Constructive Modal and Linear Logics
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The document discusses the history and development of modal and intuitionistic logics. It covers key figures like Frege, Hilbert, Gentzen, Prawitz, Martin-Löf, and Girard and how their work contributed to logic systems. It also discusses more recent work on constructive modal logics by researchers like Nerode, Simpson, and the Intuitionistic Modal Logic and Applications community. The document examines challenges in developing intuitionistic modal logics and potential solutions using category theory and modal cubes. It notes the importance of modal logic due to its applications in formal methods and computing.
![Russel
– ...there is no one
fundamental logical
notion of necessity,
nor consequently of
possibility. If this
conclusion is valid,
the subject of
modality ought to be
banished from logic,
since propositions are
simply true or false...
– [Russell, 1905]
–](https://image.slidesharecdn.com/topos2oct2021-211029190555/85/Constructive-Modal-and-Linear-Logics-16-320.jpg)
![DanaScott
One often hears that modal
(or some other) logic is
pointless because it can be
translated into some simpler
language in a first- order way.
Take no notice of such
arguments. There is no
weight to the claim that the
original system must
therefore be replaced by the
new one. What is essential is
to single out important
concepts and to investigate
their properties.
[Advice on Modal Logic, 71]](https://image.slidesharecdn.com/topos2oct2021-211029190555/85/Constructive-Modal-and-Linear-Logics-17-320.jpg)
Constructive Modal and Linear Logics
- 1. Constructive Modal & Linear Logics Valeria de Paiva
- 2. – Build collaborations with friends – Clarify your research for yourself – TICAMORE: Translating and dIscovering CAlculi for MOdal and RElated logics, 7th meeting https://ticamore.lo gic.at/
- 3. ResearchClusters Dialectica Spaces, Linear Logics Intuitionistic Modal Logics Natural Language Inference Lexical Resources
- 4. Logics Frege: quantifiers FOL Hilbert: proofs as mathematical objects axiom systems Gentzen: sequents and Natural Deduction Prawitz: normalization of Natural Deduction Martin-Löf: type theory Girard: system F Linear Logic
- 5. Logics Frege: quantifiers & analytic philosophy 1879 Hilbert: proofs as mathematical objects & axiom systems 1900 Gentzen: sequents and Natural Deduction 1934 Prawitz: normalization of Natural Deduction 1965 Martin-Löf: type theory& dependent types 1972 Girard: System F and Linear Logic 1987
- 8. Hilbert CPL
- 13. Martin-Löf – Intuitionistic type Theory – Dependent types – Universes
- 14. Girard – Normalization of System F – Linear Logic 1987 – GoI, Ludics, etc…
- 15. Logics Frege: quantifiers FOL Hilbert: proofs as mathematical objects axiom systems Gentzen: sequents and Natural Deduction Prawitz: normalization of Natural Deduction Martin-Loef: type theory Girard: system F Linear Logic
- 16. Russel – ...there is no one fundamental logical notion of necessity, nor consequently of possibility. If this conclusion is valid, the subject of modality ought to be banished from logic, since propositions are simply true or false... – [Russell, 1905] –
- 17. DanaScott One often hears that modal (or some other) logic is pointless because it can be translated into some simpler language in a first- order way. Take no notice of such arguments. There is no weight to the claim that the original system must therefore be replaced by the new one. What is essential is to single out important concepts and to investigate their properties. [Advice on Modal Logic, 71]
- 18. van Bentham: Modalities – Box A= A is necessarily the case, A holds for all times, A is obligatory,… Dia A = A is possibly the case, A holds at some time, A is permitted, …c – Temporal logic, knowledge operators, BDI models, denotational semantics, effects, security modelling and verification, natural language understanding and inference, databases, etc.. – Bisimulation right notion of morphism. Correspondece theory, Modal logic about structures, not operators
- 19. Nerode: Constructive Modalities – Modalities over an Intuitionistic basis: ∧ ∨ → ¬ Constructive modalities ought to be twice as useful? Usual phenomenon: classical facts can be ‘constructivized’ in many different ways.
- 20. Simpson: Intuitionistic Modalities – Operators Box , Diamond (like for all/exists) not inter-definable – How do these two modalities interact? – Depends on expected behavior and on tools you want to use. Solutions add to syntax: hypersequents, labelled deduction systems, (linear) nested sequents, tree- sequents, all add some semantics to syntax (many ways...)
- 21. IMLA IMLA: Intuitionistic Modal Logic andApplications since 1999 functional programmers, AI scientists, philosophical logicians talking to each other and cooperating 01 Not attained? Communities still largely talking past each other 02 Incremental work on intuitionistic modal logics continues, as well as some of the big research programmes that started it 03 Does it make sense to try to change this status quo? 04
- 22. I expected –Curry-Howard for a BIG collection of intuitionistic modal logics –Design space for intuitionistic modal logic, for classical logic and how to move from intuitionistic modal to classic modal –Applications of modal type systems –Fully worked out dualities for systems –On-the-shelf implementations for proof search/proof normalization
- 23. Why did I think it would be easy? – Early successes: systems CS4, Lax, CK – CS4: On an Intuitionistic Modal Logic (Studia Logica 2000, conference 1992) – DIML: Explicit Substitutions for Constructive Necessity (with Neil Ghani and Eike Ritter), ICALP 1998 – Lax Logic: ComputationalTypes from a Logical Perspective (with Benton, Bierman, JFP 1998) – CK: Basic Constructive Modal Logic. (with Bellin and Ritter, M4M 2001), Kripke semantics for CK (with Mendler 2005)
- 24. ConstructiveS4 Used by G ̈odel and Girard Usual intuitionistic axioms plus MP, Nec rules
- 25. Why it isn’t easy… –Natural Deduction is problematic, as discovered byWadler and others for LL. –Issue is PROMOTION rule
- 26. Substitution! Any proof using Promotion Any Modus Ponens proof finishing in one assumption BUT cannot apply PROMOTION anymore!! Abramsky’s “Computational Interpretation of Linear Logic” (1993), a calculus that does not satisfy substitution
- 28. Divergences
- 29. ModalCubes
- 30. Why it Matters – Most applications of formal methods use Modal logic Most applications of formal methods in computing use Modal logic
- 31. What’s Next?