Editing Second-order logic (section)

==History and disputed value==

Predicate logic was introduced to the mathematical community by [[Charles Sanders Peirce|C. S. Peirce]], who coined the term ''second-order logic'' and whose notation is most similar to the modern form (Putnam 1982). However, today most students of logic are more familiar with the works of [[Gottlob Frege|Frege]], who published his work several years prior to Peirce but whose works remained less known until [[Bertrand Russell]] and [[Alfred North Whitehead]] made them famous.  Frege used different variables to distinguish quantification over objects from quantification over properties and sets; but he did not see himself as doing two different kinds of logic.  After the discovery of [[Russell's paradox]] it was realized that something was wrong with his system.  Eventually logicians found that restricting Frege's logic in various ways—to what is now called [[First-order predicate calculus|first-order logic]]—eliminated this problem: sets and properties cannot be quantified over in first-order logic alone.  The now-standard hierarchy of orders of logics dates from this time.

It was found that [[set theory]] could be formulated as an axiomatized system within the apparatus of first-order logic (at the cost of several kinds of [[completeness (logic)|completeness]], but nothing so bad as Russell's paradox), and this was done (see [[Zermelo–Fraenkel set theory]]), as sets are vital for [[mathematics]].  [[Arithmetic]], [[mereology]], and a variety of other powerful logical theories could be formulated axiomatically without appeal to any more logical apparatus than first-order quantification, and this, along with [[Kurt Gödel|Gödel]] and [[Thoralf Skolem|Skolem]]'s adherence to first-order logic, led to a general decline in work in second (or any higher) order logic.{{Citation needed|date=January 2010}}

This rejection was actively advanced by some logicians, most notably [[W. V. Quine]].  Quine advanced the view{{Citation needed|date=January 2010}} that in predicate-language sentences like ''Fx'' the "''x''" is to be thought of as a variable or name denoting an object and hence can be quantified over, as in "For all things, it is the case that . . ." but the "''F''" is to be thought of as an ''abbreviation'' for an incomplete sentence, not the name of an object (not even of an [[abstract object]] like a property).  For example, it might mean " . . . is a dog."  But it makes no sense to think we can quantify over something like this. (Such a position is quite consistent with Frege's own arguments on the [[concept and object|concept-object]] distinction). So to use a predicate as a variable is to have it occupy the place of a name, which only individual variables should occupy. This reasoning has been rejected by [[George Boolos]].{{citation needed|date=April 2020}}

In recent years{{when|date=October 2017}} second-order logic has made something of a recovery, buoyed by Boolos' interpretation of second-order quantification as [[plural quantification]] over the same domain of objects as first-order quantification (Boolos 1984). Boolos furthermore points to the claimed [[nonfirstorderizability]] of sentences such as "Some critics admire only each other" and "Some of Fianchetto's men went into the warehouse unaccompanied by anyone else", which he argues can only be expressed by the full force of second-order quantification.  However, [[generalized quantifier|generalized quantification]] and [[branching quantification|partially ordered]] (or branching) quantification may suffice to express a certain class of purportedly nonfirstorderizable sentences as well and these do not appeal to second-order quantification.