natural deduction metalanguage, practical foundations
type theory (dependent, intensional, observational type theory, homotopy type theory)
computational trinitarianism = propositions as types +programs as proofs +relation type theory/category theory
Modal type theory is a flavor of type theory with type formation rules for modalities, hence type theory which on propositions reduces to modal logic.
Following (Moggi91, Benton-Bierman-de Paiva 95, Kobayashi 97) modal type theory is specifically understood as being a type theory equipped with (co-)monads on its type system, representing the intended modalities. Since monads in computer science embody a notion of computation, some authors also speak of computational type theory here (Benton-Bierman-de Paiva 95, Fairtlough-Mendler 02).
According to (Benton-Bierman-de Paiva 95, p. 1-2) this matches well with the default interpretation of (S4) modal logic as being about the modality $T$ of “possibility”:
The starting point for Moggi’s work is an explicit semantic distinction between computations and values. If $A$ is an object which interprets the values of a particular type, then $T(A)$ is the object which models computation of that type $A$. $[...]$ For a wide variety of notions of computation, the unary operator $T(-)$ turns out to have the categorical structure of a strong monad on an underlying cartesian closed category of values. $[...]$ On a purely intuitive level and particularly if one thinks about non-termination, there is certainly something appealing about the idea that a computation of type $A$ represents the possibility of a value of type $A$.
When the underlying type theory is homotopy type theory these modalities are a “higher” generalization of traditional modalities, with “higher” in the sense of higher category theory: they have categorical semantics in (∞,1)-categories given by (∞,1)-monads. See (Shulman 12, Rijke, Shulman, Spitters ) for definition of such higher modalities, and see at reflective subuniverse.
At least in many cases, modalities in type theory are identified with monads or comonads on the underlying type universe, or on the subuniverse of propositions.
See for instance (Benton-Bierman-de Paiva, section 3.2), (Kobayashi), (Gabbay-Nanevski, section 8), (Gaubault-Larrecq, Goubault, section 5.1), (Park-Harper, section 2.6), (Shulman) as examples of the first, and (Moggi, def. 4.7), (Awodey-Birkedal, section 4.2) as examples of the second.
As a special case of the modalities-are-monads relation, a Grothendieck topology on the site underlying a presheaf topos is equivalently encoded in the sheafification monad $PSh(C) \to Sh(C) \hookrightarrow PSh(C)$ induced by the sheaf topos geometric embedding. More generally, any geometric subtopos is equivalently represented by a left-exact idempotent monad.
When restricted to act on (-1)-truncated objects (i.e. subterminal objects or more generally monomorphisms), this becomes a universal closure operator. When internalized as an operation on the subobject classifier, this becomes the corresponding Lawvere-Tierney operator. This modal perspective on sheafification was maybe first made explicit by Bill Lawvere:
A Grothendieck ‘topology’ appears most naturally as a modal operator of the nature ‘it is locally the case that’ (Lawvere).
Here, “It is locally the case that [X is metrizable/Euclidean/etc],” corresponds to, “X is locally metrizable/Euclidean/etc,” in the usual parlance.
More discussion is in (Goldblatt), where this kind of modality is called a geometric modality.
For higher toposes, it is no longer true in general that a subtopos is determined by its behavior on the $(-1)$-truncated objects, but we can still regard the entire sheafification monad as a higher modality in the internal homotopy type theory.
The canonical monad on a local topos gives rise to a closure modality on propositions in its internal language, as described in (Awodey-Birkedal).
See at necessity and possibility the section Possible worlds via dependent type theory
By adding to homotopy type theory three (higher) modalities that encode discrete types and codiscrete types and hence implicitly a non-(co)-discrete notion of cohesion one obtained cohesive homotopy type theory. Adding furthermore modalities for infinitesimal (co)discreteness yields differential homotopy type theory.
The clear identification of modal operators on types with monads (see at monad (in computer science)) is due to
This was observed (independently) to systematically yield constructive modal logic in (see also at computational type theory)
P.N. Benton , G.M. Bierman , Valeria de Paiva, Computational Types from a Logical Perspective I (1995) (web)
M. Fairlough, Michael Mendler, Propositional lax logic, Information and computation 137(1):1-33 (1997)
Satoshi Kobayashi, Monad as modality, Theoretical Computer Science, Volume 175, Issue 1, 30 (1997), Pages 29–74
See also the type-theoretic generalization of adjoint logic, as discussed in
The modal systems “CL” and “PLL” in (Benton-Bierman-Paiva) and (Fairlough-Mendler), respectively, turn out to be equivalent to the geometric modality of Goldblatt. The system CS4 in (Kobayashi) yields a constructive version of S4 modal logic.
