Generality and Existence 1: Quantification and Free Logic

Restall, Greg

doi:10.1017/s175502031800031x

Cited by 8 publications

(11 citation statements)

References 19 publications

(42 reference statements)

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“…Besides the non-logical symbols, the following (logical) symbols are available: the falsum (⊥), the implication (→), the universal quantifier (∀) the equality symbol (=). 24 Furthermore, there are countably many first order variables v k available as well as the usual auxiliary symbols. The symbols x, y, z (also with subscripts) are meta-variables for the variables.…”

Section: Preliminariesmentioning

confidence: 99%

“…23 In order to serve as a blueprint for arbitrary concrete formal languages, the paradigmatic language L remains underdetermined. 24 The mentioned logical symbols are sufficient from a classical point of view; for the intuitionistic calculus, the alphabet is extended by the missing logical constants and the relevant definitions are easily generalised with respect to these symbols.…”

Section: Preliminariesmentioning

confidence: 99%

“…As the terms represent statements, it seems reasonable to call such term occurrences apparent, in contradistinction to our term occurrences which are real. An interesting limit case is provided by Restall[24]: proper terms t may occur in a derivation as well as existence statements E(t). The terms and the statements are logically equivalent; the introduction of both types of expressions is motivated by subtle philosophical reasons.…”

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confidence: 99%

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The Calculus of Natural Calculation

Gazzari

2021

Stud Logica

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The calculus of Natural Calculation is introduced as an extension of Natural Deduction by proper term rules. Such term rules provide the capacity of dealing directly with terms in the calculus instead of the usual reasoning based on equations, and therefore the capacity of a natural representation of informal mathematical calculations. Basic proof theoretic results are communicated, in particular completeness and soundness of the calculus; normalisation is briefly investigated. The philosophical impact on a proof theoretic account of the notion of meaning is considered.

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Section: Preliminariesmentioning

confidence: 99%

Section: Preliminariesmentioning

confidence: 99%

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confidence: 99%

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The Calculus of Natural Calculation

Gazzari

2021

Stud Logica

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“…A suitable criterion of definitional success-occasionally emerging, albeit in different forms, in the literature, particularly when logics are presented by means of sequent calculi-is invertibility [1,10,30]. I shall not go into a detailed account of why invertibility is a formal property the obtaining of which is indicative of harmony; an explicit endorsement and analysis of invertibility as a criterion of harmony can be found in [15,16].…”

Section: Multiple Conclusion As Epiphenomena Of the Connectivesmentioning

confidence: 99%

Hopeful Monsters: A Note on Multiple Conclusions

Dicher

2018

Erkenn

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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, multiple-conclusion derivations. Therefore, such derivations are completely natural and can safely be used in proof-theoretic semantics. Keywords multiple conclusions • defining rules • structure of derivations • logical inferentialism Delta: [.. . ] He seems engrossed in the production of monstrosities. But monstrosities never foster growth, either in world of nature or in the world of thought. Gamma: Geneticists can easily refute that. Have you not heard that mutations producing monstrosities play a considerable role in macro-evolution? They call such monstrous mutants 'hopeful monsters'. Lakatos, Proofs and refutations, pp. 21-22 1 Preamble An argument has one or more premises and one conclusion. Thus goes the orthodox stance. In this paper, I will defend the heretical claim that an argument may have

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“…Now, it has been suggested that the meaning‐determining rule for a logical constant is the invertible rule in the set of its I‐(R) and E‐(L) rules (cf. Došen, ; Avron, ; Restall, ). If we accept this criterion of definability, then it seems that we have to accept that additive and multiplicative rules define different connectives.…”

Section: Context Generality In Sequent Calculimentioning

confidence: 99%

On a Generality Condition in Proof‐Theoretic Semantics

Dicher

2017

Theoria

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In the recent literature on proof‐theoretic 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 results and that it is ill‐supported by the arguments brought in its favour.

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Generality and Existence 1: Quantification and Free Logic

Cited by 8 publications

References 19 publications

The Calculus of Natural Calculation

The Calculus of Natural Calculation

Hopeful Monsters: A Note on Multiple Conclusions

On a Generality Condition in Proof‐Theoretic Semantics

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