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
A modality in philosophy and formally in formal logic/type theory expresses a certain mode (or “moment” as in Hegel 12) of being.
According to (Kant 1900) (see WP) the four “categories” are
Quantity
Quality
Relation
Modality
and the modalities contain the three pairs of opposites
possibility - impossibility
necessity - Zufälligkeit
In formal logic and type theory modalities are formalized by modal operator or closure operator $\sharp$, that send propositions/types $X$ to new propositions/types $\sharp X$, satisfying some properties.
Adding such modalities to propositional logic or similar produces what is called modal logic. Here operators that are meant to formalize necessity and possibility (S4 modal logic) are maybe most famous. Adding modalities more generally to type theory yields modal type theory. See there for more details.
The categorical semantics of these modalities is that $\sharp$ is interpreted an idempotent monad/comonad on the category of contexts.
This has a refinement to homotopy type theory, where the categorical semantics of a higher modality or homotopy modality as an idempotent (infinity,1)-monad (Shulman 12, HoTTBook, def. 7.7.5).
Typical notation (e.g. SEP, Reyes 91, but not Hermida 10) is as follows:
a co-modality represented by an idempotent comonad is typically denoted by $\Box$, following the traditional example of necessity in modal logic;
a modality represented by an idempotent monad is typically denoted by $\lozenge$ or (less often) by $\bigcirc$, following the traditional example of possibility in modal logic.
When adjunctions between modalities matter (adjoint modalities), then some authors (Reyes 91, p. 367 RRZ 04, p. 116, Hermida 10, p.11) use $\lozenge$ for a left adjoint of a $\Box$. That leaves $\bigcirc$ as the natural choice of notation for a right adjoint (if any) of a $\Box$-modality.
This way for instance for cohesion with shape modality $\dashv$ flat comodality $\dashv$ sharp modality the generic notation would be:
necessity $\dashv$ possibility
unit type modality, empty type co-modality (nothing $\dashv$ being)
shape modality $\dashv$ flat modality $\dashv$ sharp modality
reduction modality $\dashv$ infinitesimal shape modality $\dashv$ infinitesimal flat modality
Discussion in formal logic and homotopy type theory (modal type theory):
Univalent Foundations Project, section 7.7 of Homotopy Type Theory -- Univalent Foundations of Mathematics
German Wikipedia, Modalität (Philosophie))
Stanford Encyclopedia of Philosophy, Modal Logic
Gonzalo Reyes, A topos-theoretic approach to reference and modality, Notre Dame J. Formal Logic Volume 32, Number 3 (1991), 359-391 (Euclid)
Reyes/Reyes/Zolfaghari, Generic Figures and Their Glueings 2004, Polimetrica
Claudio Hermida, section 3.3. of A categorical outlook on relational modalities and simulations, 2010 (pdf)