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.
Thingspace is a set of points, in the example
(Sunny, Cool, Weekday)is a point.Conceptspace is a set of sets of points from Thingspace, so
{ (Sunny, Cool, Weekday), (Sunny, Cool, Weekend) }is a concept.In general, if your Thingspace has
npoints, the corresponding Conceptspace will have2^nconcepts. To keep things a little simpler, let’s use a smaller Thingspace with only four points, which we’ll just label with numbers:{1, 2, 3, 4}. So{1}would be a concept, as would{1,2}and{2, 4}.Some concepts include others:
{1}is a subset of{1, 2}, capturing the idea of one concept being an abstraction (superset) of another.This makes Conceptspace a partially ordered set, which means it has a Hasse diagram:(from Wolfram Demonstrations Project)
Every time you move up the diagram (e.g. from
{1}to{1, 2}) you move from more concrete to more abstract in Conceptspace, or from more specific to less specific. This up-and-down ordering is what gives you the order part. The fact that not all concepts are related by moving in one direction up or down the diagram is why it’s only a partial order, and why lattice is a better description than ladder (although technically lattice in the lattice theory sense requires the poset to have extra features, but a poset that is a powerset ordered by inclusion is always a lattice.)Hopefully the visual diagram helps make the way order (up-and-down the lattice) and concept specificity (subset/superset relation) more clear.
Yes, that visual diagram is very helpful. Thank you! I think I mostly get it now.
It’s just saying that
There’s more and less general categories. E.g. “Sunny day” is more general than “Sunny and cool” because if a day is S&C then it’s also C, but there’s also days that are Sunny but not Cool.
Often, if you take two categories, neither one is strictly more general than the other one. E.g. “Sunny and cool” and “Cool and buggy”. There are days that are S&C but not C&B; and there are also days that are C&B but not S&C.
You can take unions and intersections. The intersection of “Sunny and cool” and “Cool and buggy” is “Sunny and cool and buggy”. Intersections give more specific (less abstract) categories; they add more constraints, so fewer possible worlds satisfy all those constraints, so you’re talking about some more specific category of possible worlds. The union of “Sunny and cool” and “Cool and buggy” is “Cool; and also, sunny or buggy or both”. Unions give less specific (more abstract) categories, because they include all of the possible worlds from either of the two categories.
If you want to get more specific, you want to start talking about a smaller category. So you want to go downward (i.e. to a smaller set, included inside the bigger set) in the lattice. But there’s multiple ways to do that. E.g. to be more specific than “Cool; and also, sunny or buggy or both”, you could talk about “Sunny and cool”, or you could talk about “Cool and buggy”.
(This is far from everything that “abstract”, “specific”, “category”, and “concept” actually mean, but it’s something.)
Adding to @TsviBT.
“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.”
Sounds to me like you’re on the right track. The claim made is that concepts can be ordered in terms of their abstractness. For example, the concept day would be taken to be more abstract than the concept sunny day in that day abstracts from the weather by admitting both sunny and cloudy days.
The order of concepts is ‘partial’ in that not all concepts can be compared by abstraction: for example, neither sunny nor day by themselves is more abstract than the other. So, unlike the familiar ‘total’ orderings that we see with, say, numbers, in which any two numbers can be compared/ordered by ‘less than’, the abstraction ordering on concepts is only ‘partial’ in that some pairs of concepts cannot be compared.
Hm. I think that all makes sense.
Now I’m wondering whether specificity can be measured in a sort of absolute sense rather than in a relative sense.
You mention that sunny day is more specific than day because it adds weather. Or as others have mentioned, because the set of points it includes is a subset of the set of points that day includes.
But what about the concept of “pizza that is warm, has an exterior that is thin and crispy, an interior that is warm, chewy, and fresh, a thin layer of tomato sauce that is mildly sweet and acidic, and small dollups of fresh mozzarella cheese that is cool and soft”? Can we say that this pizza concept is specific? That it’s more specific than day or even sunny day, despite being distant in Thingspace from the day and sunny day concepts?
Intuitively the answers to those questions seem to be “yes”, but I’m not sure why, at least not formally.
Maybe it’s about “potential for confusion”. If I said “the pizza was good”, the good pizza concept is it’s easy to be confused about what the concept is pointing too (thick or thin crust? red or white sauce?). I suppose this is because there are a lot points clustered together in the good pizza concept.
Alternatively, if I said “last weekend”, I used a lot less words than I did in my “pizza that is warm...” ramble, but I still think “last weekend” is quite specific. I suppose because even though I’m only specifying two “things”—“last” and “weekend”—the potential for confusion is low. There are only two days “last weekend” can be referring to.
They’re all interesting questions.
One point I’d make is that I tried to limit my discussion to ‘abstraction’ as it relates to concepts, as that seemed most pertinent to the quote you cited. The opposite of abstract is concrete. I’m not entirely sure that we should confound the abstract/concrete comparison with that of general/specific.
“pizza that is warm, has an exterior that is thin and crispy, an interior that is warm, chewy, and fresh, a thin layer of tomato sauce that is mildly sweet and acidic, and small dollups of fresh mozzarella cheese that is cool and soft”
Let’s say you’ve just described a ‘Napolitana’ pizza. Is “large Napolitana” specific in your sense? I can see a definitional regression emerging. For this reason, I suspect that specificity/generality might be best kept to comparisons relative to some ‘basis’ concepts, rather than made in some absolute sense.
“Last weekend” has its own issues, being relative to the day it is uttered. Indeed, to account for any day of utterance, it might be a difficult concept to articulate precisely. As a ‘relative’ (or ‘relational’) concept, it might be general in that it ranges over many instances.
There is the further issue of vagueness. Many of the concepts you use (eg, warm, thin, etc.) are not precise. This issue is exacerbated with good, which is, in addition, inherently subjective.
Rather than imagining a single concept boundary, maybe try imagining the entire ontology (division of the set of states into buckets) at once. Imagine a really fine-grained ontology that splits the set of states into lots of different buckets, and then imagine a really coarse-grained ontology that lumps most states into just a few buckets. And then imagine a different coarse-grained ontology that draws different concept boundaries than the first, so that in order to describe the difference between the two you have to talk in the fine-grained ontology.
The “unique infinum” of two different ontologies is the most abstract ontology you can still use to specify the differences between the first two.
Consider the set 2Tof concepts aka subsets in the Thingspace T. A concept A is a specification of another concept B if A⊆B. This allows one to partially compare concepts by specificity, whether A is more specific than B, less specific, they are equal or incomparable.
In addition, for any two concepts B and C we find that B∩C is a subset both of B and C. Therefore, it is a specification of both. Similarly, any concept D which is a specification both of B and C is also a specification of B∩C.
Additionally, B and C are specifications of B∪C, and any concept D, such that B and C are specifications of D, contains B and C. Therefore, D contains their union.
Thus for any two concepts B and C we find a unique supremum of specification B∩C and a unique infimum of specification B∪C.
There also exist many other lattices. Consider, for example, the set Z2 where we declare that (a,b)≤(c,d) if a≤b,c≤d. Then for any pairs (a,b),(c,d),(e,f) s.t.(a,b)≤(e,f) and (c,d)≤(e,f) we also know that (max(a,c),max(b,d))≤(e,f), while (a,b)≤(max(a,c),max(b,d)) and (c,d)≤(max(a,c),max(b,d)). Therefore, (max(a,c),max(b,d)) is the unique supremum for (a,b) and (c,d). Similarly, (min(a,c),min(b,d)) is the unique infimum.
I hope that these examples help.