The Justification of the Logical Laws Revisited

The proof-theoretic analysis of logical semantics undermines the received view of proof theory as being concerned with symbols devoid of meaning, and of model theory as the sole branch of logical theory entitled to access the realm of semantics. The basic tenet of proof-theoretic semantics is that meaning is given by some rules of proofs, in terms of which all logical laws can be justified and the notion of logical consequence explained. In this paper an attempt will be made to unravel some aspects of the issue and to show that this justification as it stands is untenable, for it relies on a formalistic conception of meaning and fails to recognise the fundamental distinction between semantic definitions and rules of inference. It is also briefly suggested that the profound connection between meaning and proofs should be approached by first reconsidering our very notion of proof.     

The original sin of proof-theoretic semantics

Proof-theoretic semantics is an alternative to model-theoretic semantics. It aims at explaining the meaning of the logical constants in terms of the inference rules that govern their behaviour in proofs. We argue that this must be construed as the task of explaining these meanings relative to a logic, i.e., to a consequence relation. Alas, there is no agreed set of properties that a relation must have in order to qualify as a consequence relation. Moreover, the association of a consequence relation to a logical calculus is not as straightforward as it may seem. We show that these facts are problematic for the proof-theoretic project but the problems can be solved. Our thesis is that the consequence relation relevant for proof-theoretic semantics is the one given by the sequent-to-sequent derivability relation in Gentzen systems.

     

What is the Meaning of Proofs?

Journal of Philosophical Logic - The origins of proof-theoretic semantics lie in the question of what constitutes the meaning of the logical connectives and its response: the rules of inference...     

The categorical and the hypothetical: a critique of some fundamental assumptions of standard semantics

The hypothetical notion of consequence is normally understood as the transmission of a categorical notion from premisses to conclusion. In model-theoretic semantics this categorical notion is ??truth??, in standard proof-theoretic semantics it is ??canonical provability??. Three underlying dogmas, (I) the priority of the categorical over the hypothetical, (II) the transmission view of consequence, and (III) the identification of consequence and correctness of inference are criticized from an alternative view of proof-theoretic semantics. It is argued that consequence is a basic semantical concept which is directly governed by elementary reasoning principles such as definitional closure and definitional reflection, and not reduced to a categorical concept. This understanding of consequence allows in particular to deal with non-wellfounded phenomena as they arise from circular definitions.     

一类命题逻辑的一般弱框架择类语义

命题逻辑的一般弱框架择类语义是相干邻域语义的变形,其特点是：采用择类运算来刻画逻辑常项;语义运算与逻辑联结词之间有清晰的对应关系,可以从整体上处理一类逻辑,具有普适性。本文将这种语义用于一类B、C、K、W命题逻辑,包括相干逻辑R及其线性片段、直觉主义逻辑及其BCK片段等,并借助典范框架和典范赋值,证明了这些逻辑系统的可靠性和完全性。     

Proof-Theoretic Semantics,a Problem with Negation and Prospects for Modality

This paper discusses proof-theoretic semantics, the project of specifying the meanings of the logical constants in terms of rules of inference governing them. I concentrate on Michael Dummett’s and Dag Prawitz’ philosophical motivations and give precise characterisations of the crucial notions of harmony and stability, placed in the context of proving normalisation results in systems of natural deduction. I point out a problem for defining the meaning of negation in this framework and prospects for an account of the meanings of modal operators in terms of rules of inference.     

Validity Concepts in Proof-theoretic Semantics

The standard approach to what I call “proof-theoretic semantics”, which is mainly due to Dummett and Prawitz, attempts to give a semantics of proofs by defining what counts as a valid proof. After a discussion of the general aims of proof-theoretic semantics, this paper investigates in detail various notions of proof-theoretic validity and offers certain improvements of the definitions given by Prawitz. Particular emphasis is placed on the relationship between semantic validity concepts and validity concepts used in normalization theory. It is argued that these two sorts of concepts must be kept strictly apart.     

Proof-Theoretic Semantics for Subsentential Phrases

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.     

A Constructive Type-Theoretical Formalism for the Interpretation of Subatomically Sensitive Natural Language Constructions

The analysis of atomic sentences and their subatomic components poses a special problem for proof-theoretic approaches to natural language semantics, as it is far from clear how their semantics could be explained by means of proofs rather than denotations. The paper develops a proof-theoretic semantics for a fragment of English within a type-theoretical formalism that combines subatomic systems for natural deduction [20] with constructive (or Martin-L?f) type theory [8, 9] by stating rules for the formation, introduction, elimination and equality of atomic propositions understood as types (or sets) of subatomic proof-objects. The formalism is extended with dependent types to admit an interpretation of non-atomic sentences. The paper concludes with applications to natural language including internally nested proper names, anaphoric pronouns, simple identity sentences, and intensional transitive verbs.     

Proof,Meaning and Paradox: Some Remarks

Topoi - In the present paper, the Fregean conception of proof-theoretic semantics that I developed elsewhere will be revised so as to better reflect the different roles played by open and closed...     

Proof-theoretic pluralism

Starting from a proof-theoretic perspective, where meaning is determined by the inference rules governing logical operators, in this paper we primarily aim at developing a proof-theoretic alternative to the model-theoretic meaning-invariant logical pluralism discussed in Beall and Restall (Logical pluralism, Oxford University Press, Oxford, 2006). We will also outline how this framework can be easily extended to include a form of meaning-variant logical pluralism. In this respect, the framework developed in this paper—which we label two-level proof-theoretic pluralism—is much broader in scope than the one discussed in Beall and Restall’s book.

