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 n points, the corresponding Conceptspace will have 2^n concepts. 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:
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.
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.
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.