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Welcome to The Nonlinear Library, where we use Text-to-Speech software to convert the best writing from the Rationalist and EA communities into audio. This is: What is a definition, how can it be extrapolated?, published by Stuart Armstrong on March 14, 2023 on The AI Alignment Forum.
What is a definition? Philosophy has, ironically, a large number of definitions of definitions, but three of them are especially relevant to ML and AI safety.
There is the intensional definition, where concepts are defined logically in terms of other concepts (“bachelors are unmarried males”). There is also the extensional definition, which proceeds by listing all the members of a set (“the countries in the European Union are those listed here”).
Much more relevant, though with a less developed philosophical analysis, is the ostensive definition. This is where you point out examples of a concept, and let the viewer generalise from them. This is in large part how we all learnt concepts as children: examples and generalisation. In many cultures, children have a decent grasp of “dog” just from actual and video examples - and that’s the definition of “dog” we often carry into adulthood.
We can use ostensive definitions for reasoning and implications. For example, consider the famous syllogism, “Socrates is human”, “humans are mortal” imply “Socrates is mortal”. “Socrates is human” means that we have an ostensive definition of what humans are, and Socrates fits it. Then “humans are mortal” means that we’ve observed that the set of “human” seems to be mainly a subset of the set of “mortals”. So we can ostensively define humans as mortal (note that we are using definitions as properties: having the property of “being mortal” means that one is inside the ostensive definition of “mortals”). And so we can conclude that Socrates is likely mortal, without waiting till he’s dead.
Distinctions: telling what from non-what
There’s another concept that I haven’t seen articulated, which is what I’ll call the “distinction”. This does not define anything, but is sufficient to distinguish between an element of a set from non-members.
To formalise "the distinction", let Ω be the universe of possible objects, and E⊂Ω the “environment” of objects we expect to encounter. An ostensive definition starts with a list S⊂E of examples, and generalises to a “natural” category SE with S⊂SE⊂E - we are aiming to "carve reality at the joints", and get an natural extension of the examples. So, for example, E might be the entities in our current world, S might be the example of dogs we’ve seen, and SE the set of all dogs.
Then, for any set T⊂E, we can define the “distinction” dT,E which maps T to 1 (“True”) and its complement E∖T to 0 (“False”). So dSE,E would be a distinction that identifies all the dogs in our current world.
Mis-definitions
A lot of confusion around definition seems to come from mistaking distinctions for definitions. To illustrate, consider the idea of defining maleness as "possessing the Y chromosome". As a distinction, it's serviceable: there's a strong correlation between having that chromosome and being ostensively male.
But it is utterly useless as a definition of maleness. For instance, it would imply that nobody before the 20th century had any idea what maleness was. Oh, sure, they may have referred to something as "maleness" - something to do with genitalia, voting rights, or style of hats - but those are mere correlates of the true definition of maleness, which is the Y chromosome. It would also imply that all "male" birds are actually female, and vice-versa.
Scott had a description of maleness here: “Absolutely typical men have Y chromosomes, have male genitalia, appreciate manly things like sports and lumberjackery, are romantically attracted to women, personally identify as male, wear male clothing like blue jeans, sing baritone in the opera, et cetera.”
Is this a definition? I’d say not; it’s not a definition, it’s a reminder of the properties of o...