General Proof Theory Research Articles

Kant’s Die falsche Spitzfindigkeit and Proof-theoretic Semantics

According to Schroeder-Heister 2018, proof-theoretic semantics is ‘an alternative to truth-condition semantics. It is based on the fundamental assumption that the central notion in terms of which meanings are assigned to certain expressions of our language, in particular to logical constants, is that of proof rather than truth. In this sense proof-theoretic semantics is semantics in terms of proof. Proof-theoretic semantics also means the semantics of proofs, i.e. the semantics of entities which describe how we arrive at certain assertions given certain assumptions.' This text advocates that proof-theoretic semantics, as described above, was an approach to semantical issues in logic that appeared in mainstream philosophical literature at least a couple of centuries before the tradition of general proof theory (cf. Prawitz 1971) came into being. This is done by means of an interpretive analysis of Kant’s 1762 essay Die falsche Spitzfindigkeit der vier syllogistischen Figuren, in which two theses are argued: first, that the issue of justifying the logical validity of inferences is approached by Kant in this text, in a strong sense, in terms of proofs; and second, that the purpose of his effort is to establish a point concerning the semantical equivalence between certain distinct valid inferences.

FACETS OF INTENSIONALITY

The goal of this paper is to introduce the reader to the distinction between intensional and extensional as a distinction between different approaches to meaning. We will argue that despite the common belief, intensional aspects of mathematical notions can be, and in fact have been successfully described in mathematics. One that is for us particularly interesting is the notion of deduction as depicted in general proof theory. Our considerations result in defending a) the importance of a rule-based semantical approach and b) the position according to which non-reductive and somewhat circular explanations play an essential role in describing intensionality in mathematics.

Composition of Deductions within the Propositions-As-Types Paradigm

Kosta Dosen argued in his papers Inferential Semantics Dosen (in Inferential semantics, Springer, Berlin 2015) and On the Paths of Categories Dosen (in On the paths of categories, Springer, Cham 2016) that the propositions-as-types paradigm is less suited for general proof theory because—unlike proof theory based on category theory—it emphasizes categorical proofs over hypothetical inferences. One specific instance of this, Dosen points out, is that the Curry–Howard isomorphism makes the associativity of deduction composition invisible. We will show that this is not necessarily the case.

General Proof Theory: Introduction

This special issue on general proof theory collects papers resulting from the conference on general proof theory held in November 2015 in Tubingen.

The Fundamental Problem of General Proof Theory

I see the question what it is that makes an inference valid and thereby gives a proof its epistemic power as the most fundamental problem of general proof theory. It has been surprisingly neglected in logic and philosophy of mathematics with two exceptions: Gentzen’s remarks about what justifies the rules of his system of natural deduction and proposals in the intuitionistic tradition about what a proof is. They are reviewed in the paper and I discuss to what extent they succeed in answering what a proof is. Gentzen’s ideas are shown to give rise to a new notion of valid argument. At the end of the paper I summarize and briefly discuss an approach to the problem that I have proposed earlier.

Gödel on Deduction

This is an examination, a commentary, of links between some philosophical views ascribed to Godel and general proof theory. In these views deduction is of central concern not only in predicate logic, but in set theory too, understood from an infinitistic ideal perspective. It is inquired whether this centrality of deduction could also be kept in the intensional logic of concepts whose building Godel seems to have taken as the main task of logic for the future.

Reprint of: A more general general proof theory

In this paper it is suggested to generalize our understanding of general (structural) proof theory and to consider it as a general theory of two kinds of derivations, namely proofs and dual proofs. The proposal is substantiated by (i) considerations on assertion, denial, and bi-lateralism, (ii) remarks on compositionality in proof-theoretic semantics, and (iii) comments on falsification and co-implication. The main formal result of the paper is a normal form theorem for the natural deduction proof system N2Int of the bi-intuitionistic logic 2Int. The proof makes use of the faithful embedding of 2Int into intuitionistic logic with respect to validity and shows that conversions of dual proofs can be sidestepped.

A more general general proof theory

In this paper it is suggested to generalize our understanding of general (structural) proof theory and to consider it as a general theory of two kinds of derivations, namely proofs and dual proofs. The proposal is substantiated by (i) considerations on assertion, denial, and bi-lateralism, (ii) remarks on compositionality in proof-theoretic semantics, and (iii) comments on falsification and co-implication. The main formal result of the paper is a normal form theorem for the natural deduction proof system N2Int of the bi-intuitionistic logic 2Int. The proof makes use of the faithful embedding of 2Int into intuitionistic logic with respect to validity and shows that conversions of dual proofs can be sidestepped.

