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I would like to understand why SOL and FOL differ on being committed to abstract objects. It seems one difference is the existence of classes for SOL. Can classes not be made concrete? It seems like Field succeeded in something like that. I don’t think FOL is committed (is it?), so why do we treat the existence of SOL objects differently than those of FOL? Surely outside of explicitly saying objects of SOL are abstract objects, would one be so committed ontologically. Rarely does language commit oneself to a certain ontology. In the case where SOL was committed, why was FOL not?

One and only edit: Here’s broadly what I mean. Didn’t Husserl say everything including math and logic come from base sense precepts (symbolic vs actual?)? Since senses and sense precepts are not abstract, because we have direct epistemic access, SOL never requires abstract objects.

“A number of nominalists have been persuaded by Benacerraf’s (1973) epistemological challenge about reference to abstract objects and concluded that sentences with terms making apparent reference to them—such as mathematical statements—are either false or lack a truth value. They argue that those sentences must be paraphrasable without vocabulary that commits one to any sort of abstract entity. These proposals sometimes suggest that statements about abstract objects are merely instrumental; they serve only to help us establish conclusions about concrete objects. Field’s fictionalism (1980, 1989) has been influential in this regard. Field reconstructed Newtonian physics using second-order logic and quantification over (concrete) regions of space-time. A completely different tactic for avoiding the commitment to abstract, mathematical objects is put forward in Putnam (1967) and Hellman (1989), who separately reconstructed various mathematical theories in second-order modal logic. On their view, abstract objects aren’t in the range of the existential quantifier at the actual world (hence, we can’t say that they exist), but they do occur in the range of the quantifier at other possible worlds, where the axioms of the mathematical theory in question are true.” 1

“Yet appeal to second-order logic in the philosophy of mathematics is by no means uncontroversial. A first objection is that the ontological commitment of second-order logic is higher than the ontological commitment of first-order logic. After all, use of second-order logic seems to commit us to the existence of abstract objects: classes.”2

J Kusin
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  • In a nutshell because SOL quantifies over properties , sets, clsses of individuale and all them are "abstracts". – Mauro ALLEGRANZA Feb 17 '22 at 17:34
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    I think the key word is "seems". FOL similarly quantifies over abstract objects, sets and numbers, and so "seems" to commit one to their existence, by Quine's criterion. Field's fictionalization strategy removes the commitment by providing a fictional paraphrase. Since SOL quantifies over more it requires more fictionalization work to dispel the seeming, *if* one accepts Quine's terms. Of course, many now reject Quine's terms, including his former students, that is reject the idea that linguistic structure commits one to anything at all. – Conifold Feb 17 '22 at 19:02
  • @MauroALLEGRANZA & Conifold: but why/how did the linguistic structure of FOL or SOL ever commit us to abstract objects? Hasn’t there always been a competition between possible interpretations of symbols on paper and word sounds as formalist, platonic, or otherwise? Why did these few instances (FOL and SOL) *commit* us to anything when there’s been disputes about language for thousands of years? Again I see it as unless a theory says it references abstract objects explicitly, the interpretational windows has always been wide open. – J Kusin Feb 17 '22 at 20:30
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    Quine's proposal was to ignore what people themselves say about their commitments, for that is often too vague and weasely, and extract something objective, that they cannot get out of easily, from the global structure of their talk. So, from Quine's point of view, we are interested not in what formalists, etc., have to say on what they are committed to, but in what their *theories* as notated are "committed" to objectively, regardless of their wishes and self-interpretations. This was popular at one time. – Conifold Feb 18 '22 at 00:17
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    The much more expressive SOL seemingly has another sort of same abstract domain containing *all* sets of real numbers which is really multi-sorted FOL, for example. But this spec cannot exactly turn it to a FOL as the Löwenheim–Skolem theorem implies that there's *countably* infinite collection of sets of numbers whose members called "internal sets", such that the domain consisting of "internal sets" satisfies exactly the same FO sentences as are satisfied by the domain of all sets of real numbers which is uncountable. Thus SOL is at a higher abstract level than FOL without Henkin semantics... – Double Knot Feb 18 '22 at 02:47
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    FOL has quantifiers that quantify over variables, but there is no assumption about what those variables are. Someone who rejects abstract objects can use FOL and simply refuse to quantify over abstract objects. But in SOL, you quantify over predicates, which are themselves abstract objects. You can't say "I'm going to use SOL but not quantify over abstract objects" because the very thing that distinguishes SOL from FOL is that it quantifies over abstract objects. Simply by using SOL, you are in a sense committed to quantifying over abstract objects. – David Gudeman Feb 18 '22 at 04:25
  • See also [Second-Order Logic](https://plato.stanford.edu/entries/logic-higher-order/#InfaPoweSecoOrdeLogi): "Second-order logic hides in its semantics some of the most difficult problems of set theory. In Philosophy of Logic [1970], W. V. Quine summed up a popular opinion among mathematical logicians by referring to second-order logic as “set theory in sheep’s clothing”." – Mauro ALLEGRANZA Feb 18 '22 at 07:41
  • @DavidGudeman I guess what I'm confused at: isn't there always at the very least a formalist interpretation too when symbols are being written? – J Kusin Feb 18 '22 at 15:49
  • I see what you mean now. I'm no longer up on this area, but I'm 95% sure I've read a proof that quantifying over parameterized WFFs is strictly less powerful than quantifying over predicates because there are uncountable predicates and only countable WFFs. Your proposed formalist interpretation of SOL sounds like quantifying over parameterized WFFs. I think this proof involved lambda expressions in FOL, in case that helps your search. – David Gudeman Feb 19 '22 at 07:38

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