassing diverse areas such as fuzzy control, fuzzy databases, and fuzzy mathematics. This book provides an accessible and up-to-date introduction to this fast-growing and increasingly popular area. It focuses in particular on the development and applications of "proof-theoretic" presentations of fuzzy logics; the result of more than ten years of intensive work (...) by researchers in the area, including the authors. In addition to providing alternative elegant presentations of fuzzy logics, proof-theoretic methods are useful for addressing theoretical problems and developing efficient deduction and decision algorithms. Proof-theoretic presentations also place fuzzy logics in the broader landscape of non-classical logics, revealing deep relations with other logics studied in Computer Science, Mathematics, and Philosophy. The book builds methodically from the semantic origins of fuzzy logics to proof-theoretic presentations such as Hilbert and Gentzen systems, introducing both theoretical and practical applications of these presentations. (shrink)

Apologies. The purpose of the following talk is to give an overview of the present state of aims, methods and results in Pure Proof Theory. Shortage of time forces me to concentrate on my very personal views. This entails that I will emphasize the work which I know best, i.e., work that has been done in the triangle Stanford, Munich and Münster. I am of course well aware that there are as important results coming from outside this triangle (...) and I apologize for not displaying these results as well.Moreover the audience should be aware that in some points I have to oversimplify matters. Those who complain about that are invited to consult the original papers.1.1. General. Proof theory startedwithHilbert's Programme which aimed at a finitistic consistency proof for mathematics.By Gödel's Theorems, however, we know that we can neither formalize all mathematics nor even prove the consistency of formalized fragments by finitistic means. Inspite of this fact I want to give some reasons why I consider proof theory in the style of Gentzen's work still as an important and exciting field of Mathematical Logic. I will not go into applications of Gentzen's cut-elimination technique to computer science problems—this may be considered as applied proof theory—but want to concentrate on metamathematical problems and results. In this sense I am talking about Pure Proof Theory.Mathematicians are interested in structures. There is only one way to find the theorems of a structure. Start with an axiom system for the structure and deduce the theorems logically. These axiom systems are the objects of proof-theoretical research. Studying axiom systems there is a series of more or less obvious questions. (shrink)

I consider two cognitive phenomena: inquiring and justifying, as complementary processes running in opposite directions. I explain on an example that the former process is driven by questions and the latter is a codification of the results of the first one. Traditionally, proof theory focuses on the latter process, and thus describes the former, at best, as an example of a backward proof search. I argue that this is not the best way to analyze cognitive processes driven by (...) questions, and that proof-theoretical analysis of questions can bring mutual benefits for both proof theory and erotetic logic. In the second part of my paper (sect. 6.3) I introduce the structural rules of weakening and contraction. Results concerning their admissibility are discussed. I also sketch an erotetic interpretation of the rule of weakening as reducing information noise, and that of contraction as securing information. (shrink)

Until not too many years ago, all logics except classical logic (and, perhaps, intuitionistic logic too) were considered to be things esoteric. Today this state of a airs seems to have completely been changed. There is a growing interest in many types of nonclassical logics: modal and temporal logics, substructural logics, paraconsistent logics, non-monotonic logics { the list is long. The diversity of systems that have been proposed and studied is so great that a need is felt by many researchers (...) to try to put some order in the present logical jungle. Thus Cl91], Ep90] and Wo88] are three recent books in which an attempt is made to develop a general theoretical framework for the study of logics. On the more pragmatic side, several systems have been developed with the goal of providing a computerized logical framework in which many di erent logical systems can be implemented in a uniform way. An example is the Edinburgh LF( HHP91]). (shrink)

In view of an enhancement of our implementation on the computer, we explore the possibility of an algorithmic optimization of the various proof-theoretic techniques employed by Kohlenbach for the synthesis of new effective uniform bounds out of established qualitative proofs in Numerical Functional Analysis. Concretely, we prove that the method of “colouring” some of the quantifiers as “non-computational” extends well to ε-arithmetization, elimination-of-extensionality and model-interpretation.

