We expose some basic elements of a style of programming supported by functional languages like Haskell by relating them to a coherent set of notions and techniques from Curry’s work in combinatory logic and formal systems, and their algebraic and categorical interpretations. Our account takes the form of a commentary to a simple fragment of Haskell code attempting to isolate the conceptual sources of the linguistic abstractions involved.

Studies of Aristotle’s syllogistic system, since Corcoran’s deductionist interpretation supplanted Łukasiewicz’ axiomaticist interpretation, misrepresent Aristotle’s logic in two important respects. Following Corcoran, they take indirect deduction to occur only once in a deduction discourse; they then obviate the system having a reductio rule. Second, they represent reduction as a deductive process for deriving ‘imperfect’ syllogisms from ‘perfect’ syllogisms to impose an axiomatic interpretation on the logic. Denying that Aristotle's logic admits of a reductio rule results from this misrepresentation of reduction. (...) This discussion shows that the defects imputed to Aristotle's logic, and systems devised to resolve them, result from misunderstanding reduction, which itself results from misapprehending Prior Analytics expressly to identify in a metadiscourse deduction rules and not deductions per se. (shrink)

This collection of papers, celebrating the contributions of Swedish logician Dag Prawitz to Proof Theory, has been assembled from those presented at the Natural Deduction conference organized in Rio de Janeiro to honour his seminal research. Dag Prawitz’s work forms the basis of intuitionistic type theory and his inversion principle constitutes the foundation of most modern accounts of proof-theoretic semantics in Logic, Linguistics and Theoretical Computer Science. The range of contributions includes material on the extension of natural deduction with higher-order (...) rules, as opposed to higher-order connectives, and a paper discussing the application of natural deduction rules to dealing with equality in predicate calculus. The volume continues with a key chapter summarizing work on the extension of the Curry-Howard isomorphism, via methods of category theory that have been successfully applied to linear logic, as well as many other contributions from highly regarded authorities. With an illustrious group of contributors addressing a wealth of topics and applications, this volume is a valuable addition to the libraries of academics in the multiple disciplines whose development has been given added scope by the methodologies supplied by natural deduction. The volume is representative of the rich and varied directions that Prawitz work has inspired in the area of natural deduction. (shrink)

This volume is dedicated to Prof. Dag Prawitz and his outstanding contributions to philosophical and mathematical logic. Prawitz's eminent contributions to structural proof theory, or general proof theory, as he calls it, and inference-based meaning theories have been extremely influential in the development of modern proof theory and anti-realistic semantics. In particular, Prawitz is the main author on natural deduction in addition to Gerhard Gentzen, who defined natural deduction in his PhD thesis published in 1934. The book opens with an (...) introductory paper that surveys Prawitz's numerous contributions to proof theory and proof-theoretic semantics and puts his work into a somewhat broader perspective, both historically and systematically. Chapters include either in-depth studies of certain aspects of Dag Prawitz's work or address open research problems that are concerned with core issues in structural proof theory and range from philosophical essays to papers of a mathematical nature. Investigations into the necessity of thought and the theory of grounds and computational justifications as well as an examination of Prawitz's conception of the validity of inferences in the light of three “dogmas of proof-theoretic semantics” are included. More formal papers deal with the constructive behaviour of fragments of classical logic and fragments of the modal logic S4 among other topics. In addition, there are chapters about inversion principles, normalization of proofs, and the notion of proof-theoretic harmony and other areas of a more mathematical persuasion. Dag Prawitz also writes a chapter in which he explains his current views on the epistemic dimension of proofs and addresses the question why some inferences succeed in conferring evidence on their conclusions when applied to premises for which one already possesses evidence. (shrink)

In this paper we argue that an account of proof-theoretic harmony based on reductions and expansions delivers an inferentialist picture of meaning which should be regarded as intensional, as opposed to other approaches to harmony that will be dubbed extensional. We show how the intensional account applies to any connective whose rules obey the inversion principle first proposed by Prawitz and Schroeder-Heister. In particular, by improving previous formulations of expansions, we solve a problem with quantum-disjunction first posed by Dummett. As (...) recently observed by Schroeder-Heister, however, the specification of an inversion principle cannot yield an exhaustive account of harmony. The reason is that there are more collections of elimination rules than just the one obtained by inversion which we are willing to acknowledge as being in harmony with a given collection of introduction rules. Several authors more or less implicitly suggest that what is common to all alternative harmonious collection of rules is their being interderivable with each other. On the basis of considerations about identity of proofs and formula isomorphism, we show that this is too weak a condition for a given collection of elimination rules to be in harmony with a collection of introduction rules, at least if the intensional picture of meaning we advocate is not to collapse on an extensional one. (shrink)

