[Question] What does “lattice of abstraction” mean?

[Adam Zerner]

I’ve been thinking about specificity recently and decided to re-read SotW: Be Specific. In that post Eliezer writes the following:

S. I. Hayakawa called this the ladder of abstraction. I’m not sure if understanding the following section will really help with the skill of Being Specific, or help anyone construct exercises for the skill of being specific. But a better theoretical understanding does sometimes prove useful. So I will now digress to explain that abstraction isn’t really a ladder, but a lattice.

I think I understand some of what he’s saying. I think about it in terms of drawing boundaries around points in Thingspace. So the concept of “Sunny Days” is drawing a boundary around the points:

• { Sunny, Cool, Weekday }

• { Sunny, Cool, Weekend }

• { Sunny, Hot, Weekday }

• { Sunny, Hot, Weekend }

And the concept of “Sunny Cool Days” is drawing a narrower boundary around the points:

• { Sunny, Cool, Weekday }

• { Sunny, Cool, Weekend }

And so we can say that “Sunny Cool Days” is more specific than “Sunny Days” because it draws a narrower boundary.

But I still have no clue what a lattice is. Wikipedia’s description was very intimidating:

A lattice is an abstract structure studied in the mathematical subdisciplines of order theory and abstract algebra. It consists of a partially ordered set in which every pair of elements has a unique supremum (also called a least upper bound or join) and a unique infimum (also called a greatest lower bound or meet). An example is given by the power set of a set, partially ordered by inclusion, for which the supremum is the union and the infimum is the intersection. Another example is given by the natural numbers, partially ordered by divisibility, for which the supremum is the least common multiple and the infimum is the greatest common divisor.

Maybe a lattice is simply what I described: a boundary around points in Thingspace. But I get a sense that “lattice” involves order in some way, and I am not seeing how order fits in to the question of how specific a concept is. I also think it’s plausible that there are other aspects of lattices that are relevant to the discussion of specificity that I am missing.