Using labelled formulae, a cut-free sequent calculus for intuitionistic propositional logic is presented, together with an easy cut-admissibility proof; both extend to cover, in a uniform fashion, all intermediate logics characterised by frames satisfying conditions expressible by one or more geometric implications. Each of these logics is embedded by the Gödel–McKinsey–Tarski translation into an extension of S4. Faithfulness of the embedding is proved in a simple and general way by constructive proof-theoretic methods, without appeal to semantics (...) other than in the explanation of the rules. (shrink)

Structural proof theory is a branch of logic that studies the general structure and properties of logical and mathematical proofs. This book is both a concise introduction to the central results and methods of structural proof theory, and a work of research that will be of interest to specialists. The book is designed to be used by students of philosophy, mathematics and computer science. The book contains a wealth of results on proof-theoretical systems, including extensions of (...) such systems from logic to mathematics, and on the connection between the two main forms of structural proof theory - natural deduction and sequent calculus. The authors emphasize the computational content of logical results. A special feature of the volume is a computerized system for developing proofs interactively, downloadable from the web and regularly updated. (shrink)

In a classical 1976 paper, Solovay proved the arithmetical completeness of the modal logic GL; provability of a formula in GL coincides with provability of its arithmetical interpretations of it in Peano Arithmetic. In that paper, he also provided an axiomatic system GLS and proved arithmetical completeness for GLS; provability of a formula in GLS coincides with truth of its arithmetical interpretations in the standard model of arithmetic. Proof theory for GL has been studied intensively up to the present (...) day. However, it might sound somewhat strange that no proof theory for GLS was ever developed nor even suggested thus far, except for the axiomatic system offered by Solovay. In this paper, we develop a proof theory for GLS based on the sequent calculus method. We provide a sequent calculus for GLS and prove the cut-elimination and some standard consequences of it: the interpolation and definability theorems. As another consequence of cut-elimination, we also prove the equivalence of GL and GLS with respect to a special form of formulas which we call Gödel sentences, using a purely proof-theoretical method. (shrink)

In our previous work we proved the conjecture NP = PSPACE by advanced proof theoretic methods that combined Hudelmaier’s cut-free sequent calculus for minimal logic with the horizontal compressing in the corresponding minimal Prawitz-style natural deduction. In this Addendum we show how to prove a weaker result NP = coNP without referring to HSC. The underlying idea is to omit full minimal logic and compress only “naive” normal tree-like ND refutations of the existence of Hamiltonian cycles in (...) given non-Hamiltonian graphs, since the Hamiltonian graph problem in NPcomplete. Thus, loosely speaking, the proof of NP = coNP can be obtained by HSC-elimination from our proof of NP = PSPACE. (shrink)

This book presents a detailed treatment of ordinal combinatorics of large sets tailored for independence results. It uses model theoretic and combinatorial methods to obtain results in proof theory, such as incompleteness theorems or a description of the provably total functions of a theory. In the first chapter, the authors first discusses ordinal combinatorics of finite sets in the style of Ketonen and Solovay. This provides a background for an analysis of subsystems of Peano Arithmetic as well (...) as for combinatorial independence results. Next, the volume examines a variety of proofs of Gödel's incompleteness theorems. The presented proofs differ strongly in nature. They show various aspects of incompleteness phenomena. In additon, coverage introduces some classical methods like the arithmetized completeness theorem, satisfaction predicates or partial satisfaction classes. It also applies them in many contexts. The fourth chapter defines the method of indicators for obtaining independence results. It shows what amount of transfinite induction we have in fragments of Peano arithmetic. Then, it uses combinatorics of large sets of the first chapter to show independence results. The last chapter considers nonstandard satisfaction classes. It presents some of the classical theorems related to them. In particular, it covers the results by S. Smith on definability in the language with a satisfaction class and on models without a satisfaction class. Overall, the book's content lies on the border between combinatorics, proof theory, and model theory of arithmetic. It offers readers a distinctive approach towards independence results by model-theoretic methods. (shrink)

We give a simple proof-theoretic argument showing that Glivenko’s theorem for propositional logic and its version for predicate logic follow as an easy consequence of the deduction theorem, which also proves some Glivenko type theorems relating intermediate predicate logics between intuitionistic and classical logic. We consider two schemata, the double negation shift (DNS) and the one consisting of instances of the principle of excluded middle for sentences (REM). We prove that both schemata combined derive classical logic, while each (...) one of them provides a strictly weaker intermediate logic, and neither of them is derivable from the other. We show that over every intermediate logic there exists a maximal intermediate logic for which Glivenko’s theorem holds. We deduce as well a characterization of DNS, as the weakest (with respect to derivability) scheme that added to REM derives classical logic. (shrink)

A method is described for obtaining conjunctive normal forms for logics using Gentzen-style rules possessing a special kind of strong invertibility. This method is then applied to a number of prominent fuzzy logics using hypersequent rules adapted from calculi defined in the literature. In particular, a normal form with simple McNaughton functions as literals is generated for łukasiewicz logic, and normal forms with simple implicational formulas as literals are obtained for Gödel logic, Product logic, and Cancellative hoop logic.

