Harmony and Autonomy in Classical Logic

Read, Stephen

doi:10.1023/a:1004787622057

Cited by 110 publications

(42 citation statements)

References 11 publications

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“…Read [28] has, following a suggestion of Schroeder-Heister (with Appendix B of Prawitz's [26] and Ekman's Paradox 5 [7] in mind), proposed as a logical constant a nullary operator • (aka R, for "Russell") with the single (but impure) introduction rule…”

Section: Is Bullet a Logical Constant?mentioning

confidence: 99%

Some Remarks on Proof-Theoretic Semantics

Dyckhoff

2015

Advances in Proof-Theoretic Semantics

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This is a tripartite work. The first part is a brief discussion of what it is to be a logical constant, rejecting a view that allows a particular self-referential "constant" • to be such a thing in favour of a view that leads to strong normalisation results. The second part is a commentary on the flattened version of Modus Ponens, and its relationship with rules of type theory. The third part is a commentary on work (joint with Nissim Francez) on "general elimination rules" and harmony, with a retraction of one of the main ideas of that work, i.e. the use of "flattened" general elimination rules for situations with discharge of assumptions. We begin with some general background on general elimination rules.

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Section: Is Bullet a Logical Constant?mentioning

confidence: 99%

Some Remarks on Proof-Theoretic Semantics

Dyckhoff

2015

Advances in Proof-Theoretic Semantics

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“…31 The proof requires no principles not available in minimal logic, and so avoids the objections to the use of EFQ in the attempt to recover the grounds for assertion mooted in Section 8. In particular, it finally gives a positive answer to Prawitz's Conjecture: the set of inference rules warranted by the meaning conferred on the logical constants by the set of I-rules for each logic S are derivable from the addition to those rules of the GE-rules generated by them.…”

Section: Generalized Local Completenessmentioning

confidence: 97%

“…In [31] it was shown how the rule ∀I can be thought of heuristically as a restriction to a finitary setting of the infinitary rule ∀I ∞ :…”

Section: Generalized Local Completenessmentioning

confidence: 99%

General-Elimination Stability

Jacinto

Read

2016

Stud Logica

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Abstract.General-elimination 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 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.

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“…(Milne (1994)) and (Read (2000)). It is fair to say that they all deviate in some way from Dummett's and Prawitz' harmony. 11 Or 2A, in modal logic.…”

Section: A Problem With Negationmentioning

confidence: 98%

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

Kürbis

2013

J Philos Logic

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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.

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Harmony and Autonomy in Classical Logic

Cited by 110 publications

References 11 publications

Some Remarks on Proof-Theoretic Semantics

Some Remarks on Proof-Theoretic Semantics

General-Elimination Stability

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

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