# A judgmental reconstruction of modal logic

@article{Pfenning2001AJR, title={A judgmental reconstruction of modal logic}, author={Frank Pfenning and Rowan Davies}, journal={Mathematical Structures in Computer Science}, year={2001}, volume={11}, pages={511 - 540} }

We reconsider the foundations of modal logic, following Martin-Löf's methodology of distinguishing judgments from propositions. We give constructive meaning explanations for necessity and possibility, which yields a simple and uniform system of natural deduction for intuitionistic modal logic that does not exhibit anomalies found in other proposals. We also give a new presentation of lax logic and find that the lax modality is already expressible using possibility and necessity. Through a…

## 369 Citations

A judgmental analysis of linear logic

- Philosophy
- 2003

We reexamine the foundations of linear logic, developing a system of natural deduction following Martin-L¨ of’s separation of judgments from propositions. Our construction yields a clean and elegant…

Proof-Theoretic Semantics, a Problem with Negation and Prospects for Modality

- Philosophy
- 2021

This paper discusses proof-theoretic semantics, the project of specifying the meanings of the logical constants in terms of rules of inference governing them, and gives precise characterisations of the crucial notions of harmony and stability placed in the context of proving normalisation results in systems of natural deduction.

Categorical Judgments in a Logical Framework 15-816 Modal Logic

- Philosophy
- 2010

Categorical judgments possess a context-clearing property, making them difficult to express elegantly in the LF logical framework because the context of available LF hypotheses grows monotonically.…

Proof-Theoretic Semantics, a Problem with Negation and Prospects for Modality

- PhilosophyJ. Philos. Log.
- 2015

This paper discusses proof-theoretic semantics, the project of specifying the meanings of the logical constants in terms of rules of inference governing them, and gives precise characterisations of the crucial notions of harmony and stability placed in the context of proving normalisation results in systems of natural deduction.

A Linear-logical Reconstruction of Intuitionistic Modal Logic S4

- PhilosophyFSCD
- 2019

An extension of intuitionistic multiplicative exponential linear logic with a modality combining the S4 modality and the exponential modality is introduced, and it is shown that it admits a sound translation from IS4.

Hypothetical Logic of Proofs

- PhilosophyLogica Universalis
- 2014

The Hypothetical Logic of Proofs is developed, a reformulation of LP based on judgemental reasoning that enjoys arithmetical soundness and completeness and is capable of reflecting its own proofs.

A Judgmental Deconstruction of Modal Logic

- Philosophy
- 2009

The modalities and© of necessary and lax truth described by Pfenning and Davies can be seen to arise from the same pair of adjoint logical operators F and U , which pass in both directions between…

Contextual modal type theory

- PhilosophyTOCL
- 2008

The consequences of relativizing contextual modal logic and its type-theoretic analogue to explicitly specified contexts are investigated.

The Focused Constraint Inverse Method for Intuitionistic Modal Logics

- Philosophy, Computer Science
- 2010

A focused inverse method for proof search in a variety of intuitionistic modal logics such as K, D, T, S4 and S5 that have recently been proposed to reason about authorization and information flow security.

## References

SHOWING 1-10 OF 63 REFERENCES

A Computational Interpretation of Modal Proofs

- Philosophy
- 1996

Proof theory of modal logics, though largely studied since the fifties, has always been a delicate subject, the main reason being the apparent impossibility to obtain elegant, natural systems for…

Propositional Lax Logic

- Computer Science, PhilosophyInf. Comput.
- 1997

This paper investigates a peculiar intuitionistic modal logic, called Propositional Lax Logic (PLL), which has promising applications to the formal verification of computer hardware, and investigates some of its proof-theoretic properties and defines a new class of fallible two-frame Kripke models for PLL.

Computational Types from a Logical Perspective

- MathematicsJ. Funct. Program.
- 1998

This paper shows that the computational lambda calculus also arises naturally as the term calculus corresponding to a novel intuitionistic modal propositional logic and gives natural deduction, sequent calculus and Hilbert-style presentations of this logic and proves strong normalisation and confluence results.

On a Modal -calculus for S4 ?

- Philosophy, Computer Science
- 1995

It is shown that, with respect to provability, this formulation of a proof term calculus for the intuitionistic modal logic S4 is equivalent to other formulations in the literature, sketch a simple type checking algorithm, and prove subject reduction and the existence of canonical forms for well-typed terms.

An Intuitionistic Modal Logic with Applications to the Formal Verification of Hardware

- Computer Science, PhilosophyCSL
- 1994

A concrete and rather natural class of models from hardware verification such that the modality O models correctness up to timing constraints, and further shows soundness and completeness for several classes of fallible two-frame Kripke models.

A modal analysis of staged computation

- Computer SciencePOPL '96
- 1996

The main technical result is a conservative embedding of Nielson & Nielson's two-level functional language in the language Mini-ML, thus proving that binding-time correctness is equivalent to modal correctness on this fragment.

On a Modal λ-Calculus for S 4 ?

- Philosophy, Computer Science

We present λ→2, a concise formulation of a proof term calculus for the intuitionistic modal logic S4 that is well-suited for practical applications. We show that, with respect to provability, it is…

Logic Programming in a Fragment of Intuitionistic Linear Logic

- Computer ScienceInf. Comput.
- 1994

The intuitionistic notion of context is refined by using a fragment of J.-Y. Girard's (Theor. Comput. Sci., vol.50, p.1-102, 1987) linear logic that includes additive and multiplicative conjunction,…

Logic programming in a fragment of intuitionistic linear logic

- Computer Science[1991] Proceedings Sixth Annual IEEE Symposium on Logic in Computer Science
- 1991

The intuitionistic notion of context is refined by using a fragment of J.-Y. Girard's (Theor. Comput. Sci., vol.50, p.1-102, 1987) linear logic that includes additive and multiplicative conjunction,…