# Skolemization

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

In general, the functions and constants symbols are new ones added to the language  for the purpose of satisfying these formulas, and are often denoted by the formula they realize, for instance $c_{\exists x\phi(x)}$.

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 $\forall^{1}x\forall^{1}y\exists^{1}z\phi(x,y,z)$ is equivalent     to $\exists^{2}F\forall^{1}x\forall^{1}y\phi(x,y,F(x,y))$.

Title Skolemization Skolemization 2013-03-22 12:59:13 2013-03-22 12:59:13 Henry (455) Henry (455) 5 Henry (455) Definition msc 03B15 msc 03B10 Skolem function Skolem constant