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IlyaShpitser comments on Conceptual Analysis and Moral Theory - Less Wrong
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There is an entire chapter in Pearl's Causality book devoted to the rabbit-hole of defining what 'actual cause' means. (Note: the definition given there doesn't work, and there is a substantial literature discussing why and proposing fixes).
The counterargument to your post is that some seemingly fuzzy concepts actually have perfect intuitive consensus (e.g. almost everyone will classify any example as either concept X or not concept X the same way). This seems to be the case with 'actual cause.' As long as intuitive consensus continues to hold, the argument goes, there is hope of a concise logical description of it.
Maybe the concept of "infinity" is a sort of success story. People said all sorts of confused and incompatible things about infinity for millennia. Then finally Cantor found a way to work with it sensibly. His approach proved to be robust enough to survive essentially unchanged even after the abandonment of naive set theory.
But even that isn't an example of philosophers solving a problem with conceptual analysis in the sense of the OP.
Thanks for the Causality heads-up.
Can you name an example or two?
Well, as I said, 'actual cause' appears to be one example. The literature is full of little causal stories where most people agree that something is an actual cause of something else in the story -- or not. Concepts which have already been formalized include concepts which are both used colloquially in "everyday conversation" and precisely in physics (e.g. weight/mass).
One could argue that 'actual cause' is in some sense not a natural concept, but it's still useful in the sense that formalizing the algorithm humans use to decide 'actual cause' problems can be useful for automating certain kinds of legal reasoning.
The Cyc project is a (probably doomed) example of a rabbit-hole project to construct an ontology of common sense. Lenat has been in that rabbit-hole for 27 years now.