(1) I understood that in the language of first order logic quantification cannot be over sets (there are no predicate variables that can be bound by quantifiers), whereas the language of second-order logic extends the language of first-order logic by allowing quantification of predicate symbols and function symbols (where these symbols may be assigned sets as their extensions http://plato.stanford.edu/entries/logic-higher-order/ ).

(2) I understand that in first order set theory (which uses a first order language) the (first order) variables x,y and z range over sets.

But suppose we have some mathematical theory, T, which concerns some kind of entity. Externally to T, these entities might be able to be understood as sets of sets (but lets suppose that they are not defined as such in the formal theory, T). They would then form the domain of T. Could we, in that case, use FO to quantify over such elements of the domain in this mathematical theory? If so, then it would seem the distinction between FO quantification and SO quantification becomes theory relative in a way which is rather strange. We could quantify over such an entity in a first order way (even though those entities outside the theory are definable as sets of sets). While Full SO quantification allows quantification over arbitrary subsets of the domain, certain of those subsets might be elements of the domain of our hypothetical mathematical theory, T, and so in T we might quantify over these elements, treating them as substitutable for FO variables. But then what does the distinction between FO and SO quantification amount to in the precise case of the hypothetical theory T? Does it simply amount to the fact that SO quantifiers can range over arbitrary subsets of the domain of T?

  • $\begingroup$ Like in your previous example regarding : individuals like "John" [that are] sets of sets [I presume : complex organisms made of structure made of cells]. So we might want to quantify over individuals (for all x, y such that x and y have the same name, x and y have the same birthday). But when we do so, names like "John" denote elements of the domain. So is this a case of FO quantification or SO quantification?" the domain is made of individuals like John, and thus the quantifiers range over them. In that "universe" there are no cells nor set of cells... $\endgroup$ – Mauro ALLEGRANZA Apr 28 '15 at 15:00
  • $\begingroup$ Thus, the logic is definitively FO. The issue is mainly with infinite domain, and thus is relevant mainly for mathematics: in FOL we can "define" only a countable number of formulae and thus any FOL theory can "speak of" at most of a countable number of properties of individuals, while it is well-known that an infinite domain has an uncountable number of subset, and thus the possible set of individual are uncountable. $\endgroup$ – Mauro ALLEGRANZA Apr 28 '15 at 15:03

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