Explicit mentioning of type theory equipped with such a monad as modal type theory or computational type theory is in
Discussion of modal operators explicitly in dependent type theory (and with a brief mentioning of the relation to monads) is in
Aleksandar Nanevski, Frank Pfenning, Brigitte Pientka, Contextual Modal Type Theory (2005) (web, slides)
Valeria de Paiva, Eike Ritter, Fibrational Modal Type Theory, Electronic Notes in Theoretical Computer Science Volume 323, 11 July 2016, Pages 143–161 (doi:10.1016/j.entcs.2016.06.010)
Proceedings of the Tenth Workshop on Logical and Semantic Frameworks, with Applications (LSFA 2015)
A survey of the field of modal type theory is in the collections
M. Fairtlough, M. Mendler, Eugenio Moggi (eds.), Modalities in Type Theory, Mathematical Structures in Computer Science, Vol. 11, No. 4, (2001)
Valeria de Paiva, Rajeev Goré, Michael Mendler, Modalities in constructive logics and type theories, Special issue of the Journal of Logic and Computation, editorial pp. 439-446, Vol. 14, No. 4, Oxford University Press, (2004) (pdf)
Valeria de Paiva, Brigitte Pientka (eds.) Intuitionistic Modal Logic and Applications (IMLA 2008), . Inf. Comput. 209(12): 1435-1436 (2011) (web)
The historically first modal type theory, the interpretation of sheafification as a modality in the internal language of a Grothendieck topos goes back to the notion of Lawvere-Tierney operator
reviewed in
Modal type theory with an eye towards homotopy type theory is discussed in
Formalization of modalities in homotopy type theory is discussed
Mike Shulman, Higher modalities, talk at UF-IAS-2012, October 2012 (pdf)
Egbert Rijke, Mike Shulman, Bas Spitters, Modalities in homotopy type theory arXiv
Mike Shulman, http://homotopytypetheory.org/2015/07/05/modules-for-modalities/ (2015)
Egbert Rijke, Bas Spitters, around def. 1.11 of Sets in homotopy type theory (arXiv:1305.3835)
Kevin Quirin and Nicolas Tabareau, Lawvere-Tierney Sheafification in Homotopy Type Theory, Workshop talk 2015 (pdf), Kevin Quirin thesis
and specifically in cohesive homotopy type theory in
Urs Schreiber, Michael Shulman, Quantum gauge field theory in Cohesive homotopy type theory, in Ross Duncan, Prakash Panangaden (eds.) Proceedings 9th Workshop on Quantum Physics and Logic, Brussels, Belgium, 10-12 October 2012 (arXiv:1408.0054)
Michael Shulman, Brouwer’s fixed-point theorem in real-cohesive homotopy type theory (arXiv:1509.07584)
Monadic modal type theory with idempotent monads/monadic reflection is discussed in
Andrzej Filinski, Representing Layered Monads, POPL 1999. (pdf)
Andrzej Filinski, On the Relations between Monadic Semantics, TCS 375:1-3, 2007. (pdf)
Andrzej Filinski, Monads in Action, POPL 2010. (pdf)
Oleg Kiselyov and Chung-chieh Shan, Embedded Probabilistic Programming. Working conference on domain-specific languages, (2009) (pdf)
See also
Frank Pfenning, Towards modal type theory (2000) (pdf)
Frank Pfenning, Intensionality, Extensionality, and Proof Irrelevance in Modal Type Theory, Pages 221–230 of: Symposium on Logic in Computer Science (2001) (web)
Giuseppe Primiero, A multi-modal dependent type theory (pdf)
Murdoch Gabbay, Aleksandar Nanevski, Denotation of contextual modal type theory (CMTT): syntax and metaprogramming (pdf)
A modality in the internal language of a local topos is discussed in section 4.2 of
Steve Awodey, Lars Birkedal, Elementary axioms for local maps of toposes, Journal of Pure and Applied Algebra, 177(3):215-230, (2003) (ps, pdf )
Jean Goubault-Larrecq, Éric Goubault, On the geometry of intuitionistic S4 proofs, Homology, homotopy and applications vol 5(2) (2003)
Sungwoo Park, Robert Harper, A modal language for Effects (2004) (web)
Dan Licata, Robert Harper, A Monadic Formalization of ML5 (arXiv:1009.2793)
A list of related references is also kept at