This book continues from where the authors' previous book, Structural Proof Theory, ended. It presents an extension of the methods of analysis of proofs in pure logic to elementary axiomatic systems and to what is known as philosophical logic. A self-contained brief introduction to the proof theory of pure logic is included that serves both the mathematically and philosophically oriented reader. The method is built up gradually, with examples drawn from theories of order, lattice theory and elementary (...) geometry. The aim is, in each of the examples, to help the reader grasp the combinatorial behaviour of an axiom system, which typically leads to decidability results. The last part presents, as an application and extension of all that precedes it, a proof-theoretical approach to the Kripke semantics of modal and related logics, with a great number of new results, providing essential reading for mathematical and philosophical logicians. (shrink)

Familiar proof theoretical and especially automated deduction methods sometimes accept infinity where, in fact, it can be omitted. Our first example deals with the infinite supply of individual variables admitted in 1-order deductions, the second one deals with infinite-branching rules in sequent calculi with number-theoretical induction. The contents of Section 1 summarize and extend basic ideas and results published elsewhere, whereas basic ideas and results of Section 2 are exposed for the first time in the present paper. We (...) consider classical formalisms only, to which constructive formalisms can be reduced by Shanin-style interpretations. (shrink)

We estimate the derivation lengths of functionals in Gödel's system \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $T$\end{document} of primitive recursive functionals of finite type by a purely recursion-theoretic analysis of Schütte's 1977 exposition of Howard's weak normalization proof for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $T$\end{document}. By using collapsing techniques from Pohlers' local predicativity approach to proof theory and based on the Buchholz-Cichon and Weiermann 1994 approach to subrecursive hierarchies (...) we define a collapsing f unction\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} ${\cal D}:T\to \omega$\end{document} so that for (closed) terms \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $a,b$\end{document} of Gödel's \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $T$\end{document} we have: If \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $a$\end{document} reduces to \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $b$\end{document} then \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $\omega>{\cal D}(a)>{\cal D}(b).$\end{document} By one uniform proof we obtain as corollaries: A derivation lengths classification for functionals in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $T$\end{document}, hence new proof of strongly uniform termination of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $T$\end{document}. A new proof of the Kreisel's classific ation of the number-theoretic functions which can be defined in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $T$\end{document}, hence a classification of the provably total functions of Peano Arithmetic. A new proof of Tait's results on weak normalization for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $T$\end{document}. A new proof of Troelstra's result on strong normalization for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $T$\end{document}. Additionally, a slow growing analysis of Gödel's \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $T$\end{document} is obtained via Girard's hierarchy comparison theorem. This analyis yields a contribution to two open pro blems posed by Girard in part two of his book on proof theory. For the sake of completeness we also mention the Howard Schütte bound on derivation lengths for the simple typed \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $\lambda$\end{document}-calculus. (shrink)

An Introduction to Proof Theory provides an accessible introduction to the theory of proofs, with details of proofs worked out and examples and exercises to aid the reader's understanding. It also serves as a companion to reading the original pathbreaking articles by Gerhard Gentzen. The first half covers topics in structural proof theory, including the Gödel-Gentzen translation of classical into intuitionistic logic, natural deduction and the normalization theorems, the sequent calculus, including cut-elimination and mid-sequent theorems, and various applications (...) of these results. The second half examines ordinal proof theory, specifically Gentzen's consistency proof for first-order Peano Arithmetic. The theory of ordinal notations and other elements of ordinal theory are developed from scratch, and no knowledge of set theory is presumed. The proof methods needed to establish proof-theoretic results, especially proof by induction, are introduced in stages throughout the text. Mancosu, Galvan, and Zach's introduction will provide a solid foundation for those looking to understand this central area of mathematical logic and the philosophy of mathematics. (shrink)

In this paper we apply proof theoretic methods used for classical systems in order to obtain upper bounds for systems in partial logic. We focus on a truth predicate interpreted in a Kripke style way via strong Kleene; whereas the aim is to connect harmoniously the partial version of Kripke–Feferman with its intended semantics. The method we apply is based on infinitary proof systems containing an ω-rule.

