SkolemizationMathworldPlanetmath is a way of removing existential quantifiersMathworldPlanetmath from a formulaMathworldPlanetmathPlanetmath. Variables bound by existential quantifiers which are not inside the scope of universal quantifiers can simply be replaced by constants: x[x<3] can be changed to c<3, with c a suitable constant.

When the existential quantifier is inside a universal quantifier, the bound variableMathworldPlanetmath must be replaced by a Skolem function of the variables bound by universal quantifiers. Thus x[x=0y[x=y+1]] becomes x[x=0x=f(x)+1].

In general, the functions and constants symbols are new ones added to the languagePlanetmathPlanetmath for the purpose of satisfying these formulas, and are often denoted by the formula they realize, for instance cxϕ(x).

This is used in second order logic to move all existential quantifiers outside the scope of first order universal quantifiers. This can be done since second order quantifiers can quantify over functions. For instance 1x1y1zϕ(x,y,z) is equivalentMathworldPlanetmathPlanetmathPlanetmathPlanetmath to 2F1x1yϕ(x,y,F(x,y)).

Title Skolemization
Canonical name Skolemization
Date of creation 2013-03-22 12:59:13
Last modified on 2013-03-22 12:59:13
Owner Henry (455)
Last modified by Henry (455)
Numerical id 5
Author Henry (455)
Entry type Definition
Classification msc 03B15
Classification msc 03B10
Defines Skolem function
Defines Skolem constant