Editing First-order logic (section)

===Algebraizations===
An alternate approach to the semantics of first-order logic proceeds via [[abstract algebra]]. This approach generalizes the  [[Lindenbaum–Tarski algebra]]s of propositional logic.  There are three ways of eliminating quantified variables from first-order logic that do not involve replacing quantifiers with other variable binding term operators:
*[[Cylindric algebra]], by [[Alfred Tarski]], et al.;
*[[Polyadic algebra]], by [[Paul Halmos]];
*[[Predicate functor logic]], primarily by [[Willard Van Orman Quine|Willard Quine]].
These [[algebra]]s are all [[lattice (order)|lattices]] that properly extend the [[two-element Boolean algebra]].

Tarski and Givant (1987) showed that the fragment of first-order logic that has no [[atomic sentence]] lying in the scope of more than three quantifiers has the same expressive power as [[relation algebra]].<ref>[[Chris Brink|Brink, C.]], Kahl, W., & [[Gunther Schmidt|Schmidt, G.]], eds., ''Relational Methods in Computer Science'' ([[Berlin]] / [[Heidelberg]]: [[Springer Science+Business Media|Springer]], 1997), [https://books.google.com/books?id=p0qqCAAAQBAJ&pg=PA32&redir_esc=y#v=onepage&q&f=false pp. 32–33].</ref>{{rp|32–33}} This fragment is of great interest because it suffices for [[Peano arithmetic]] and most [[axiomatic set theory]], including the canonical [[Zermelo–Fraenkel set theory]] (ZFC). They also prove that first-order logic with a primitive [[ordered pair]] is equivalent to a relation algebra with two ordered pair [[projection function]]s.<ref>Anon., ''[[Mathematical Reviews]]'' ([[Providence, Rhode Island|Providence]]: [[American Mathematical Society]], 2006), p. 803.</ref>{{rp|803}}