Intuitionistic propositional logic with Galois connections

@article{Dzik2010IntuitionisticPL,
  title={Intuitionistic propositional logic with Galois connections},
  author={Wojciech I Dzik and Jouni J{\"a}rvinen and Michiro Kondo},
  journal={Log. J. IGPL},
  year={2010},
  volume={18},
  pages={837-858}
}
In this work, an intuitionistic propositional logic with a Galois connection (IntGC) is introduced. In addition to the intuitionistic logic axioms and inference rule of modus ponens, the logic contains only two rules of inference mimicking the performance of Galois connections. Both Kripke-style and algebraic semantics are presented for IntGC, and IntGC is proved to be complete with respect to both of these semantics. We show that IntGC has the finite model property and is decidable, but… 

Intuitionistic logic with two Galois connections combined with Fischer Servi axioms

TLDR
Both Kripke-style and algebraic semantics are presented for Int2GC and Int2 GC+FS, and the logics are proved to be complete with respect to both of these semantics, and it is proved that rough lattice-valued fuzzy sets defined on complete Heyting algebras are proper algebraic models for Int1GC+FS.

Fitch-Style Modal Lambda Calculi

TLDR
It is shown that Fitch-style modal deduction calculi have good computational properties for a variety of intuitionistic modal logics, and can be extended a la tense logic with the left adjoint of necessity, and are then complete for the categorical semantics.

Intuitionistic modal logic with a galois connection has the finite model property1

We show that the intuitionistic propositional logic with a Galois connection (IntGC), introduced by the authors, has the finite model property.

Representing expansions of bounded distributive lattices with Galois connections in terms of rough sets

Modes of Adjointness

TLDR
It seems a worthy enterprise to trace the concept of adjunction back to the origins of the algebraic semantics of modal logic and to make explicit its ubiquity in this branch of mathematics.

Characterizing intermediate tense logics in terms of Galois connections

TLDR
A uniform way of defining for every logic L, intermediate between intuitionistic and classical logics, the corresponding intermediate minimal tense logic is proposed by building the fusion of two copies of intermediate logic with a Galois connection and interlinking their operators by two Fischer Servi axioms.

Constructive Temporal Logic, Categorically

TLDR
The second author does owe Grigori an intellectual debt, but not at the stage that he would approve of it, especially given all his work on Dynamic Topological Logic with Kremer and others, which might be related to what is described here.

Algebraic Representation, Dualities and Beyond

In this research chapter, dualities and representations of various kinds associated with the semantics of rough sets are explained, critically reviewed, new proofs have been proposed, open problems

On Fuzzy Preordered Sets and Monotone Galois Connections

In this work, we focus on the study of necessary and sufficient conditions in order to ensure the existence (under some constraints) of monotone Galois connections between fuzzy preordered sets.

References

SHOWING 1-10 OF 19 REFERENCES

Logics from Galois connections

On an Intuitionistic Modal Logic

TLDR
This paper considers an intuitionistic variant of the modal logic S4 (which it is called IS4), and places particular importance on the natural deduction formulation of IS4— this formulation has several important metatheoretic properties.

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

TLDR
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.

A Primer on Galois Connections

TLDR
The rudiments of the theory of Galois connections (or residuation theory, as it is sometimes called) are provided, together with many examples and applications, that can be used as an effective research tool throughout mathematics and related areas.

Intuitionistic tense and modal logic

  • W. Ewald
  • Philosophy
    Journal of Symbolic Logic
  • 1986
TLDR
Construction des analogues intuitionnistes des principaux systemes de la logique temporelle classique d'augmenter les analogues de l'homme de l’invention intellectuelle d’un £10,000 ou â €15,000 d‚‚¬20,000 en France.

Positive modal logic

TLDR
A set of postulates for the minimal normal modal logicK+ without negation or any kind of implication is given, and it is shown thatK+ is complete with respect to the usual Kripke-style semantics.

A unifying study between modal-like operators, topologies and fuzzy sets

Discrete Dualities for Heyting Algebras with Operators

Discrete dualities are presented for Heyting algebras with various modal operators, for Heyting algebras with an external negation, for symmetric Heyting algebras, and for Heyting-Brouwer algebras.

L-fuzzy sets