We show that every MALL proof-structure [9] satisfies the property of softness, originally a categorical notion introduced by Joyal. Furthermore, we show that the notion of hereditary softness precisely captures Girard’s algebraic restriction of the technical condition on proof-structures. Relying on this characterization, we prove a MALL+Mix sequentialization theorem by a proof-theoretical method, using Girard’s notion of jump. Our MALL+Mix correctness criterion subsumes the Danos/Fleury-Retoré criterion [6] for MLL+Mix.

This chapter focuses on the development of Gerhard Gentzen's structural proof theory and its connections with intuitionism. The latter is important in proof theory for several reasons. First, the methods of Hilbert's old proof theory were limited to the “finitistic” ones. These methods proved to be insufficient, and they were extended by infinitistic principles that were still intuitionistically meaningful. It is a general tendency in proof theory to try to use weak principles. A second (...) reason for the importance of intuitionism for proof theory is that the proof-theoretical study of intuitionistic logic has become a prominent part of logic, with applications in computer science. (shrink)

We use model-theoretic methods described in [3] to obtain ordinal analyses of a number of theories of first- and second-order arithmetic, whose proof-theoretic ordinals are less than or equal to Γ0.

A natural deduction formulation is given for the intermediate logic called MH by Gabbay in [4]. Proof-theoretic methods are used to show that every deduction can be normalized, that MH is the weakest intermediate logic for which the Glivenko theorem holds, and that the Craig-Lyndon interpolation theorem holds for it.

This paper deals with a collection of concerns that, over a period of time, led the author away from the Routley–Meyer semantics, and towards proof- theoretic approaches to relevant logics, and indeed to the weak relevant logic MC of meaning containment.

Proofs and countermodels are the two sides of completeness proofs, but, in general, failure to find one does not automatically give the other. The limitation is encountered also for decidable non-classical logics in traditional completeness proofs based on Henkin’s method of maximal consistent sets of formulas. A method is presented that makes it possible to establish completeness in a direct way: For any given sequent either a proof in the given logical system or a countermodel in the corresponding frame (...) class is found. The method is a synthesis of a generation of calculi with internalized relational semantics, a Tait–Schütte–Takeuti style completeness proof, and procedures to finitize the countermodel construction. Finitizations for intuitionistic propositional logic are obtained through the search for a minimal derivation, through pruning of infinite branches in search trees by means of a suitable syntactic counterpart of semantic filtration, or through a proof-theoretic embedding into an appropriate provability logic. A number of examples illustrates the method, its subtleties, challenges, and present scope. (shrink)

We introduce new proof systems G[β] and G ext[β], which are equivalent to the standard equational calculi of λβ- and λβη- conversion, and which may be qualified as ‘analytic’ because it is possible to establish, by purely proof-theoretical methods, that in both of them the transitivity rule admits effective elimination. This key feature, besides its intrinsic conceptual significance, turns out to provide a common logical background to new and comparatively simple demonstrations—rooted in nice proof-theoretical properties of (...) transitivity-free derivations—of a number of well-known and central results concerning β- and βη-reduction. The latter include the Church–Rosser theorem for both reductions, the Standardization theorem for β- reduction, as well as the Normalization (Leftmost reduction) theorem and the Postponement of η-reduction theorem for βη-reduction. (shrink)

ABSTRACT In this paper we state and prove Herbrand's properties for two modal systems, namely T and S4, thus adapting a previous result obtained for the system D [CIA 86a] to such theories. These properties allow the first order extension?along the lines of [CIA 91]?of the resolution method defined in [ENJ 86] for the corresponding propositional modal systems. In fact, the Herbrand-style procedures proposed here treat quantifiers in a uniform way, that suggests the definition of a restricted notion of unification (...) for quantifier free modal formulae obtained by means of a Skolem-like procedure. The differences among the three systems considered, D, T and S4, are taken into account by preliminary propositional transformations of the formulae, in such a way that the problem of quantifier elimination can be treated within the same pattern in the different systems. The methods employed to prove the fundamental theorems are purely proof-theoretical and make use of the formalization of the considered modal theories in terms of sequent calculi, where the cut elimination theorem holds. (shrink)

