### Abstract

In this paper we introduce Commutative/Non-Commutative Logic (CNC logic) and two categorical models for CNC logic. This work abstracts Benton’s Linear/Non-Linear Logic [4] by removing the existence of the exchange structural rule. One should view this logic as composed of two logics; one sitting to the left of the other. On the left, there is intuitionistic linear logic, and on the right is a mixed commutative/non-commutative formalization of the Lambek calculus. Then both of these logics are connected via a pair of monoidal adjoint functors. An exchange modality is then derivable within the logic using the adjunction between both sides. Thus, the adjoint functors allow one to pull the exchange structural rule from the left side to the right side. We then give a categorical model in terms of a monoidal adjunction, and then a concrete model in terms of dialectica Lambek spaces.

Original language | English (US) |
---|---|

Pages (from-to) | 43-89 |

Number of pages | 47 |

Journal | Electronic Proceedings in Theoretical Computer Science, EPTCS |

Volume | 292 |

DOIs | |

State | Published - Apr 15 2019 |

Event | 2018 Joint International Workshop on Linearity and Trends in Linear Logic and Applications, Linearity-TLLA 2018 - Oxford, United Kingdom Duration: Jul 7 2018 → Jul 8 2018 |

### ASJC Scopus subject areas

- Software

### Cite this

*Electronic Proceedings in Theoretical Computer Science, EPTCS*,

*292*, 43-89. https://doi.org/10.4204/EPTCS.292.4

**On the Lambek calculus with an exchange modality.** / Jiang, Jiaming; Eades, Harley D; de Paiva, Valeria.

Research output: Contribution to journal › Conference article

*Electronic Proceedings in Theoretical Computer Science, EPTCS*, vol. 292, pp. 43-89. https://doi.org/10.4204/EPTCS.292.4

}

TY - JOUR

T1 - On the Lambek calculus with an exchange modality

AU - Jiang, Jiaming

AU - Eades, Harley D

AU - de Paiva, Valeria

PY - 2019/4/15

Y1 - 2019/4/15

N2 - In this paper we introduce Commutative/Non-Commutative Logic (CNC logic) and two categorical models for CNC logic. This work abstracts Benton’s Linear/Non-Linear Logic [4] by removing the existence of the exchange structural rule. One should view this logic as composed of two logics; one sitting to the left of the other. On the left, there is intuitionistic linear logic, and on the right is a mixed commutative/non-commutative formalization of the Lambek calculus. Then both of these logics are connected via a pair of monoidal adjoint functors. An exchange modality is then derivable within the logic using the adjunction between both sides. Thus, the adjoint functors allow one to pull the exchange structural rule from the left side to the right side. We then give a categorical model in terms of a monoidal adjunction, and then a concrete model in terms of dialectica Lambek spaces.

AB - In this paper we introduce Commutative/Non-Commutative Logic (CNC logic) and two categorical models for CNC logic. This work abstracts Benton’s Linear/Non-Linear Logic [4] by removing the existence of the exchange structural rule. One should view this logic as composed of two logics; one sitting to the left of the other. On the left, there is intuitionistic linear logic, and on the right is a mixed commutative/non-commutative formalization of the Lambek calculus. Then both of these logics are connected via a pair of monoidal adjoint functors. An exchange modality is then derivable within the logic using the adjunction between both sides. Thus, the adjoint functors allow one to pull the exchange structural rule from the left side to the right side. We then give a categorical model in terms of a monoidal adjunction, and then a concrete model in terms of dialectica Lambek spaces.

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U2 - 10.4204/EPTCS.292.4

DO - 10.4204/EPTCS.292.4

M3 - Conference article

AN - SCOPUS:85065757400

VL - 292

SP - 43

EP - 89

JO - Electronic Proceedings in Theoretical Computer Science, EPTCS

JF - Electronic Proceedings in Theoretical Computer Science, EPTCS

SN - 2075-2180

ER -