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\section*{graded modality}

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

Classic [[modalities]] are defined in absolute terms, so that, for instance, [[necessity and possibility]] are understood in terms of a state of affairs obtaining in all or some world. Some \hyperlink{philosophy}{philosophers} in the early 1970s looked to generalize these to graded versions so that a proposition may be said to be, say, quite likely, highly likely, and so on. Since then, linguists and computer scientists have developed these ideas, the latter expanding the topic to include those modalities relating to resources and [[side effects]].

A \textbf{graded modality} is an indexed family of [[modalities]] with some additional [[structure]] on the indices which mirrors the structure of the axioms/proof rules. A graded modality is a [[graded monad]] with idempotent components.

For example, the [[exponential modality]] of [[linear logic]] $!$ has a graded counterpart in [[bounded linear logic]], where $!$ is replaced with a family of modalities $!_n$ indexed by the natural numbers (the reuse bound).

\hypertarget{references}{}\subsection*{{References}}\label{references}

\begin{itemize}%
\item Brunel, Aloïs and Gaboardi, Marco and Mazza, Damiano and Zdancewic, Steve, \emph{A Core Quantitative Coeffect Calculus}, (\href{https://doi.org/10.1007/978-3-642-54833-8_19}{doi}) (\href{https://lipn.univ-paris13.fr/~mazza/papers/CoreQuantCoeff.pdf}{pdf})


\item Dan R. Ghica and Alex I. Smith, \emph{Bounded Linear Types in a Resource Semiring}, (\href{https://doi.org/10.1007/978-3-642-54833-8_18}{doi}) (\href{https://www.cs.bham.ac.uk/~drg/papers/esop14.pdf}{pdf})


\item [[Dominic Orchard]], Vilem-Benjamin Liepelt, Harley Eades III, \emph{Quantitative Program Reasoning with Graded Modal Types}, (\href{https://www.cs.kent.ac.uk/people/staff/dao7/publ/granule-icfp19.pdf}{pdf})


\item [[Dominic Orchard]], Vilem-Benjamin Liepelt, \emph{Gram: A linear functional language with graded modal types}, (\href{https://www.cs.ox.ac.uk/conferences/fscd2017/preproceedings_unprotected/TLLA_Orchard.pdf}{extended abstract})


\item Marco Gaboardi, Shin-ya Katsumata, Dominic Orchard, Flavien Breuvart, Tarmo Uustalu, \emph{Combining Effects and Coeffects via Grading}, (\href{https://www.cs.kent.ac.uk/people/staff/dao7/publ/combining-effects-and-coeffects-icfp16.pdf}{pdf})


\item Shin-ya Katsumata, \emph{A Double Category Theoretic Analysis of Graded Linear Exponential Comonads}, (\href{https://doi.org/10.1007/978-3-319-89366-2_6}{doi})


\item Soichiro Fujii, Shin-ya Katsumata, Paul-André Melliès, \emph{Towards a Formal Theory of Graded Monads}, (\href{https://doi.org/10.1007/978-3-662-49630-5_30}{doi}) (\href{https://www.irif.fr/~mellies/papers/fossacs2016-final-paper.pdf}{pdf})



\end{itemize}
A formal framework for type theories with a collection of modalities is in

\begin{itemize}%
\item [[Daniel Licata]], [[Mike Shulman]], and [[Mitchell Riley]], \emph{A Fibrational Framework for Substructural and Modal Logics (extended version)}, in Proceedings of 2nd International Conference on Formal Structures for Computation and Deduction (FSCD 2017) (\href{http://drops.dagstuhl.de/opus/volltexte/2017/7740/}{doi: 10.4230/LIPIcs.FSCD.2017.25}, \href{http://dlicata.web.wesleyan.edu/pubs/lsr17multi/lsr17multi-ex.pdf}{pdf})

\end{itemize}
For discussion in [[philosophy]] and linguistics, see

\begin{itemize}%
\item Patrick Grosz, \emph{Grading Modality: A New Approach to Modal Concord and its Relatives}, (\href{https://www.univie.ac.at/sub14/proc/grosz.pdf}{paper})


\item Daniel Lassiter, \emph{Graded modality}, (\href{https://web.stanford.edu/~danlass/Lassiter-graded-modality-draft.pdf}{paper})


\item Daniel Lassiter, \emph{Graded Modality: Qualitative and Quantitative Perspectives}, OUP, (\href{https://global.oup.com/academic/product/graded-modality-9780198701354?cc=us&lang=en&}{website})



\end{itemize}
 Earlier work in philosophy includes

\begin{itemize}%
\item Lou Goble, 1970. Grades Of Modality. Logique Et Analyse 13, pp. 323-334, (\href{http://virthost.vub.ac.be/lnaweb/ojs/index.php/LogiqueEtAnalyse/article/view/405}{paper}).


\item Kit Fine, 1972. In so many possible worlds. Notre Dame Journal of Formal Logics 13: 516–520.


\item David Lewis, 1973. Counterfactuals. Oxford: Blackwell



\end{itemize}
[[!redirects graded modalities]]



\end{document}