Paul Cohen’s method of forcing, together with Saul Kripke’s related semantics for modal and intuitionistic logic, has had profound effects on a number of branches of mathematical logic, from set theory and model theory to constructive and categorical logic. Here, I argue that forcing also has a place in traditional Hilbert-style proof theory, where the goal is to formalize portions of ordinary mathematics in restricted axiomatic theories, and study those theories in constructive or syntactic terms. I will discuss the (...) aspects of forcing that are useful in this respect, and some sample applications. The latter include ways of obtaining conservation results for classical and intuitionistic theories, interpreting classical theories in constructive ones, and constructivizing model-theoretic arguments. (shrink)

In 1969, De Jongh proved the “maximality” of a fragment of intuitionistic predicate calculus forHA. Leivant strengthened the theorem in 1975, using proof-theoretical tools (normalisation of infinitary sequent calculi). By a refinement of De Jongh's original method (using Beth models instead of Kripke models and sheafs of partial combinatory algebras), a semantical proof is given of a result that is almost as good as Leivant's. Furthermore, it is shown thatHA can be extended to Higher Order Heyting Arithmetic+all trueΠ (...) 2 0 -sentences + transfinite induction over primitive recursive well-orderings. As a corollary of the proof, maximality of intuitionistic predicate calculus is established wrt. an abstract realisability notion defined over a suitable expansion ofHA. (shrink)

In this paper a proof of the normal form theorem for the closed terms of Girard's system F is given by using a computability method à la Tait. It is worth noting that most of the standard consequences of the normal form theorem can be obtained using this version of the theorem as well. From the proof-theoretical point of view the interest of the proof is that the definition of computable derivation here used does not seem to (...) be well founded. MSC: 03F05, 03B15. (shrink)

At the turn of the nineteenth century, mathematics exhibited a style of argumentation that was more explicitly computational than is common today. Over the course of the century, the introduction of abstract algebraic methods helped unify developments in analysis, number theory, geometry, and the theory of equations; and work by mathematicians like Dedekind, Cantor, and Hilbert towards the end of the century introduced set-theoretic language and infinitary methods that served to downplay or suppress computational content. This shift (...) in emphasis away from calculation gave rise to concerns as to whether such methods were meaningful, or appropriate to mathematics. The discovery of paradoxes stemming from overly naive use of set-theoretic language and methods led to even more pressing concerns as to whether the modern methods were even consistent. This led to heated debates in the early twentieth century and what is sometimes called the “crisis of foundations.” In lectures presented in 1922, David Hilbert launched his Beweistheorie, or Proof Theory, which aimed to justify the use of modern methods and settle the problem of foundations once and for all. This, Hilbert argued, could be achieved as follows. (shrink)

Inspired by Pohlers' local predicativity approach to Pure Proof Theory and Howard's ordinal analysis of bar recursion of type zero we present a short, technically smooth and constructive strong normalization proof for Gödel's system T of primitive recursive functionals of finite types by constructing an ε 0 -recursive function [] 0 : T → ω so that a reduces to b implies [a] $_0 > [b]_0$ . The construction of [] 0 is based on a careful analysis of (...) the Howard-Schütte treatment of Gödel's T and utilizes the collapsing function ψ: ε 0 → ω which has been developed by the author for a local predicativity style proof-theoretic analysis of PA. The construction of [] 0 is also crucially based on ideas developed in the 1995 paper "A proof of strongly uniform termination for Gödel's T by the method of local predicativity" by the author. The results on complexity bounds for the fragments of T which are obtained in this paper strengthen considerably the results of the 1995 paper. Indeed, for given n let T n be the subsystem of T in which the recursors have type level less than or equal to n+2. (By definition, case distinction functionals for every type are also contained in T n .) As a corollary of the main theorem of this paper we obtain (reobtain?) optimal bounds for the T n -derivation lengths in terms of ω n+2 -descent recursive functions. The derivation lengths of type one functionals from T n (hence also their computational complexities) are classified optimally in terms of $ -descent recursive functions. In particular we obtain (reobtain?) that the derivation lengths function of a type one functional a ∈ T 0 is primitive recursive, thus any type one functional a in T 0 defines a primitive recursive function. Similarly we also obtain (reobtain?) a full classification of T 1 in terms of multiple recursion. As proof-theoretic corollaries we reobtain the classification of the IΣ n+1 -provably recursive functions. Taking advantage from our finitistic and constructive treatment of the terms of Gödel's T we reobtain additionally (without employing continuous cut elimination techniques) that PRA + PRWO ( $\varepsilon_0) \vdash \Pi^0_2$ - Refl(PA) and PRA + PRWO ( $\omega_{n+2}) \vdash \Pi^0_2$ - Refl(I Σ n+1 ), hence PRA + PRWO( $\epsilon_0) \vdash$ Con(PA) and PRA + PRWO( $\omega_{n+2}) \vdash$ Con(IΣ n+1 ). For programmatic reasons we outline in the introduction a vision of how to apply a certain type of infinitary methods to questions of finitary mathematics and recursion theory. We also indicate some connections between ordinals, term rewriting, recursion theory and computational complexity. (shrink)

