Citations of:
A Note on Harmony
Journal of Philosophical Logic 41 (3):613628 (2012)
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Inferentialism claims that expressions are meaningful by virtue of rules governing their use. In particular, logical expressions are autonomous if given meaning by their introductionrules, rules specifying the grounds for assertion of propositions containing them. If the eliminationrules do no more, and no less, than is justified by the introductionrules, the rules satisfy what Prawitz, following Lorenzen, called an inversion principle. This connection between rules leads to a general form of eliminationrule, and when the rules have this form, they may (...) 

Arguments, the story goes, have one or more premises and only one conclusion. A contentious generalisation allows arguments with several disjunctively connected conclusions. Contentious as this generalisation may be, I will argue nevertheless that it is justified. My main claim is that multiple conclusions are epiphenomena of the logical connectives: some connectives determine, in a certain sense, multipleconclusion derivations. Therefore, such derivations are completely natural and can safely be used in prooftheoretic semantics. 

We present our calculus of higherlevel rules, extended with propositional quantification within rules. This makes it possible to present general schemas for introduction and elimination rules for arbitrary propositional operators and to define what it means that introductions and eliminations are in harmony with each other. This definition does not presuppose any logical system, but is formulated in terms of rules themselves. We therefore speak of a foundational account of prooftheoretic harmony. With every set of introduction rules a canonical elimination (...) 

Inspired by the grammar of natural language, the paper presents a variant of firstorder logic, in which quantifiers are not sentential operators, but are used as subnectors . A quantified term formed by a subnector is an argument of a predicate. The logic is defined by means of a meaningconferring naturaldeduction proofsystem, according to the prooftheoretic semantics program. The harmony of the I/Erules is shown. The paper then presents a translation, called the Frege translation, from the defined logic to standard (...) 

The paper suggests a revision of the notion of harmony, a major necessary condition in prooftheoretic semantics for a naturaldeduction proofsystem to qualify as meaning conferring, when moving to a bilateral proofsystem. The latter considers both forces of assertion and denial as primitive, and is applied here to positive logics, lacking negation altogether. It is suggested that in addition to the balance between (positive) introduction and elimination rules traditionally imposed by harmony, a balance should be imposed also on: (i) negative (...) 

This paper presents rules of inference for a binary quantifier I for the formalisation of sentences containing definite descriptions within intuitionist positive free logic. I binds one variable and forms a formula from two formulas. Ix[F, G] means ‘The F is G’. The system is shown to have desirable prooftheoretic properties: it is proved that deductions in it can be brought into normal form. The discussion is rounded up by comparisons between the approach to the formalisation of definite descriptions recommended (...) 

In this dissertation, we shall investigate whether Tennant's criterion for paradoxicality(TCP) can be a correct criterion for genuine paradoxes and whether the requirement of a normal derivation(RND) can be a prooftheoretic solution to the paradoxes. Tennant’s criterion has two types of counterexamples. The one is a case which raises the problem of overgeneration that TCP makes a paradoxical derivation nonparadoxical. The other is one which generates the problem of undergeneration that TCP renders a nonparadoxical derivation paradoxical. Chapter 2 deals with (...) 



This special issue collects together nine new essays on logical consequence :the relation obtaining between the premises and the conclusion of a logically valid argument. The present paper is a partial, and opinionated,introduction to the contemporary debate on the topic. We focus on two inﬂuential accounts of consequence, the modeltheoretic and the prooftheoretic, and on the seeming platitude that valid arguments necessarilypreserve truth. We brieﬂy discuss the main objections these accounts face, as well as Hartry Field’s contention that such objections (...) 



The paper suggests a revision of the notion of harmony, a major necessary condition in prooftheoretic semantics for a naturaldeduction proofsystem to qualify as meaning conferring, when moving to a bilateral proofsystem. The latter considers both forces of assertion and denial as primitive, and is applied here to positive logics, lacking negation altogether. It is suggested that in addition to the balance between introduction and elimination rules traditionally imposed by harmony, a balance should be imposed also on: negative introduction and (...) 

The paper presents a prooftheoretic semantics (PTS) for a fragment of natural language, providing an alternative to the traditional modeltheoretic (Montagovian) semantics (MTS), whereby meanings are truthcondition (in arbitrary models). Instead, meanings are taken as derivabilityconditions in a dedicated naturaldeduction (ND) proofsystem. This semantics is effective (algorithmically decidable), adhering to the meaning as use paradigm, not suffering from several of the criticisms formulated by philosophers of language against MTS as a theory of meaning. In particular, Dummett’s manifestation argument does not (...) 

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 introductionrules whch give expressions their meaning and the eliminationrules should accord harmoniously with the meaning so given. It is proposed here that the (...) 









A logical constant is weakly disharmonious if its elimination rules are weaker than its introduction rules. Substructural weak disharmony is the weak disharmony generated by structural restrictions on the eliminations. I argue that substructural weak disharmony is not a defect of the constants which exhibit it. To the extent that it is problematic, it calls into question the structural properties of the derivability relation. This prompts us to rethink the issue of controlling the structural properties of a logic by means (...) 

A firstorder language with a defined identity predicate is proposed whose apparatus for atomic predication is sensitive to grammatical categories of natural language. Subatomic natural deduction systems are defined for this naturalistic firstorder language. These systems contain subatomic systems which govern the inferential relations which obtain between naturalistic atomic sentences and between their possibly composite components. As a main result it is shown that normal derivations in the defined systems enjoy the subexpression property which subsumes the subformula property with respect (...) 

A comparison is given between two conditions used to define logical constants: Belnap's uniqueness and Hacking's deducibility of identicals. It is shown that, in spite of some surface similarities, there is a deep difference between them. On the one hand, deducibility of identicals turns out to be a weaker and less demanding condition than uniqueness. On the other hand, deducibility of identicals is shown to be more faithful to the inferentialist perspective, permitting definition of genuinely prooftheoretical concepts. This kind of (...) 

The paper exposes the relevance of permuting conversions (in naturaldeduction systems) to the role of such systems in the theory of meaning known as prooftheoretic 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, (...) 

In the recent literature on prooftheoretic semantics, there is mention of a generality condition on defining rules. According to this condition, the schematic formulation of the defining rules must be maximally general, in the sense that no restrictions should be placed on the contexts of these rules. In particular, context variables must always be present in the schematic rules and they should range over arbitrary collections of formulae. I argue against imposing such a condition, by showing that it has undesirable (...) 

Generalelimination harmony articulates Gentzen’s idea that the eliminationrules 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 Erules allow one to infer no more and no less than the Irules justify. Pfenning and Davies call the rules locally complete if the Erules are strong enough to allow one to infer the original judgement. A method is given (...) 

The paper studies the extension of harmony and stability, major themes in prooftheoretic semantics, from singleconclusion naturaldeduction systems to multiple conclusions naturaldeduction, independently of classical logic. An extension of the method of obtaining harmoniouslyinduced general elimination rules from given introduction rules is suggested, taking into account substructurality. Finally, the reductions and expansions of the multiple conclusions naturaldeduction representation of classical logic are formulated. 