     

Matrix representations for structural strengthenings of a propositional logic

The aim of this paper is to show that the operations of forming direct products and submatrices suffice to construct exhaustive semantics for all structural strengthenings of the consequence determined by a given class of logical matrices.     

Classical harmony: Rules of inference and the meaning of the logical constants

The thesis that, in a system of natural deduction, the meaning of a logical constant is given by some or all of its introduction and elimination rules has been developed recently in the work of Dummett, Prawitz, Tennant, and others, by the addition of harmony constraints. Introduction and elimination rules for a logical constant must be in harmony. By deploying harmony constraints, these authors have arrived at logics no stronger than intuitionist propositional logic. Classical logic, they maintain, cannot be justified from this proof-theoretic perspective. This paper argues that, while classical logic can be formulated so as to satisfy a number of harmony constraints, the meanings of the standard logical constants cannot all be given by their introduction and/or elimination rules; negation, in particular, comes under close scrutiny.     

On Reduction Rules,Meaning-as-use,and Proof-theoretic Semantics

The intention here is that of giving a formal underpinning to the idea of ‘meaning-is-use’ which, even if based on proofs, it is rather different from proof-theoretic semantics as in the Dummett–Prawitz tradition. Instead, it is based on the idea that the meaning of logical constants are given by the explanation of immediate consequences, which in formalistic terms means the effect of elimination rules on the result of introduction rules, i.e. the so-called reduction rules. For that we suggest an extension to the Curry– Howard interpretation which draws on the idea of labelled deduction, and brings back Frege’s device of variable-abstraction to operate on the labels (i.e., proof-terms) alongside formulas of predicate logic. Presented by Heinrich Wansing     

Proof-Theoretic Semantics, Self-Contradiction, and the Format of Deductive Reasoning

From the point of view of proof-theoretic semantics, it is argued that the sequent calculus with introduction rules on the assertion and on the assumption side represents deductive reasoning more appropriately than natural deduction. In taking consequence to be conceptually prior to truth, it can cope with non-well-founded phenomena such as contradictory reasoning. The fact that, in its typed variant, the sequent calculus has an explicit and separable substitution schema in form of the cut rule, is seen as a crucial advantage over natural deduction, where substitution is built into the general framework.     

Harmonious logic: Craig’s interpolation theorem and its descendants

Though deceptively simple and plausible on the face of it, Craig’s interpolation theorem (published 50 years ago) has proved to be a central logical property that has been used to reveal a deep harmony between the syntax and semantics of first order logic. Craig’s theorem was generalized soon after by Lyndon, with application to the characterization of first order properties preserved under homomorphism. After retracing the early history, this article is mainly devoted to a survey of subsequent generalizations and applications, especially of many-sorted interpolation theorems. Attention is also paid tomethodological considerations, since the Craig theorem and its generalizations were initially obtained by proof-theoretic arguments while most of the applications are model-theoretic in nature. The article concludes with the role of the interpolation property in the quest for “reasonable” logics extending first-order logic within the framework of abstract model theory. For Bill Craig, with great appreciation for his fundamental contributions to our subject, and for his perennially open, welcoming attitude and fine personality that enhances every encounter.     

Formal Ontologies and Coherent Spaces

The paper contains a short summary – oriented by a logical point of view – of a joint work on Formal Ontologies. We shall show how Formal Ontologies correspond to Coherent Spaces, and operations on Formal Ontologies correspond to operations on corresponding Coherent Spaces. So, we are offering a new way to establish the semantics of Formal Ontologies. Surely, we are giving a contribution towards a geometrical treatment of Formal Ontologies (as decidable organizations of digital data).     

Hilbert's epsilon as an operator of indefinite committed choice

Hilbert and Bernays avoided overspecification of Hilbert's ε-operator. They axiomatized only what was relevant for their proof-theoretic investigations. Semantically, this left the ε-operator underspecified. After briefly reviewing the literature on semantics of Hilbert's epsilon operator, we propose a new semantics with the following features: We avoid overspecification (such as right-uniqueness), but admit indefinite choice, committed choice, and classical logics. Moreover, our semantics for the ε simplifies proof search and is natural in the sense that it mirrors some cases of referential interpretation of indefinite articles in natural language.     

RULE-CIRCULARITY AND THE JUSTIFICATION OF DEDUCTION

I examine Paul Boghossian's recent attempt to argue for scepticism about logical rules. I argue that certain rule-and proof-theoretic considerations can avert such scepticism. Boghossian's 'Tonk Argument' seeks to justify the rule of tonk-introduction by using the rule itself. The argument is subjected here to more detailed proof-theoretic scrutiny than Boghossian undertook. Its sole axiom, the so-called Meaning Postulate for tonk, is shown to be false or devoid of content. It is also shown that the rules of Disquotation and of Semantic Ascent cannot be derived for sentences with tonk dominant. These considerations deprive Boghossian's scepticism of its support.     

Why Conclusions Should Remain Single

This paper argues that logical inferentialists should reject multiple-conclusion logics. Logical inferentialism is the position that the meanings of the logical constants are determined by the rules of inference they obey. As such, logical inferentialism requires a proof-theoretic framework within which to operate. However, in order to fulfil its semantic duties, a deductive system has to be suitably connected to our inferential practices. I argue that, contrary to an established tradition, multiple-conclusion systems are ill-suited for this purpose because they fail to provide a ‘natural’ representation of our ordinary modes of inference. Moreover, the two most plausible attempts at bringing multiple conclusions into line with our ordinary forms of reasoning, the disjunctive reading and the bilateralist denial interpretation, are unacceptable by inferentialist standards.