quantifier algebra

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

 $\exists:P(V)\to B^{B}$

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

1. 1.

the pair $(B,\exists(I))$ is a monadic algebra for each subset $I\subseteq V$,

2. 2.

$\exists(\varnothing)=I_{B}$, the identity function on $B$, and

3. 3.

$\exists(I\cup J)=\exists(I)\circ\exists(J)$, for any $I,J\in P(V)$.

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

Think of $V$ as a set of variables  and $B$ a set of propositional functions closed under  the usual logical connectives. From this, $\exists(I)$ in the first condition can be viewed as the existential quantifier  $\exists$ bounding a set $I$ of variables. The second condition stipulates that, when no variables are bound by $\exists$, then $\exists$ has no effect on the propositional functions. The last condition states that the order and frequency of the variables bound by $\exists$ does not affect the outcome ($\exists x_{2},x_{1},x_{2}$ is the same as $\exists x_{1}\exists x_{2}$).

Remarks

References

Title quantifier algebra QuantifierAlgebra 2013-03-22 17:49:04 2013-03-22 17:49:04 CWoo (3771) CWoo (3771) 10 CWoo (3771) Definition msc 03G15 MonadicAlgebra PolyadicAlgebra locally finite