Talk:Quantifier elimination

A question about the Example

Is not the given example quantified as well by "For real a, b, and c," and this quantification applies also to the sentence on the right-hand side of the biconditional? If so, then the sentence on the right-hand side is not free of quantifiers. — Preceding unsigned comment added by Aliotra (talk • contribs) 21:25, 26 July 2026 (UTC)

Fixed. D.Lazard (talk) 14:14, 27 July 2026 (UTC)

Proposed move

What would people think of moving this title to "elimination of quantifiers", which now redirects to this page? Michael Hardy 18:59, 13 July 2007 (UTC)

"Quantifier elimination" has far more Google scholar hits, for what that's worth. Algebraist 18:37, 5 September 2008 (UTC)

The phrase "quantifier elimination is what I use and find used. I have some lecture notes on quantifier elimination here, which may be useful:

 http://lara.epfl.ch/dokuwiki/sav09:lecture_09

Vkuncak (talk) 15:53, 21 June 2009 (UTC)

Formula / Sentence

Changed this over because QE is for all formulae not just sentences.

Note that for any complete theory $T$ (in a language with a constant symbol $a$) we have that every sentence is equivalent mod $T$ to either $a=a$ or $\lnot (a=a)$. Thehalfone (talk) 09:59, 8 February 2009 (UTC)

Skolemization and Herbrandization

I think that Skolemization and Herbrandization should be mentioned or at least linked to. —Preceding unsigned comment added by 90.156.82.87 (talk) 10:07, 6 May 2011 (UTC)

ETA: Perhaps a mention that Skolemization and Herbrandization are really a different thing as they are concerned with logics whereas quantifier elimination is concerned with models? —Preceding unsigned comment added by 90.156.82.87 (talk) 14:10, 12 May 2011 (UTC)

Alternate Definition of Quantifier Elimination

John Doner and Wilfrid Hodges, in their paper "Alfred Tarski and Decidable Theories" (Journal of Symbolic Logic, vol 53, no 1, March 1988, pp 20–35), give a formal definition of "the method of eliminating quantifiers" wherein the definiens does not require (or even mention) eliminating any quantifiers. See specifically page 24 of Doner and Hodges's paper. Tarski himself, in his paper "Grundzüge des Systemenkalküls. Zweiter Teil" (Fundamenta Mathematicae, vol 26, 1936, pp 283–301), uses the phrase "sukzessiven Elimination der Operatoren" (successive elimination of quantifiers) in describing his proof that every sentence in the elementary theory of dense order is deductively equivalent to a Boolean combination of the two sentences: "There is no first element" and "There is no last element" ... even though both of these sentences, on the face of them, employ quantifiers. See specifically pages 293 and 295 of Tarski's paper. Jan Zygmunt, in a paper titled "Alfred Tarski: Auxiliary Notes on His Legacy" (in the Birkhäuser volume The Lvov-Warsaw School. Past and Present, 2018, pp 425–455), gives a succinct overview of the method of eliminating quantifiers in the context of decidable and undecidable theories, quoting both the Doner and Hodges paper and the Tarski paper, and also quoting Chang and Keisler on the subject. See specifically pages 440 and 441 of Zygmunt's paper. The scope of the concept appears to have quickly broadened as its usefulness as a tool of discovery became clearer, and thus it outgrew its original name. However the name seems to have been too catchy to be bothered changing. The article in its present form (2021-Aug-15) speaks to the concept that originally gave rise to the name. The article is silent on the broadened definition that apparently supplanted the protoconcept sometime before Tarski's 1936 paper. Grandmotherfrompeoria (talk) 16:40, 13 August 2021 (UTC)