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Monoidal logics: completeness and classical systems

Monoidal logics: completeness and classical systems Monoidal logics were introduced as a foundational framework to analyze the proof theory of logical systems. Inspired by Lambek's seminal work in categorical logic, the objective is to define logical systems in order to make explicit their categorical (monoidal) structure. In this setting, logical connectives can be proven to be functors with specific properties. Accordingly, monoidal logics allow a classification of logical systems in function of their categorical structure and the functorial properties of their connectives. As they stand, however, strong parallels can be made between monoidal logics and the broader proof-theoretical framework of display logics. In this paper, we extend the results presented in Peterson ((2016). A comparison between monoidal and substructural logics. Journal of Applied Non-Classical Logics, 26(2), 126–159) and we show that monoidal logics are sound and complete with respect to associative display logics, thus providing a completeness result with regards to the algebraic semantics of display and substructural logics. In addition, we discuss the notions of classical and intuitionistic systems. Starting from Lambek's and Grishin's analyses, we explore the role played by partial De Morgan dualities and discuss the necessary and sufficient conditions required for the definitions of classical and intuitionistic deductive systems. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Journal of Applied Non-Classical Logics Taylor & Francis

Monoidal logics: completeness and classical systems

Peterson, Clayton
Journal of Applied Non-Classical Logics , Volume 29 (2): 31 – Apr 3, 2019
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Monoidal logics: completeness and classical systems

Abstract

Monoidal logics were introduced as a foundational framework to analyze the proof theory of logical systems. Inspired by Lambek's seminal work in categorical logic, the objective is to define logical systems in order to make explicit their categorical (monoidal) structure. In this setting, logical connectives can be proven to be functors with specific properties. Accordingly, monoidal logics allow a classification of logical systems in function of their categorical structure and the...
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Publisher
Taylor & Francis
Copyright
© 2019 Informa UK Limited, trading as Taylor & Francis Group
ISSN
1958-5780
eISSN
1166-3081
DOI
10.1080/11663081.2018.1547513
Publisher site
See Article on Publisher Site

Abstract

Monoidal logics were introduced as a foundational framework to analyze the proof theory of logical systems. Inspired by Lambek's seminal work in categorical logic, the objective is to define logical systems in order to make explicit their categorical (monoidal) structure. In this setting, logical connectives can be proven to be functors with specific properties. Accordingly, monoidal logics allow a classification of logical systems in function of their categorical structure and the functorial properties of their connectives. As they stand, however, strong parallels can be made between monoidal logics and the broader proof-theoretical framework of display logics. In this paper, we extend the results presented in Peterson ((2016). A comparison between monoidal and substructural logics. Journal of Applied Non-Classical Logics, 26(2), 126–159) and we show that monoidal logics are sound and complete with respect to associative display logics, thus providing a completeness result with regards to the algebraic semantics of display and substructural logics. In addition, we discuss the notions of classical and intuitionistic systems. Starting from Lambek's and Grishin's analyses, we explore the role played by partial De Morgan dualities and discuss the necessary and sufficient conditions required for the definitions of classical and intuitionistic deductive systems.

Journal

Journal of Applied Non-Classical Logics – Taylor & Francis

Published: Apr 3, 2019

Keywords: Non-classical logics; display logics; linear logic; Lambek calculus; De Morgan logic; intuitionistic logic

References

  • A comparison between Lambek syntactic calculus and intuitionistic linear propositional logic
  • Non-commutative intuitionistic linear logic
  • Display logic
  • Continuation semantics for the Lambek-Grishin calculus
  • A unified display proof theory for bunched logic
  • Lambek calculus and substructural logics
  • Weakly distributive categories
  • Proof analysis in intermediate logics
  • Multi-type display calculus for dynamic epistemic logic
  • Linear Logic
  • Substructural logics on display
  • Bilattice logic properly displayed
  • Full intuitionistic linear logic (extended abstract)
  • The mathematics of sentence structure
  • Deductive systems and categories I
  • Contrary-to-duty reasoning: A categorical approach
  • A comparison between monoidal and substructural logics
  • Falsification, natural deduction and bi-intuitionistic logic
  • The calculus of the weak “law of excluded middle”