Gödel’s Natural Deduction

This is a companion to a paper by the authors entitled “Godel on deduction”, which examined the links between some philosophical views ascribed to Godel and general proof theory. When writing that other paper, the authors were not acquainted with a system of natural deduction that Godel presented with the help of Gentzen’s sequents, which amounts to Jaśkowski’s natural deduction system of 1934, and which may be found in Godel’s unpublished notes for the elementary logic course he gave in 1939 at the University of Notre Dame. Here one finds a presentation of this system of Godel accompanied by a brief reexamination in the light of the notes of some points concerning his interest in sequents made in the preceding paper. This is preceded by a brief summary of Godel’s Notre Dame course, and is followed by comments concerning Godel’s natural deduction system.

Ekman’s Paradox

Prawitz observed that Russell’s paradox in naive set theory yields a derivation of absurdity whose reduction sequence loops. Building on this observation, and based on numerous examples, Tennant claimed that this looping feature, or more generally, the fact that derivations of absurdity do not normalize, is characteristic of the paradoxes. Striking results by Ekman show that looping reduction sequences are already obtained in minimal propositional logic, when certain reduction steps, which are prima facie plausible, are considered in addition to the standard ones. This shows that the notion of reduction is in need of clarification. Referring to the notion of identity of proofs in general proof theory, we argue that reduction steps should not merely remove redundancies, but must respect the identity of proofs. Consequentially, we propose to modify Tennant’s paradoxicality test by basing it on this refined notion of reduction.

GÖDEL’S NOTRE DAME COURSE

AbstractThis is a companion to a paper by the authors entitled “Gödel’s natural deduction,” which presented and made comments about the natural deduction system in Gödel’s unpublished notes for the elementary logic course he gave at the University of Notre Dame in 1939. In that earlier paper, which was itself a companion to a paper that examined the links between some philosophical views ascribed to Gödel and general proof theory, one can find a brief summary of Gödel’s notes for the Notre Dame course. In order to put the earlier paper in proper perspective, a more complete summary of these interesting notes, with comments concerning them, is given here.

Models of Deduction*

In standard model theory, deductions are not the things one models. But in general proof theory, in particular in categorial proof theory, one finds models of deductions, and the purpose here is to motivate a simple example of such models. This will be a model of deductions performed within an abstract context, where we do not have any particular logical constant, but something underlying all logical constants. In this context, deductions are represented by arrows in categories involved in a general adjoint situation. To motivate the notion of adjointness, one of the central notions of category theory, and of mathematics in general, it is first considered how some features of it occur in set-theoretical axioms and in the axioms of the lambda calculus. Next, it is explained how this notion arises in the context of deduction, where it characterizes logical constants. It is shown also how the categorial point of view suggests an analysis of propositional identity. The problem of propositional identity, i.e., the problem of identity of meaning for propositions, is no doubt a philosophical problem, but the spirit of the analysis proposed here will be rather mathematical. Finally, it is considered whether models of deductions can pretend to be a semantics. This question, which as so many questions having to do with meaning brings us to that wall that blocked linguists and philosophers during the whole of the twentieth century, is merely posed. At the very end, there is the example of a geometrical model of adjunction. Without pretending that it is a semantics, it is hoped that this model may prove illuminating and useful.

On the Proof-theoretic Foundation of General Definition Theory

A general definition theory should serve as a foundation for the mathematical study of definitional structures. The central notion of such a theory is a precise explication of the intuitively given notion of a definitional structure. The purpose of this paper is to discuss the proof theory of partial inductive definitions as a foundation for this kind of a more general definition theory. Among the examples discussed is a suggestion for a more abstract definition of lambda-terms (derivations in natural deduction) that could provide a basis for a more systematic definitional approach to general proof theory.

A parameterised hierarchy of argumentation semantics for extended logic programming and its application to the well-founded semantics

Argumentation has proved a useful tool in defining formal semantics for assumption-based reasoning by viewing a proof as a process in which proponents and opponents attack each others arguments by undercuts (attack to an argument's premise) and rebuts (attack to an argument's conclusion). In this paper, we formulate a variety of notions of attack for extended logic programs from combinations of undercuts and rebuts and define a general hierarchy of argumentation semantics parameterised by the notions of attack chosen by proponent and opponent. We prove the equivalence and subset relationships between the semantics and examine some essential properties concerning consistency and the coherence principle, which relates default negation and explicit negation. Most significantly, we place existing semantics put forward in the literature in our hierarchy and identify a particular argumentation semantics for which we prove equivalence to the paraconsistent well-founded semantics with explicit negation, WFSX. Finally, we present a general proof theory, based on dialogue trees, and show that it is sound and complete with respect to the argumentation semantics.

