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Wikipedia says: First-order logic quantifies only variables that range over individuals (elements of the domain of discourse); second-order logic, in addition, also quantifies over relations. For example, the second-order sentence ${\displaystyle \forall P\,\forall x(x\in P\lor x\notin P)} \forall P\,\forall x(x\in P\lor x\notin P)$.

Who has first published this doctrin? And why has it been accepted?

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Modern logic began with Boole, Peirce and Frege.

Frege's system (1879) was the first "fully formalized" system of mathematical logic and was a Second- and Higher-order system of Predicate Logic: quantification over predicate of any order was allowed.

Subsequently, the first modern math lofic textbook: Hilbert & Ackermann's Grundzüge der Theoretischen Logik (1928) codified the distinction between first-order: quantification over individuals but not predicates, and second- and higher-order.

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  • $\begingroup$ Thanks for the nice references. They partially answer the second part of my question. They do not answer however this question: In a theory without urelements everything (sets, relations, functions, predicates) is an element of some set and as such an individual. Could you extend your answer to this aspect? $\endgroup$ – Wilhelm Dec 27 '17 at 11:15
  • $\begingroup$ @Wilhelm - "In a theory without urelements everything" is a set: relations, functions, numbers. The "usual" set theories are first-order; thus, the individual variables are quantified and they stand for sets. Teht have only one predicate symbol: the binary relation . $\endgroup$ – Mauro ALLEGRANZA Dec 27 '17 at 11:39
  • $\begingroup$ I fully agree with you. Therefore I do not comprehend what second-order logic should be good for. Everything is a set and as such an individual and subject to first-order logic, better simply denoted as "logic". $\endgroup$ – Wilhelm Dec 28 '17 at 8:14
  • $\begingroup$ @Wilhelm - SOL is more "expressive" than FOL; there are well-known example of mathematical properties that are not expressible in FOL. $\endgroup$ – Mauro ALLEGRANZA Dec 28 '17 at 8:56
  • $\begingroup$ Can you give an example? As far as I am informed it is impossible to quantify over all real numbers in FOL. I do not agree. I quantify over all real numbers and over all relations and functions I like. $\endgroup$ – Wilhelm Dec 28 '17 at 9:24

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