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I have read that a second-order logic can help one define equality by quantifying over all predicates such as what is done in the following definition:

(x=y):⟺[∀P:P(x)⟺P(y)]

By contrast a first-order logic with identity would define "=" as a primitive logical symbol.

Besides this benefit, but using it as an example of a benefit, what are other uses or benefits of a second-order logic over a first-order logic?

Han de Bruijn (https://math.stackexchange.com/users/96057/han-de-bruijn), Leibniz' Law and that good old riddle, URL (version: 2013-12-18): https://math.stackexchange.com/q/608947

Frank Hubeny
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    It depends on what counts as a benefit. Second order logic is believed to be closer to our natural reasoning. In particular, one can formulate induction in a natural way for all predicates instead of postulating it for each predicate one at a time as in the first order induction schema. One can also exclude the non-standard models of arithmetic by using it. [Boolos argued](https://www.jstor.org/stable/2026308) that natural language constructions with plural quantification are modeled by the second order logic. On the other hand, it is technically unwieldy, there is no good proof theory for it. – Conifold Sep 01 '19 at 22:57
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    See [Is first-order logic the only fundamental logic?](https://philosophy.stackexchange.com/questions/3318/is-first-order-logic-fol-the-only-fundamental-logic) – Mauro ALLEGRANZA Sep 02 '19 at 07:15
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    And see [How did first-order logic come to be the dominant formal logic](https://philosophy.stackexchange.com/questions/2617/how-did-first-order-logic-come-to-be-the-dominant-formal-logic) – Mauro ALLEGRANZA Sep 02 '19 at 07:18
  • As for natural language constructions which Conifold brought up, not only plural NPs, but also many other perfectly common natural language expressions can only straightforwardly be translated in the language of second order logic: for instance, quantifiers like *more than* or *most* can be proven not to be first-order definable; and adverbs (e.g. *slowly(walk)*) as well many ordinary adjectives/nouns/verbs (e.g. *color(red)* or *is-a-property(color)*) are properties of properties (of properties) and thus higher-order concepts. – Natalie Clarius Sep 02 '19 at 09:25

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Second Order Logic has much stronger expressive power.

Many propositions can be formalized in second order language, but can't be expressed in first order language.

For example:enter image description here

Xi Li
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  • Good answer, though it might be worth adding that since all these examples are of _monadic_ second-order predicates, all of them can be expressed in first-order logic with plural quantification. – Bumble Apr 12 '21 at 13:49