Disclaimer: if you are using a definition in a nonmathematical piece of writing, you are probably making a mistake; you should just get rid of the definition and instead use a few examples. This applies double to people who think they are being "rigorous" by defining things but are not actually doing any math. Nonetheless, definitions are still useful and necessary when one is ready to do math, and some pre-formal conceptual work is often needed to figure out which mathematical definitions to use; thus the usefulness of this post.

Suppose I’m negotiating with a landlord about a pet, and in the process I ask the landlord what counts as a “big dog”. The landlord replies “Well, any dog that's not small”. I ask what counts as a “small dog”. The landlord replies “Any dog that's not big”.

Obviously this is “not a proper definition”, in some sense. If [...]

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