
TL;DR
This paper proves that Tennant's logical system, known as Core logic, entails a paradox called the Core logic paradox, highlighting a fundamental inconsistency within this logical framework.
Contribution
It introduces and formally proves the existence of the Core logic paradox within Tennant's intuitionistic relevant logic system.
Findings
Identification of the Core logic paradox in Tennant's system
Formal proof demonstrating the paradox's entailment
Implications for the consistency of Core logic
Abstract
This paper provides a proof that Tennant's logical system entails a paradox that is called Core logic paradox, in reference to the new name given by Tennant to his intuitionistic relevant logic.
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Taxonomy
TopicsLogic, Reasoning, and Knowledge · Philosophy and Theoretical Science
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The Core Logic Paradox
Joseph Vidal-Rosset
Université de Lorraine, Département de philosophie, Archives Poincaré, UMR 7117 du CNRS, bd. Libération, 54000 Nancy - France
Abstract
This paper provides a proof that Tennant’s logical system entails a paradox that is called “Core logic paradox”, in reference to the new name given by Tennant to his intuitionistic relevant logic [2, 3]. The inconsistency theorem on which this paper ends is preceded by lemmas that prove syntactically and semantically that rule is strongly admissible in the sequent calculus for propositional Core logic. This rule plays a key role in a consistency test on which Tennant’s logical system fails.
1 Two basic claims of Core logic
The idiosyncrasy of Tennant’s logical system lies mainly in this pair of claims:
- (1)
The First Lewis Paradox that is the sequent is not provable in Core logic, therefore the formula
[TABLE]
is claimed as being valid in Core logic.111Tennant [5, page 195] wrote , because of the completeness claimed for Core logic, formula (1) is justified. 2. (2)
Every intuitionistic logical theorem is provable in Core logic, therefore
[TABLE]
is provable in Core logic.
The slogan “relevance at the level of the turnstile” [5, p. 15, p. 41, p. 121, p. 263] explains this surprising feature of Tennant’s logical system.
The proofs of lemmas and theorem given in the next section are based on the following rules of sequent calculus for propositional Core logic.
The reference list from the paper itself. Each links out to its DOI / PubMed record.
- 1[1] R. Dyckhoff & S. Negri – “Admissibility of structural rules for contraction-free systems of intuitionistic logic”, Journal of Symbolic Logic 65 (2000), no. 4, p. 1499–1518.
- 2[2] N. Tennant – Anti-Realism and Logic: truth as eternal , Clarendon Press, Oxford, U.K., 1987.
- 3[3] — , “Natural Deduction and Sequent Calculus for Intuitionistic Relevant Logic”, The Journal of Symbolic Logic 52 (1987), no. 3, p. 665–680.
- 4[4] — , “The Relevance of Premises to Conclusions of Core Proofs”, Review of Symbolic Logic 8 (2015), no. 4, p. 743 – 784.
- 5[5] — , Core Logic , 1 éd., Oxford University Press, 2017.
