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There are a number of questions on this site about second-order logic (admittedly I may have asked a few more of them than is reasonable). For example, searching "second-order logic" (in quotes) currently returns 152 hits, and skimming through them most do indeed seem to be about $\mathsf{SOL}$. Math.stackexchange does have a tag for this topic, and I'd like to add one here too, but I want to get some feedback first:

Does it seem reasonable to have a "second-order-logic" tag?

Here's a possible tag description:

For questions about second-order logic, or more generally any logical system with quantification over sets/relations/functions as well as individual objects.

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    $\begingroup$ It might be a reasonable suggestion. $\endgroup$
    – Asaf Karagila Mod
    Commented Dec 15, 2020 at 22:50
  • $\begingroup$ You have my vote. $\endgroup$
    – Jordan Barrett
    Commented Dec 16, 2020 at 8:52
  • $\begingroup$ I don't think so. Most questions I've seen about this reflect misunderstandings, or are aimed at, say, distinguishing second-order arithmetic (which is a first order theory whose objects are numbers and sets of numbers) from the historical axiomitization of arithmetic in second-order logic (which is not of much use in current study). $\endgroup$
    – none
    Commented Dec 16, 2020 at 22:52
  • $\begingroup$ @none " Most questions I've seen about this reflect misunderstandings" That hasn't been my experience, and very few of the hits on the first page of the search above fit that pattern. "Or are aimed at [...]" I don't see what's wrong with that, that's an ineresting topic. $\endgroup$
    – Noah Schweber
    Commented Dec 16, 2020 at 23:33
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    $\begingroup$ I could be wrong, but it seems a bit narrow. Do you feel "logic" is too broad of a tag? How about something like "higher order logics" instead? There is sometimes interest in set theory in $n^{th}$ order logic, and also in logics in between 1st and 2nd order (examples: Chang's Conjecture has such a characterization, and also there is work on "stationary logic" and connection with large cardinals). $\endgroup$
    – Monroe Eskew
    Commented Dec 19, 2020 at 18:31
  • $\begingroup$ @MonroeEskew Now I looked at Mathematics - I see that on that site there are two separate tags (higher-order-logic) and (second-order-logic). The tag-info basically makes one of them subset of the other - for tags which are relatively close to each other this does not seem an optimal choice. $\endgroup$
    – Martin Sleziak
    Commented Dec 22, 2020 at 9:19

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Based on the feedback of the past week, I've gone ahead and created a tag (following Monroe Eskew's suggestion of "higher-order-logics" instead of "second-order-logic"). I've kept the tag description very short, and others should feel free to improve it.

I'll add this tag to relevant questions over time, but only gradually to avoid flooding the front page.

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  • $\begingroup$ I think it would be better if the description were shorter and the details were in the tag wiki. $\endgroup$
    – François G. Dorais Mod
    Commented Dec 23, 2020 at 4:44

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