quantifier algebra

A quantifier algebra is a triple (B,V,), where B is a Boolean algebraMathworldPlanetmath, V is a set, and is a function


from the power setMathworldPlanetmath of V to the set of functions on B, such that

  1. 1.

    the pair (B,(I)) is a monadic algebra for each subset IV,

  2. 2.

    ()=IB, the identity function on B, and

  3. 3.

    (IJ)=(I)(J), for any I,JP(V).

The cardinality of V is called the degree of the quantifier algebra (B,V,).

Think of V as a set of variablesMathworldPlanetmath and B a set of propositional functions closed underPlanetmathPlanetmath the usual logical connectives. From this, (I) in the first condition can be viewed as the existential quantifierMathworldPlanetmath bounding a set I of variables. The second condition stipulates that, when no variables are bound by , then has no effect on the propositional functions. The last condition states that the order and frequency of the variables bound by does not affect the outcome (x2,x1,x2 is the same as x1x2).


  • A monadic algebra is a quantifier algebra where V={x}, a singleton, and a Boolean algebra is just a quantifier algebra with V=.

  • In classical first order logic, the set of variables bound by a quantifier appearing in a formulaMathworldPlanetmathPlanetmath is finite. Any variable not bound by the quantifier is considered free, as far as the scope of the quantifier is concerned. This basically says that every propositional function in the classical first order logic has a finite number variables. Translated into the languagePlanetmathPlanetmath of quantifier algebras, this means that

    for each pB, there is a finite IV, such that (V-I)(p)=p.

    Any quantifier algebra satisfying the above condition is said to be locally finitePlanetmathPlanetmath.

    Alternatively, a set IV is called a support of pB if (V-I)(p)=p. The intersectionMathworldPlanetmath of all supports of p is called the support of p, denoted by Supp(p). B is locally finite iff every element of B has a finite support, or that Supp(p) is finite.

  • Quantifier algebras are a step closer in fully characterizing the “algebraMathworldPlanetmathPlanetmathPlanetmath” of predicate logic than monadic algebras. However, it is not powerful enough to address situations where a “change of variable” occurs in a propositional function, such as x(x2+z2=1) versus y(y2+z2=1). In a quatifier algebra, these two formulas are distinct, even though they are the same semantically in logic. In order take into account these additional considerations, polyadic algebras are needed.


  • 1 P. Halmos, Algebraic Logic, Chelsea Publishing Co. New York (1962).
  • 2 B. Plotkin, Universal AlgebraMathworldPlanetmathPlanetmath, Algebraic Logic, and Databases, Kluwer Academic Publishers (1994).
Title quantifier algebra
Canonical name QuantifierAlgebra
Date of creation 2013-03-22 17:49:04
Last modified on 2013-03-22 17:49:04
Owner CWoo (3771)
Last modified by CWoo (3771)
Numerical id 10
Author CWoo (3771)
Entry type Definition
Classification msc 03G15
Related topic MonadicAlgebra
Related topic PolyadicAlgebra
Defines locally finite