Along the same line as that in Ono (Ann Pure Appl Logic 161:246–250, 2009), a proof-theoretic approach to Glivenko theorems is developed here for substructural predicate logics relative not only to classical predicate logic but also to arbitrary involutive substructural predicate logics over intuitionistic linear predicate logic without exponentials QFLe. It is shown that there exists the weakest logic over QFLe among substructural predicate logics for which the Glivenko theorem holds. Negative translations of substructural predicate logics are studied (...) by using the same approach. First, a negative translation, called extended Kuroda translation is introduced. Then a translation result of an arbitrary involutive substructural predicate logics over QFLe is shown, and the existence of the weakest logic is proved among such logics for which the extended Kuroda translation works. They are obtained by a slight modification of the proof of the Glivenko theorem. Relations of our extended Kuroda translation with other standard negative translations will be discussed. Lastly, algebraic aspects of these results will be mentioned briefly. In this way, a clear and comprehensive understanding of Glivenko theorems and negative translations will be obtained from a substructural viewpoint. (shrink)

This paper is a logical study of valid uses of symmetry in deductive reasoning, of what underlying principles make some appeals to symmetry legitimate but others illegitimate. The issue is first motivated informally. A framework is then given covering a fairly broad range of symmetry arguments, and the formulation of symmetry provided is shown to be a valid principle of reasoning, as is a slightly stronger principle of reasoning, one that is shown to be in some sense as strong as (...) possible. The relationship between symmetry and isomorphism is discussed, and finally the framework is extended to a more general model-theoretic setting. (shrink)

Roughly, a proof of a theorem, is “pure” if it draws only on what is “close” or “intrinsic” to that theorem. Mathematicians employ a variety of terms to identify pure proofs, saying that a pure proof is one that avoids what is “extrinsic,” “extraneous,” “distant,” “remote,” “alien,” or “foreign” to the problem or theorem under investigation. In the background of these attributions is the view that there is a distance measure (or a variety of such measures) between mathematical (...) statements and proofs. Mathematicians have paid little attention to specifying such distance measures precisely because in practice certain methods of proof have seemed self- evidently impure by design: think for instance of analytic geometry and analytic number theory. By contrast, mathematicians have paid considerable attention to whether such impurities are a good thing or to be avoided, and some have claimed that they are valuable because generally impure proofs are simpler than pure proofs. This article is an investigation of this claim, formulated more precisely by proof- theoretic means. After assembling evidence from proof theory that may be thought to support this claim, we will argue that on the contrary this evidence does not support the claim. (shrink)

At the turn of the nineteenth century, mathematics exhibited a style of argumentation that was more explicitly computational than is common today. Over the course of the century, the introduction of abstract algebraic methods helped unify developments in analysis, number theory, geometry, and the theory of equations; and work by mathematicians like Dedekind, Cantor, and Hilbert towards the end of the century introduced set-theoretic language and infinitary methods that served to downplay or suppress computational content. This shift (...) in emphasis away from calculation gave rise to concerns as to whether such methods were meaningful, or appropriate to mathematics. The discovery of paradoxes stemming from overly naive use of set-theoretic language and methods led to even more pressing concerns as to whether the modern methods were even consistent. This led to heated debates in the early twentieth century and what is sometimes called the “crisis of foundations.” In lectures presented in 1922, David Hilbert launched his Beweistheorie, or Proof Theory, which aimed to justify the use of modern methods and settle the problem of foundations once and for all. This, Hilbert argued, could be achieved as follows. (shrink)