# Categorical Proof Theory of Co-Intuitionistic Linear Logic

@article{Bellin2014CategoricalPT, title={Categorical Proof Theory of Co-Intuitionistic Linear Logic}, author={Gianluigi Bellin}, journal={Log. Methods Comput. Sci.}, year={2014}, volume={10} }

To provide a categorical semantics for co-intuitionistic logic one has to face the fact, noted by Tristan Crolard, that the definition of co-exponents as adjuncts of coproducts does not work in the category Set, where coproducts are disjoint unions. Following the familiar construction of models of intuitionistic linear logic with exponential"!", we build models of co-intuitionistic logic in symmetric monoidal left-closed categories with additional structure, using a variant of Crolard's term…

## 16 Citations

### Dualized Simple Type Theory

- PhilosophyLog. Methods Comput. Sci.
- 2016

DTT is a simple type theory with perfect intuitionistic duality, and corresponds to a single-sided polarized sequent calculus, which gives a direct proof of consistency, but proves completeness by reduction to L.

### Dualized Type Theory

- Computer Science
- 2014

DTT is a type theory with perfect intuitionistic duality, and corresponds to a single-sided polarized sequent calculus, and it is proved strongly normalizing, and type preservation is proved.

### Extended Abstract : The Categorical Structure of Semi-Bilinear Intuitionistic Logic

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- 2013

In [11] T. Crolard introduced Subtractive Logic (SL) a conservative extension of intuitionistic logic, where for every connective in the logic their dual is also a connective of the logic. In…

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We reconsider the pragmatic interpretation of intuitionistic logic [21] regarded as a logic of assertions and their justifications and its relations with classical logic. We recall an extension of…

### Structuring Co-constructive Logic for Proofs and Refutations

- PhilosophyLogica Universalis
- 2016

This paper considers a topos-theoretic structure for the interpretation of co-constructive logic for proofs and refutations following Trafford and develops abstractions of elementary topoi to consider a dialogue structure between these topoi.

### A Cointuitionistic Adjoint Logic

- PhilosophyArXiv
- 2017

The dual to LNL models which are studied are shown to correspond to dual linear categories, the dual to Bierman's linear categories proposed by Bellin, and the definition of bi-LNL models is given, which are a corresponding sequent calculus, natural deduction, and term assignment for dual LNLmodels.

### A General Glivenko-Gödel Theorem for Nuclei

- PhilosophyMFPS
- 2021

This work generalises Glivenko’s theorem from double negation to an arbitrary nucleus, from provability in a calculus to an inductively generated abstract consequence relation, and from propositional logic to any set of objects whatsoever.

### Pragmatic and dialogic interpretations of bi-intuitionism. Part I

- Philosophy
- 2014

It is claimed that some conceptual refinements suffice to make their “pragmatic interpretation” a bona fide representation of intuitionism, and sketches a meaning-asuse interpretation of co-intuitionism that seems to fulfil the requirements of Dummett and Prawitz’s justificationist approach.

### A refutation calculus for intuitionistic logic

- Philosophy
- 2017

Classically, logical consequence can be equivalently defined as truth transmission or as “backward” falsity transmission, in the following sense: that a consequence statement Γ ` ∆ holds can be…

### A pragmatic dialogic interpretation of bi-intuitionism

- Philosophy
- 2014

We consider a " polarized " version of bi-intuitionistic logic [9, 7, 10, 11] as a logic of assertions and hypotheses and show that it supports a " rich proof theory " and an interesting categorical…

## References

SHOWING 1-10 OF 43 REFERENCES

### A TERM ASSIGNMENT FOR DUAL INTUITIONISTIC LOGIC

- Mathematics
- 2006

We study the proof-theory of co-Heyting algebras and present a calculus of continuations typed in the disjunctive–subtractive fragment of dual intuitionistic logic. We give a single-assumption…

### A Mixed Linear and Non-Linear Logic: Proofs, Terms and Models (Extended Abstract)

- Philosophy, MathematicsCSL
- 1994

Intuitionistic linear logic regains the expressive power of intuitionistic logic through the ! (‘of course’) modality and an associated notion of categorical model in which the ! modality is modelled by a comonad satisfying certain extra conditions.

### Dual Intuitionistic Logic Revisited

- Computer ScienceTABLEAUX
- 2000

It is shown that a previously reported generalised display framework does deliver the required cut-free display calculus and the structural rule necessary to turn this display calculus into one for classical logic is pinpointed.

### Full Intuitionistic Linear Logic (extended abstract)

- Philosophy, Computer ScienceAnn. Pure Appl. Log.
- 1993

### Relating Categorical Semantics for Intuitionistic Linear Logic

- MathematicsAppl. Categorical Struct.
- 2005

It is pointed out that mere soundness and completeness of a linear typed calculus with respect to a class of categorical models are not sufficient to identify the most appropriate class uniquely.

### Completeness Results for Intuitionistic and Modal Logic in a Categorical Setting

- Philosophy, MathematicsAnn. Pure Appl. Log.
- 1995

### A Term Calculus for Intuitionistic Linear Logic

- Computer Science, MathematicsTLCA
- 1993

This paper considers the problem of deriving a term assignment system for Girard's Intuitionistic Linear Logic for both the sequent calculus and natural deduction proof systems and explores the relationship between these and considers their computational content.

### A Formulae-as-Types Interpretation of Subtractive Logic

- MathematicsJ. Log. Comput.
- 2004

A very natural restriction is defined of the λμ-calculus which is closed under reduction and whose type system is a constructive restriction of the Classical Natural Deduction and extended conservatively to Subtractive Logic.

### On the π-calculus and Co-intuitionistic Logic. Notes on Logic for Concurrency and λP Systems

- Computer ScienceFundam. Informaticae
- 2014

It is argued that translations of typed functional languages in concurrent and distributed systems such as π-calculi or λP systems are best typed with co-intuitionistic logic, where some features of computations match the logical properties in a natural way.