> Non-Boolean logics are rare, and relations more general than partial order are almost never useful.
That's a bold claim! In set theory, a relation is just some set of pairs made from some base set of elements. I guess the sentence should say "orderings more exotic than the partial order" instead or something, to be more specific.
For example, if I'm implementing a Sokoban engine, I might define a "step" relation on game states, informally saying that two game states are "step"-related if you can get from one state to the other in a single move. From another perspective, this is just an implicit definition of the edges in a large or infinite graph of game states. That's because the concept of a set-theoretic relation is in a sense isomorphic to the concept of a graph-theoretic edge set: A relation is a subset of the possible pairs of elements, just like an edge set is a subset of the possible pairs of vertices. And just like a relation can be symmetric ("for all a,b, if a is related to b, then b is related to a"), so can an edge set be symmetric or undirected.
And of course intervals are comparable so long as a comparing relationship be defined on them — one might argue that it's never going to be a sensible or useful one, but one could make that same argument for a comparison relationship between skyscrapers that simply compares their height.
If we look at medieval, or even Victorian, times, people (at least the elite and the middle classes who aped them) were obsessed with putting total orders on all social interactions. Thankfully we live in the twenty-first century, where our social relations remain transitive and reflexive, but we attempt to replace antisymmetric interactions with (at least on paper) symmetric ones.
> While in C++, a sorting algorithm will run on anything with the operator< reloaded with no correctness guaranteed or even pinky-promised.
Though this claim is correct, it's important to state that C++20 also gained language support for working with weaker equivalence/ordering relations. Details:
There is no reason why a language or proof system cannot use different axiomatic bases of orders, as long as they have different identities (names).
The partial order of intervals, where any overlap means equality, is perfectly respectable, and can sit alongside other theory fragments about total/partial/strict orders.
The well-known algebra of intervals (1983), with an enlarged set of axioms and operators, may also be introduced, to subsume this proliferation of theory fragments:
The algebra is almost trivial: anyone who spends 5 minutes scribbling on a sheet of paper would come up with the same solution.
There is also a beautiful symmetrical state machine of one interval smoothly sliding over another, which I leave as another 5-minute exercise for the reader with aforementioned sheet of paper.
P.S. Sometimes such professed 'issues' are just flourishes of pedagogical knives, which seem dangerous to the naive student, but the resolution is just to shoot the prophet of doom. For example, the many varied implementations of Fibonacci series can explain the differences between iterative/(tail-)recursive/functional/memo-ized implementations, while hiding the facts that:
(1) nobody really needs Fibonacci numbers, unless you are unlucky enough to be modelling the number of immortal rabbits, in a world without foxes, but with an infinite series of cabbages;
(2) the inside secret is that there's a closed-form expression called the Binet Formula (1843):
The mystical prophets never tell their students about this, even though the facts have been well-known for 40-180 years, because it would hurt the business of prophet-ing.
> The partial order of intervals, where any overlap means equality, is perfectly respectable
Not really, since then it wouldn't be an equivalence relation. Equality in mathematics is weird, but whatever it is in a given context, it's always an equivalence relation.
I don't want to trivialize this article, but I feel like the whole thing can be reduced to the statement "define 'smaller'".
Instead it reads like a "this is what 'smaller' means for intervals" at which point I just want to comment "why?" and add a "[citation needed]" tag.
It all depends on what your interval represents, or how you can agree that it should be treated. Does it relate to confidence intervals? Does it summarise a distribution? Should it relate to stochastic dominance? Should it represent conflicting evidence between true and false? Should it be treated as a fuzzy membership function?
All these interpretations would lead to different, equally valid interpretations for the term 'smaller'.
That's a bold claim! In set theory, a relation is just some set of pairs made from some base set of elements. I guess the sentence should say "orderings more exotic than the partial order" instead or something, to be more specific.
For example, if I'm implementing a Sokoban engine, I might define a "step" relation on game states, informally saying that two game states are "step"-related if you can get from one state to the other in a single move. From another perspective, this is just an implicit definition of the edges in a large or infinite graph of game states. That's because the concept of a set-theoretic relation is in a sense isomorphic to the concept of a graph-theoretic edge set: A relation is a subset of the possible pairs of elements, just like an edge set is a subset of the possible pairs of vertices. And just like a relation can be symmetric ("for all a,b, if a is related to b, then b is related to a"), so can an edge set be symmetric or undirected.