Relations between Concepts: Subsethood and Implication

It’s nice to have a mathematical definition of concepts in a conceptual space. It’s also nice that we can create new concepts based on old ones, for instance by intersecting them. But being able to talk about the relation of two concepts is certainly also useful. Last time, we talked about the size of a concept. We can use the size of concept to figure out that the concept of “animal” is more general than the concept of “Granny Smith” – simply because it is larger.

But there are also other ways of describing the relation of two concepts. Two of them, namely subsethood and implication, will be presented in today’s blog post.

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What’s the size of a concept?

A few weeks ago, I got the notification that my paper “Measuring Relations between Concepts in Conceptual Spaces” [1] (preprint available here) was accepted at the British SGAI Conference on Artificial Intelligence.

One of the question that I discuss there is posed in the title of this blog post: What’s the size of a concept?

In general, one can say that the size of a concept in a conceptual space tells you something about its specificity: Small concepts (like Granny Smith) are more specific, whereas large concepts (like fruit) are more general.

But how exactly can we measure the size of such a concept within my proposed formalization? My paper [1] gives a mathematical response to that, and today I would like to sketch the basic idea behind it.

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My implementation of conceptual spaces

As stated earlier, the goal of my PhD research is to develop a system that can autonomously learn useful concepts purely from perceptual input. For instance, the system should be able to learn the concepts of apple, banana, and pear, just by observing images of fruits and by noting commonalities and differences among these images.

So far, I have mainly been talking about the conceptual spaces framework and how we can mathematically formalize it. However, as I want to actually end up with a running system, I need to implement my formalization in a computer program. So in today’s blog post, I’d like to introduce my implementation, which is publicly available: https://github.com/lbechberger/ConceptualSpaces

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Imprecise concept boundaries with fuzzy sets

In my last blog post, I explained the underlying idea of my formalization of the conceptual spaces framework. At the end of the text, I made the remark:

There’s one little detail left, and I’ll cover it in my next blog post. It is the problem of imprecise concept boundaries.

This problem is probably best explained by an example: Suppose we would like to define the meaning of “tall person”. Let’s say we call everybody who has a height of at least 1.80 m “tall”, and everybody below this threshold “not tall”. Why is this problematic?

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