Saphiro in his "foundations without foundationalism: a case for second-order logic" defends second-order logic by claiming that talking about subsets of domain is not problematic in case of SOL. He defends it because before in the past sets theories proven to be inconsistent (Russel's paradox) but he claims that logical sets are not the same as iterative (ZFC) sets. Logical set has to be perceived as Boole sets which complements are sets them selves. Because of that they somehow should be free of contradictions and easier to work with. He didn't prove in any way how Boole set theory is not contradictory. He didn't also said what is aforementioned Boole set. So what is this set and why is it less problematic?
2026-03-25 17:34:57.1774460097
What is the difference between logical and iterative set
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Think, to start with, of logical sets (in a given context) as all being subsets of a fixed universe of discourse, a universe of non-sets (numbers or whatever).
There's a useful brief discussion of whether this limited conception of sets is really enough for second order logic in the final chapter on higher-order logics in the Oxford Handbook of Philosophy of Math and Logic (see pp. 796-797).
As the Handbook essay in fact notes, Shapiro in fact later modified his view, in a paper you can download here http://www.accionfilosofica.com/misc/1214145300crs.pdf