Thomas Piecha

Advances in Proof-Theoretic Semantics

This volume is the first ever collection devoted to the field of proof-theoretic semantics. Contributions address topics including the systematics of introduction and elimination rules and proofs of normalization, the categorial characterization of deductions, the relation between Heytings and Gentzens approaches to meaning, knowability paradoxes, proof-theoretic foundations of set theory, Dummetts justification of logical laws, Kreisels theory of constructions, paradoxical reasoning, and the defence of model theory. The field of proof-theoretic semantics has existed for almost 50 years, but the term itself was proposed by Schroeder-Heister in the 1980s. Proof-theoretic semantics explains the meaning of linguistic expressions in general and of logical constants in particular in terms of the notion of proof. This volume emerges from presentations at the Second International Conference on Proof-Theoretic Semantics in Tbingen in 2013, where contributing authors were asked to provide a self-contained description and analysis of a significant research question in this area. The contributions are representative of the field and should be of interest to logicians, philosophers, and mathematicians alike.

Failure of Completeness in Proof-Theoretic Semantics

Several proof-theoretic notions of validity have been proposed in the literature, for which completeness of intuitionistic logic has been conjectured. We define validity for intuitionistic propositional logic in a way which is common to many of these notions, emphasizing that an appropriate notion of validity must be closed under substitution. In this definition we consider atomic systems whose rules are not only production rules, but may include rules that allow one to discharge assumptions. Our central result shows that Harrop’s rule is valid under substitution, which refutes the completeness conjecture for intuitionistic logic.

Constructive semantics, admissibility of rules and the validity of Peirce's law

In his approach to proof-theoretic semantics, Sandqvist claims to provide a justification of classical logic without using the principle of bivalence. Following ideas by Prawitz, his semantics relies on the idea that logical systems extend atomic systems, so-called “bases”, with respect to which the validity of logically complex formulas is defined. We relate this approach to admissibility-based semantics and show that the latter significantly differs from the former. We also relate it to semantics based on the notion of construction, in which case the results obtained are essentially the same as Sandqvist’s. We argue that the form of rules admitted in atomic bases determines which logical rules are validated, as is the fact of whether bases are conceived as information states, which can be monotonely extended, or as non-extensible inductive definitions. This shows that the format of atomic bases is a highly relevant issue in proof-theoretic semantics.

INVERSION BY DEFINITIONAL REFLECTION AND THE ADMISSIBILITY OF LOGICAL RULES

at T¨ ubingen Abstract. The inversion principle for logical rules expresses a relationship between introduction and elimination rules for logical constants. Hallnand Schroeder-Heister (1990/91) proposed the principle of definitional reflection, which embodies basic ideas of inversion in the more general context of clausal definitions. For the context of admissibility statements, this has been further elaborated by Schroeder-Heister (2007). Using the framework of definitional reflection and its admis- sibility interpretation, we show that, in the sequent calculus of minimal propositional logic, the left introduction rules are admissible when the right introduction rules are taken as the definitions of the logical constants and vice versa. This generalises the well-known relationship between introduction and elimination rules in natural deduction to the framework of the sequent calculus.

Completeness in Proof-Theoretic Semantics

We give an overview of completeness and incompleteness results within proof-theoretic semantics. Completeness of intuitionistic first-order logic for certain notions of validity in proof-theoretic semantics has been conjectured by Prawitz. For the kind of semantics proposed by him, this conjecture is still undecided. For certain variants of proof-theoretic semantics the completeness question is settled, including a positive result for classical logic. For intuitionistic logic there are positive as well as negative completeness results, depending on which variant of semantics is considered. Further results have been obtained for certain fragments of first-order languages.

Atomic Systems in Proof-Theoretic Semantics: Two Approaches

Atomic systems are systems of rules containing only atomic formulas. In proof-theoretic semantics for minimal and intuitionistic logic they are used as the base case in an inductive definition of validity. We compare two different approaches to atomic systems. The first approach is compatible with an interpretation of atomic systems as representations of states of knowledge. The second takes atomic systems to be definitions of atomic formulas. The two views lead to different notions of derivability for atomic formulas, and consequently to different notions of proof-theoretic validity. In the first approach, validity is stable in the sense that for atomic formulas logical consequence and derivability coincide for any given atomic system. In the second approach this is not the case. This indicates that atomic systems as definitions, which determine the meaning of atomic sentences, might not be the proper basis for proof-theoretic validity, or conversely, that standard notions of proof-theoretic validity are not appropriate for definitional rule systems.

Popper's Notion of Duality and His Theory of Negations

Karl Popper developed a theory of deductive logic in the late 1940s. In his approach, logic is a metalinguistic theory of deducibility relations that are based on certain purely structural rules. Logical constants are then characterized in terms of deducibility relations. Characterizations of this kind are also called inferential definitions by Popper. In this paper, we expound his theory and elaborate some of his ideas and results that in some cases were only sketched by him. Our focus is on Poppers notion of duality, his theory of modalities, and his treatment of different kinds of negation. This allows us to show how his works on logic anticipate some later developments and discussions in philosophical logic, pertaining to trivializing (-like) connectives, the duality of logical constants, dual-intuitionistic logic, the (non-)conservativeness of language extensions, the existence of a bi-intuitionistic logic, the non-logicality of minimal negation, and to the problem of logicality in general.

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 Tübingen.

Advances in Proof-Theoretic Semantics: Introduction

As documented by the papers in this volume, which mostly result from the second conference on proof-theoretic semantics in Tubingen 2013, proof-theoretic semantics has advanced to a well-established subject in philosophical logic.

Dialogical Logic for Definitonal Reasoning and Implications as Rules

In dialogical logic, the logical constants are given a game-theoretic interpretation (see Lorenzen [15, 16], cf. Lorenz [11, 12, 13], Lorenzen and Lorenz [18] and Lorenzen [17]; for an overview see Keiff [10] and Piecha [20]). Dialogues are two-player games between a proponent and an opponent, where each of the two players can either attack claims made by the other player or defend their own claims. For example, an implication A→ B can be attacked by claiming A and is defended by claiming B. This means that in order to have a winning strategy for A→ B, the proponent must be able to argue successfully for B depending on what the opponent can put forward in defense of A. The logical constant of implication has thus been given a certain game-theoretic or dialogical interpretation, and corresponding dialogical interpretations can be given for the other logical constants as well. Here this approach will be extended in two directions: First, we want to make it possible that also definitions can be treated dialogically (cf. [19]). A definition is understood as a rule system, which specifies the meaning of atomic assertions, that is, of assertions, which do not contain any logical constant. The rules are like predicator rules [9], rules of an atomic production system (as e.g. in operative logic [14] or in logic programming [6]) or like the rules in an inductive definition (cf. Aczel [1]). Following the terminology of logic programming, such definitional rules for atomic assertions will also be called ‘clauses’. The ∗Universität Tübingen, Wilhelm-Schickard-Institut für Informatik, Sand 13, 72076 Tübingen, Germany, thomas.piecha@uni-tuebingen.de, psh@uni-tuebingen.de. This work was supported by the French-German ANR-DFG project “Hypothetical Reasoning: Its Proof-Theoretic Analysis” (HYPOTHESES), DFG grant Schr 275/16-2.