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Note:I am not competent is logic so this question may look weird to you.
So as I know there are different types of logics (first-order logic, second-order...), and the difference between them is that each next order logic can quantify over bigger set of objects, and now the question which I have is, why we need such a restriction? Why we can't make logic where we could quantify over all objects? My intuition suggests me that this would bring some paradoxes, is it the real reason, if so is it the only one? If it's not, what is the reason for such restrictions in every logic? Thanks in advance.

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  • $\begingroup$ The first full version of predicate logic (see Frege) was "high-order" (also if second-order was sufficient for Frege's project). The basic divide is between FOL, which has some very "nice" properties and Second-order and Higher-order Logic. $\endgroup$ – Mauro ALLEGRANZA Dec 2 '18 at 13:51
  • $\begingroup$ But is not "quantify over bigger set of objects" but on higher types of objects : FOL quantifies over individual objects only; SOL quantifies also on properties of indivuduals; TOL over properties of properties of individuals; and so on. See also Type Theories. $\endgroup$ – Mauro ALLEGRANZA Dec 2 '18 at 14:06
  • $\begingroup$ See Melvin Fitting, Types Tableaus and Gödel’s God, Kluwer (2002) for a modern treatment. $\endgroup$ – Mauro ALLEGRANZA Dec 2 '18 at 14:08
  • $\begingroup$ @MauroALLEGRANZA Then I formulated it in the wrong way, but the question still arises, what is the reason for such restriction? $\endgroup$ – Юрій Ярош Dec 2 '18 at 14:27
  • $\begingroup$ What "restrictions" ? If you want to use Higher-order logic with quantification on properties of every order, you can use it. $\endgroup$ – Mauro ALLEGRANZA Dec 2 '18 at 14:40

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