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1But we haven't got infinite time, so a few brief theories is definitely the way to go. Another word for axiom is assumption, and you know what happens when we ass/u/me.Scott Rowe– Scott Rowe2026-01-06 12:44:14 +00:00Commented Jan 6 at 12:44
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"What is of interest is the theory generated by their deductive closure; which may have multiple distinct axiomatizations." Is this a claim inherent in this that the same deductive conclusions can be reached through different, but similar ontologies?J D– J D2026-01-06 23:02:16 +00:00Commented Jan 6 at 23:02
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@JD I'm not sure, to be honest my answer is very mainstream model theory / proof theory, not a philosophical analysis. It's an interesting idea though. But all I'm saying is that there can be different sets of axioms that are logically equivalent.Julius Karl Hamilton– Julius Karl Hamilton2026-01-09 04:34:55 +00:00Commented Jan 9 at 4:34
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@JD There's so much to learn / research / think about, and so little time, but one thing your comment reminds me of is IEML by Pierre Levy - it is a formal ontology language based on a certain collection of ontological primitives. But long ago when I encountered it, I had the sneaking suspicion that the choice of ontological primitives was arbitrary. What matters is that you can build the same universe of concepts, even starting from a different chosen "basis". Concepts emerge by contrast from other concepts. I think this is similar to what you're saying.Julius Karl Hamilton– Julius Karl Hamilton2026-01-09 04:43:58 +00:00Commented Jan 9 at 4:43
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@JD But on the other hand... we might say the ontology is better given by the signature - this is where we state "what things exist". So we might have 2 equivalent sets of axioms but over the same signature. Or we might have 2 equivalent theories in totally different languages. And I think that's called "bi-interpretable". So yes, I would say... we can have interesting insights about different "ontologies" expressing the same "ultimate thing". Like, when I have the time I want to study NGB set theory, since you get a "class" of "all sets". Seems... ontologically intriguing.Julius Karl Hamilton– Julius Karl Hamilton2026-01-09 05:04:00 +00:00Commented Jan 9 at 5:04
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