I wanted to highlight something that wasn’t immediately obvious to me, the actual mechanism behind the non-transitivity.
(I hope you’ll forgive me for repeating you here, I’m trying to maximise my “misunderstanding surface area” and increase the chance someone corrects me.)
***
Given a subset T⊆E⊂Ω, where T is a collection of objects I’d like to define, Ebe the set of objects I expect to encounter, and Ω is the set of all possible objects.
An intensional definition identifies the set by specifying some properties that only elements of the set satisfy (and the correct intensional definition is expected to extend beyond E?)
Anextensional definition is simply telling you what every element of T
Lastly we have the ostensive definition which is given by specifying some elements of T and trusting the audience to be able to extrapolate. The extrapolation process depends on the user (it may be that there are ways of extrapolating that are more “natural”).
A distinctiondT,E is a property or collection of properties that are sufficient to distinguish the elements of the set T within E but not if we extend to the universe of all possible objects.
“Green, leafy tall things” serves as a perfectly good distinction of trees while I go for a walk at my local park, but this distinction would fail if I was to venture to Fangorn Forest.
The major idea is that the distinctions learned from an ostensive definition in one environment will not generalise to a new environment in the same way that the original ostensive definition does.
*** Mechanistically, this is occuring because the ostensive definition is not just information the examples. It’s also information about objects in the environment that are not included.
In children pointing to a “dog” in the street conveys the information that you refer to this creature as a dog, but it also conveys information that this creature is special and the other items around it are not in the category of dog. Similarly when we train a model on a classification task. We “show” the model the data, elements that are in and not in T.
Great post, really got me thinking.
I wanted to highlight something that wasn’t immediately obvious to me, the actual mechanism behind the non-transitivity.
(I hope you’ll forgive me for repeating you here, I’m trying to maximise my “misunderstanding surface area” and increase the chance someone corrects me.)
***
Given a subset T⊆E⊂Ω, where T is a collection of objects I’d like to define, Ebe the set of objects I expect to encounter, and Ω is the set of all possible objects.
An intensional definition identifies the set by specifying some properties that only elements of the set satisfy (and the correct intensional definition is expected to extend beyond E?)
An extensional definition is simply telling you what every element of T
Lastly we have the ostensive definition which is given by specifying some elements of T and trusting the audience to be able to extrapolate. The extrapolation process depends on the user (it may be that there are ways of extrapolating that are more “natural”).
A distinction dT,E is a property or collection of properties that are sufficient to distinguish the elements of the set T within E but not if we extend to the universe of all possible objects.
“Green, leafy tall things” serves as a perfectly good distinction of trees while I go for a walk at my local park, but this distinction would fail if I was to venture to Fangorn Forest.
The major idea is that the distinctions learned from an ostensive definition in one environment will not generalise to a new environment in the same way that the original ostensive definition does.
***
Mechanistically, this is occuring because the ostensive definition is not just information the examples. It’s also information about objects in the environment that are not included.
In children pointing to a “dog” in the street conveys the information that you refer to this creature as a dog, but it also conveys information that this creature is special and the other items around it are not in the category of dog. Similarly when we train a model on a classification task. We “show” the model the data, elements that are in and not in T.