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> Second order logic sort of assumes that we can simply know what finite means. First order logic doesn't allow you to wave your hands like that. OK, this I do
by lisper 2y ago
> Second order logic sort of assumes that we can simply know what finite means. First order logic doesn't allow you to wave your hands like that.
OK, this I do not understand. How does 2ndOL "sort of assume that we can simply know what finite means"? That seems like an extraordinarily imprecise characterization of 2ndOL. 2ndOL is a formal system just like 1stOL. Neither one lets you "wave your hands" about anything AFAICT.
- btilly 2y agoRead https://plato.stanford.edu/entries/logic-higher-order/ https://plato.stanford.edu/entries/logic-higher-order/ if you wish. I view the "for all properties" bit as handwaving. See particularly section 6 of that article, which at one point gives: The semantics of second-order logic depending on these hard questions of set theory puts second-order logic into the same “basket” with set theory. Criticism of set theory becomes criticism of second-order logic and vice versa. My Constructivist tendencies are due to my discomfort with accepting the validity of set-theoretic constructions with decisions based on unverifiable truths. Accepting second order logic requires accepting quantifying over similar collections of unverifiable claims. I'm going to be no more sympathetic.
- lisper 2y agoThanks.