. The term inversion principle goes back to Lorenzen who coined it in the early 1950s. It was later used by Prawitz and others to describe the symmetric relationship between introduction and elimination inferences in natural deduction, sometimes also called harmony. In dealing with the invertibility of rules of an arbitrary atomic production system, Lorenzen’s inversion principle has a much wider range than Prawitz’s adaptation to natural deduction. It is closely related to definitional reflection, which is a principle for reasoning (...) on the basis of rule-based atomic definitions, proposed by Hallnäs and Schroeder-Heister. After presenting definitional reflection and the inversion principle, it is shown that the inversion principle can be formally derived from definitional reflection, when the latter is viewed as a principle to establish admissibility. Furthermore, the relationship between definitional reflection and the inversion principle is investigated on the background of a universalization principle, called the ω- principle, which allows one to pass from the set of all defined substitution instances of a sequent to the sequent itself. (shrink)

The inversion principle for logical rules expresses a relationship between introduction and elimination rules for logical constants. Hallnäs & Schroeder-Heister 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 . Using the framework of definitional reflection and its admissibility 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 generalizes the well-known relationship between introduction and elimination rules in natural deduction to the framework of the sequent calculus. (shrink)

The inversion principle for logical rules expresses a relationship between introduction and elimination rules for logical constants. Hallnäs & Schroeder-Heister 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. Using the framework of definitional reflection and its admissibility 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 generalizes the well-known relationship between introduction and elimination rules in natural deduction to the framework of the sequent calculus. (shrink)

Inferentialism claims that expressions are meaningful by virtue of rules governing their use. In particular, logical expressions are autonomous if given meaning by their introduction-rules, rules specifying the grounds for assertion of propositions containing them. If the elimination-rules do no more, and no less, than is justified by the introduction-rules, the rules satisfy what Prawitz, following Lorenzen, called an inversion principle. This connection between rules leads to a general form of elimination-rule, and when the rules have this form, they may (...) be said to exhibit “general-elimination” harmony. Ge-harmony ensures that the meaning of a logical expression is clearly visible in its I-rule, and that the I- and E-rules are coherent, in encapsulating the same meaning. However, it does not ensure that the resulting logical system is normalizable, nor that it satisfies the conservative extension property, nor that it is consistent. Thus harmony should not be identified with any of these notions. (shrink)

Inferentialism claims that the rules for the use of an expression express its meaning without any need to invoke meanings or denotations for them. Logical inferentialism endorses inferentialism specically for the logical constants. Harmonic inferentialism, as the term is introduced here, usually but not necessarily a subbranch of logical inferentialism, follows Gentzen in proposing that it is the introduction-rules whch give expressions their meaning and the elimination-rules should accord harmoniously with the meaning so given. It is proposed here that the (...) logical expressions are those which can be given schematic rules that lie in a specific sort of harmony, general-elimination harmony, resulting from applying a certain operation, the ge-procedure, to produce ge-rules in accord with the meaning defined by the I-rules. Griffiths claims that identity cannot be given such rules, concluding that logical inferentialists are committed to ruling identity a non-logical expression. It is shown that the schematic rules for identity given in Read, slightly amended, are indeed ge-harmonious, so confirming that identity is a logical notion. (shrink)

According to logical inferentialists, the meanings of logical expressions are fully determined by the rules for their correct use. Two key proof-theoretic requirements on admissible logical rules, harmony and separability, directly stem from this thesis—requirements, however, that standard single-conclusion and assertion-based formalizations of classical logic provably fail to satisfy :1035–1051, 2011). On the plausible assumption that our logical practice is both single-conclusion and assertion-based, it seemingly follows that classical logic, unlike intuitionistic logic, can’t be accounted for in inferentialist terms. In (...) this paper, I challenge orthodoxy and introduce an assertion-based and single-conclusion formalization of classical propositional logic that is both harmonious and separable. In the framework I propose, classicality emerges as a structural feature of the logic. (shrink)

The aim of this paper is to reconsider several proposals that have been put forward in order to develop a Proof-Theoretical Semantics, from the by now classical neo-verificationist approach provided by D. Prawitz and M. Dummett in the Seventies, to an alternative, more recent approach mainly due to the work of P. Schroeder-Heister and L. Hallnäs, based on clausal definitions. Some other intermediate proposals are very briefly sketched. Particular attention will be given to the role played by the so-called Fundamental (...) Assumption. We claim that whereas, in the neo-verificationist proposal, the condition expressed by that Assumption is necessary to ensure the completeness of the justification procedure ( from the outside , so to speak), within the definitional framework it is a built-in feature of the proposal. The latter approach, therefore, appears as an alternative solution to the problem which prompted the neo-verificationists to introduce the Fundamental Assumption. (shrink)