Identity of Proofs Based on Normalization and Generality

AbstractSome thirty years ago, two proposals were made concerning criteria for identity of proofs. Prawitz proposed to analyze identity of proofs in terms of the equivalence relation based on reduction to normal formin natural deduction. Lambek worked on a normalization proposal analogous to Prawitz's, based on reduction to cut-free form in sequent systems, but he also suggested understanding identity of proofs in terms of an equivalence relation based on generality, two derivations having the same generality if after generalizing maximally the rules involved in them they yield the same premises and conclusions up to a renaming of variables. These two proposals proved to be extensionally equivalent only for limited fragments of logic.The normalization proposal stands behind very successful applications of the typed lambda calculus and of category theory in the proof theory of intuitionistic logic. In classical logic, however, it did not fare well.The generality proposal was rather neglected in logic, though related matters were much studied in pure category theory in connection with coherence problems, and there are also links to low-dimensional topology and linear algebra. This proposal seems more promising than the other one for the general proof theory of classical logic.

Resolution and the Origins of Structural Reasoning: Early Proof-Theoretic Ideas of Hertz and Gentzen

In the 1920s, Paul Hertz (1881–1940) developed certain calculi based on structural rules only and established normal form results for proofs. It is shown that he anticipated important techniques and results of general proof theory as well as of resolution theory, if the latter is regarded as a part of structural proof theory. Furthermore, it is shown that Gentzen, in his first paper of 1933, which heavily draws on Hertz, proves a normal form result which corresponds to the completeness of propositional SLD-resolution in logic programming.

Resolution and the Origins of Structural Reasoning: Early Proof-Theoretic Ideas of Hertz and Gentzen

AbstractIn the 1920s, Paul Hertz (1881–1940) developed certain calculi based on structural rules only and established normal form results for proofs. It is shown that he anticipated important techniques and results of general proof theory as well as of resolution theory, if the latter is regarded as a part of structural proof theory. Furthermore, it is shown that Gentzen, in his first paper of 1933, which heavily draws on Hertz, proves a normal form result which corresponds to the completeness of propositional SLD-resolution in logic programming.

On the idea of a general proof theory

La formulation d'un beton autoplacant (BAP) est une operation complexe qui necessite de trouver une bonne combinaison de materiaux compatibles et le dosage convenable de chacun de ses constituants afin d'obtenir une formulation repondant aux proprietes des BAP (fluidite et homogeneite). De multiples approches se sont developpees a travers le monde pour la formulation d'un beton autoplacant. La plupart des methodes de formulation sont concues actuellement de maniere empirique. Ces methodes se basent essentiellement sur des approches diphasiques du beton autoplacant. Elles sont divisees en deux categories : celles qui sont axees sur l'optimisation du volume de pate (sous forme de pate pure ou de mortier) et celles qui sont focalisees sur l'optimisation du squelette granulaire. Notre contribution est d'apprehender la problematique de la formulation des BAP de facon globale en integrant a la fois l'optimisation du volume de pate par l'ajout de filler calcaire avec trois dosages et l'optimisation du squelette granulaire par la methode graphique de Dreux - Gorisse. Les resultats obtenus montrent qu'il existe un volume de pate optimum et que la composition du squelette granulaire par la methode graphique de Dreux - Gorisse est applicable par le choix des fractions granulaires et leurs analyses granulometriques qui permet de mettre en evidence les classes manquantes Mots Cles : BAP - Optimisation - Granulats -Additions - Caracterisation The mix design of a self-compacting concrete (SCC) is a complex operation that requires finding a good combination of compatible materials, and the proper dosage of each of these components in order to obtain a formulation that meets the properties of SCC(fluidity and homogeneity). Multiple approaches have been developed in the world wide for the formulation of a self-compacting concrete. Most methods of formulation are currently designed empirically. These methods are mainly based on the fact that the BAP is a two-phase material. These methods are divided into two categories: those that focus on optimizing the volume of paste (paste form pure or mortar) and those that focus on optimizing the granular skeleton. Our contribution is to apprehend the problem formulation sec in a comprehensive manner, integrating both the optimization of the paste volume by adding the limestone filler with three dosages and optimizing the skeleton by the graphical method Dreux - Gorisse. The results obtained show that to formulate a self-compacting concrete, there is an optimum paste volume. Whereas the composition of the granular skeleton by the graphical method of Dreux - Gorisse is applicable by the choice of granular fractions and particle size analysis which allows highlighting the missing classes Keywords : SCC - Formulation - Optimization -Aggregate -Additions - Characterization