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
logic | category theory | type theory |
---|---|---|
true | terminal object/(-2)-truncated object | h-level 0-type/unit type |
proposition(-1)-truncated objecth-proposition, mere proposition
proofgeneralized elementprogram
cut rulecomposition of classifying morphisms / pullback of display mapssubstitution
cut elimination for implicationcounit for hom-tensor adjunctionbeta reduction
introduction rule for implicationunit for hom-tensor adjunctioneta conversion
logical conjunctionproductproduct type
disjunctioncoproduct ((-1)-truncation of)sum type (bracket type of)
implicationinternal homfunction type
negationinternal hom into initial objectfunction type into empty type
universal quantificationdependent productdependent product type
existential quantificationdependent sum ((-1)-truncation of)dependent sum type (bracket type of)
equivalencepath space objectidentity type
equivalence classquotientquotient type
inductioncolimitinductive type, W-type, M-type
higher inductionhigher colimithigher inductive type
completely presented setdiscrete object/0-truncated objecth-level 2-type/preset/h-set
setinternal 0-groupoidBishop set/setoid
universeobject classifiertype of types
modalityclosure operator, (idemponent) monadmodal type theory, monad (in computer science)
linear logic(symmetric, closed) monoidal categorylinear type theory/quantum computation
proof netstring diagramquantum circuit
(absence of) contraction rule(absence of) diagonalno-cloning theorem
synthetic mathematicsdomain specific embedded programming language
</table>
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, Rijke, Shulman, Spitters ).
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):
Egbert Rijke, Mike Shulman, Bas Spitters, Modalities in homotopy type theory arXiv
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)
Last revised on June 20, 2018 at 06:56:18. See the history of this page for a list of all contributions to it.