This paper studies a formalisation of intuitionistic logic by Negri and von Plato which has general introduction and elimination rules. The philosophical importance of the system is expounded. Definitions of ‘maximal formula’, ‘segment’ and ‘maximal segment’ suitable to the system are formulated and corresponding reduction procedures for maximal formulas and permutative reduction procedures for maximal segments given. Alternatives to the main method used are also considered. It is shown that deductions in the system convert into normal form and that deductions (...) in normal form have the subformula property. (shrink)

General-elimination harmony articulates Gentzen’s idea that the elimination-rules are justified if they infer from an assertion no more than can already be inferred from the grounds for making it. Dummett described the rules as not only harmonious but stable if the E-rules allow one to infer no more and no less than the I-rules justify. Pfenning and Davies call the rules locally complete if the E-rules are strong enough to allow one to infer the original judgement. A method is given (...) of generating harmonious general-elimination rules from a collection of I-rules. We show that the general-elimination rules satisfy Pfenning and Davies’ test for local completeness, but question whether that is enough to show that they are stable. Alternative conditions for stability are considered, including equivalence between the introduction- and elimination-meanings of a connective, and recovery of the grounds for assertion, finally generalizing the notion of local completeness to capture Dummett’s notion of stability satisfactorily. We show that the general-elimination rules meet the last of these conditions, and so are indeed not only harmonious but also stable. (shrink)

The paper exposes the relevance of permuting conversions (in natural-deduction systems) to the role of such systems in the theory of meaning known as proof-theoretic semantics, by relating permuting conversion to harmony, hitherto related to normalisation only. This is achieved by showing the connection of permuting conversion to the general notion of canonicity, once applied to arbitrary derivations from open assumption. In the course of exposing the relationship of permuting conversions to harmony, a general definition of the former is proposed, (...) generalising the specific cases of disjunction and existential quantifiers considered in the literature. (shrink)

The paper studies the extension of harmony and stability, major themes in proof-theoretic semantics, from single-conclusion natural-deduction systems to multiple -conclusions natural-deduction, independently of classical logic. An extension of the method of obtaining harmoniously-induced general elimination rules from given introduction rules is suggested, taking into account sub-structurality. Finally, the reductions and expansions of the multiple -conclusions natural-deduction representation of classical logic are formulated.

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. 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. (shrink)

The concept of “necessity of thought” plays a central role in Dag Prawitz’s essay “Logical Consequence from a Constructivist Point of View” (Prawitz 2005). The theme is later developed in various articles devoted to the notion of valid inference (Prawitz, 2009, forthcoming a, forthcoming b). In section 1 I explain how the notion of necessity of thought emerges from Prawitz’s analysis of logical consequence. I try to expound Prawitz’s views concerning the necessity of thought in sections 2, 3 and 4. (...) In sections 5 and 6 I discuss some problems arising with regard to Prawitz’s views. (shrink)

Many prominent writers on the philosophy of logic, including Michael Dummett, Dag Prawitz, Neil Tennant, have held that the introduction and elimination rules of a logical connective must be ‘in harmony ’ if the connective is to possess a sense. This Harmony Thesis has been used to justify the choice of logic: in particular, supposed violations of it by the classical rules for negation have been the basis for arguments for switching from classical to intuitionistic logic. The Thesis has also (...) had an influence on the philosophy of language: some prominent writers in that area, notably Dummett and Robert Brandom, have taken it to be a special case of a more general requirement that the grounds for asserting a statement must cohere with its consequences. This essay considers various ways of making the Harmony Thesis precise and scrutinizes the most influential arguments for it. The verdict is negative: all the extant arguments for the Thesis are weak, and no version of it is remotely plausible. (shrink)

This paper concentrates on how to capture harmony in sequent calculi. It starts by considering a proposal made by Tennant and some objections to it which have been presented by Steinberger. Then it proposes a different analysis which makes use of a double-line presentation of sequent calculi in the style of Dosen and it shows that this proposal is able to dismiss disharmonious operators without thereby adopting any global criterion.

Bilateralists hold that the meanings of the connectives are determined by rules of inference for their use in deductive reasoning with asserted and denied formulas. This paper presents two bilateral connectives comparable to Prior's tonk, for which, unlike for tonk, there are reduction steps for the removal of maximal formulas arising from introducing and eliminating formulas with those connectives as main operators. Adding either of them to bilateral classical logic results in an incoherent system. One way around this problem is (...) to count formulas as maximal that are the conclusion of reductio and major premise of an elimination rule and to require their removability from deductions. The main part of the paper consists in a proof of a normalisation theorem for bilateral logic. The closing sections address philosophical concerns whether the proof provides a satisfactory solution to the problem at hand and confronts bilateralists with the dilemma that a bilateral notion of stability sits uneasily with the core bilateral thesis. (shrink)