David Corfield graded modality

See nLab: graded modality

Idea

Modalities via $W \to \mathbf{1}$, and then via surjection $W \to V$. Through cospan $W \to V \leftarrow W$. But this is equivalent to an equivalence relation, so a certain span.

Modalities are push-pull through a span $W \leftarrow R \to W$. Then we might have an indexed set of relations, $R_i$. Perhaps ordered by inclusion.

More generally, when dealing with a span $A \leftarrow C \to B$, we generate an ‘intermodality’, indexed by the span.

With $A \leftarrow C \to B$, we take an $A$-property (or type) to a $B$-property (or type). $\phi$ to $b$ which allow/enforce $\phi$.

Span of plain sets, and then use $C$ and maps as index. Perhaps $A = B$, make $C$ a subobject of $A \times B$, that is, a relation.

Returning to relations, $W \leftarrow R_i \to W$ from $I \times W\times W \to 2$ or $[I, Rel(W)]$.

If a relation is a graph (variety of relation and variety of graph), then metric space is a kind of $[0, \infty]$-valued graph. With cut-offs, distance below $c$.

World types as contexts, iterated dependent sums. See it in reals, or p-adics. Two elements of a context are as close as

Cases

Case 1

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.

Case 2

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

1. There is a red apple.
2. There is an apple which seems to be red.
3. 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.

Case 3

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

Case 4

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, $P \mapsto (A \to P)$.

Case 5

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

Case 6

Families of modalities as parameterized HITs

Case 7

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.

Case 8

Could have graded reader monads too.

Case 9

Something that continues to puzzle me about analytic philosophy is its devotion to untyped logic. One conceptual tangle that ensues from this choice is what to make of quantifiers such as ‘all’. (On the one hand, we seem to quantify universally over different domains; on the other, our logic contains a single universal quantifier. Now, of course, to achieve quantification restricted to entities with property $A$, one may just write $\forall x(A(x) \to \ldots)$.

However, the quantifier variance supporters are not satisfied. The ‘every’ and ‘all’ of our speech to them mean something different depending on what follows. Now, anyone with a background in typed or sorted logics is likely to think of this variance as typed or sorted.

Can it really be as late as 2016 that Hans Halvorson notes

“To date, none of the numerous discussions of quantifier variance have noted the relevance of many-sorted logic.”

https://www.princeton.edu/~hhalvors/papers/collapse.pdf

He then proceeds to resolve the debate in these terms.

Let’s take things up a notch with dependent type theory. Then, yes, there’s a univocality to universal quantification: it arises from the dependent product type formation rule (dependent function, some call it). And, yes, there’s variation in that dependent products are indexed by types, and indeed by functions between types.

A predicate, $P$, of type $D$ is sent to the proposition $\forall x:D P(x)$.

But given a function $f: D \to E$, there’s a predicate on $E$, $\forall_f P(y)$, which asks at $e:E$, is $P$ the case at all the preimages under $f$? $\forall_f P(e)$ is true if $\forall_{d:D}(f(d) = e \to P(d))$. nLab

An example of the latter is $nation: Person \to Country$, where $\forall_{nation}Happy(y)$ is the property of a country, $y$, that all of its people are happy.

So there we are, quantifier variance aplenty. Now onto graded modalities.

Graded modalities (https://ncatlab.org/nlab/show/graded+modality) date back to philosophical work in the 70s, but the interesting work today is being done by linguists, such as @DanLassiter, and computer scientists, such as my colleague at Kent @dorchard.

One way to achieve a collection of such modalities arises out of the quantifiers we have just seen.

We’ve seen how along a map $f:D \to E$, dependence on $D$ can be mapped to dependence on $E$. Then a simple map brings us back in the opposite direction.

Composing these maps gives us an operator from $D$-dependent properties to itself. In the person-country case, it sends a predicate of a person, say, happy(x), to another such, all_compatriots_happy(x). (Details in Chap 4 of my book.)

It’s acting like a necessity operator, capturing the situation where a property holds under variation. It’s not just that you’re happy, all of your compatriots are happy too. We can then modify the mapping to give a finer discrimination, e.g., to country region.

A graded set of modalities may be derived from a series of onto mappings $f_i: D \to E_i$, each $E_i$ mapping onto the next. For an element of $D$, operators are expressing invariance under lesser or greater variation. The operators themselves compose.

Many variants on this theme are possible, but we can already begin to see, e.g., how degrees of obligatoriness, etc., can be captured by the ranges of situations in which an action is required.

The idea of a sequence of epimorphisms seemed apt since epis from a fixed object form a preorder, and so composing two of these modalities results in invariance under the coarser class. We could also consider non-symmetric relations, $R$ as subobjects of $D \times D$. Then consider the span of projections $D \leftarrow R \to D$, and dependent product along $p_1$, followed by base change along $p_2$. (A form of ‘intermodality’ https://arxiv.org/abs/2101.10490 .)

Then we’d have modalities graded by all relations, and composition via composition of relations.

Ceteris paribus, knowing the range of variation.

Let us reserve the notions of necessity and contingency for questions of the second type, namely, questions about the variation in end results that ensues from a difference in initial conditions or intervening factors. An event will be more contingent the more sensitive it is to initial conditions and intervening factors, and more necessary the less contingent it is, that is, the less sensitive it is to initial conditions and intervening factors. Thus, the defeat is more necessary if a similar defeat would have occurred in the absence of a number of conditions that in fact obtained – the storm, the sleepless night, the tactics chosen by the commander – and more contingent if changing these conditions would have altered the result significantly (see figure 9.2) (p. 122)

Since the notions of necessity and contingency assume sets of more or less similar events, their application is inherently sensitive to the descriptions we use in referring to events. To assess degree of necessity, we need to know whether the same type of event would have occurred given a certain change or intervention. Our assessment hinges, therefore, on modes of sorting and individuation, on what we consider a type, or the same type. “The war was necessary” means “a similar kind of war would have occurred in any event,” hence a judgment about the degree of necessity that should be ascribed to an event will depend on how broadly or narrowly we construe the type in question. Knowing early twentieth-century European history, we may believe a war would have started sooner or later. But if the historian uses finer distinctions – a war in 1914, a war triggered by an assassination, she might lower the level of necessity she ascribes to the war, attributing increased significance to the assassination in Sarajevo. (Ben-Menahem, Y., p 124)

Case 10

What of Spivak et al on intermodalities? These are graded by spans. $A \leftarrow C \to B$. Properties on $A$ allow and ensure properties on $B$. Can widen this to all $A$, $B$ dependent types.

Then composition possible along spans.

Comparisons for different $C$ and same $A$ and $B$, can vary power to allow, ensure.

Extra

In the wake of Kripke’s and Hintikka’s influential work, it has become a sort of default analytic choice in formal semantics to treat an enormous variety of modal expressions—of varying syntactic categories—as picking out SOME/ALL-quantifiers over restricted domains of possible worlds:may,might, can, should, ought, have to, able, certain/certainly, possible/possibly, necessary/necessarily,believe, think, know, suspect, want, wish, and many more. Importantly, a formalization along these lines implies that the concept being analyzed is not graded: SOME and ALL are all-or-nothing concepts. However, as we will see in section 3 below, the “some/all worlds” analysis has been developed for certain items in a way that also attends to the graded structure of many modal concepts—most influentially, in the work of Lewis (1973) and Kratzer (1981, 1991, 2012).

Lassiter, pp. 5-6

Related

• modality

Works

Daniel Lassiter, Graded modality, OUP, Lassiter, Graded Modality