graded modality

The decomposition of Worlds as a context gives one form of grading. With more subtle dependency graph there would be a directed category of modes. So

$\mathbf{H}_{/W}
\stackrel{\longrightarrow}{\stackrel{\longleftarrow}{\longrightarrow}}
\mathbf{H}$

factoring as

$\mathbf{H}_{/W}
\stackrel{\longrightarrow}{\stackrel{\longleftarrow}{\longrightarrow}}
\cdots
\stackrel{\longrightarrow}{\stackrel{\longleftarrow}{\longrightarrow}}
\mathbf{H}_{/W_2}
\stackrel{\longrightarrow}{\stackrel{\longleftarrow}{\longrightarrow}}
\mathbf{H}_{/W_1}
\stackrel{\longrightarrow}{\stackrel{\longleftarrow}{\longrightarrow}}
\mathbf{H}$

provides a model for graded modalities, where the grading is via the direct category of the context extension maps.

One could make sense of the distinction between Brandom’s:

- There is a red apple.
- There is an apple which seems to be red.
- There seems to be a red apple.

Decreasing amounts are being committed to. ‘Seems to seem’ = ‘seems’, so idempotent. X is P implies X seems to be P.

So two modes for confidence level, or three modes for commitment to coloured fruit, to fruit and to nothing?

Expression of lack of commitment as an effect. The ‘maybe’ monad.

Temporal operators as graded, $G(t_1)$ and $G(t_2)$, when $t_1 \lt t_2$.

Graded by contexts. Different knowledge states of agents. Curry’s construction. Comparing someone relying on $A$ with someone not, for the latter it’s like the reader monad, $P \mapsto (A \to P)$.

So can’t we see any Galois connection as a kind of modal (propositional) logic, with two modes?

Families of modalities as parameterized HITs

Similar to #1, take a type and a collection of equivalence relations. Necessity indexed by $R_i$ at a point in the set is of a property holding all over that class.

Could have graded reader monads too.

Last revised on June 25, 2021 at 04:45:48. See the history of this page for a list of all contributions to it.