Let $B$ be a Boolean algebra. An existential quantifier operator on $B$ is a function $\exists: B\to B$ such that

1. $\exists(0)=0$ ,
2. $a\le \exists(a)$ , where $a\in B$ , and
3. $\exists(a\wedge \exists(b))=\exists(a)\wedge \exists(b)$ , where $a,b\in B$ .

There is an obvious connection between an existential quantifier operator on a Boolean algebra and an existential quantifier in a first order logic:

1. A statement $\varphi(x)$ is false iff $\exists x \varphi(x)$ is false. For example, suppose $x$ is a real number. Let $\varphi(x)$ be the statement $x=x+1$ . Then $\varphi(x)$ is false no matter what $x$ is. Likewise, $\exists \varphi(x)$ is always false too.
2. $\varphi(x)$ implies $\exists x\varphi(x)$ ; in other words, if $\exists x\varphi(x)$ is false, then so is $\varphi(x)$ . For example, let $\varphi(x)$ be the statement $1<x$ , where $x\in \mathbb{R}$ . By itself, $\varphi(x)$ is neither true nor false. However $\exists x\varphi(x)$ is always true.
3. $\exists x (\varphi(x) \wedge \exists x \psi(x))$ iff $\exists x \varphi(x) \wedge \exists x \psi(x)$ . For example, suppose again $x$ is real. Let $\varphi(x)$ be the statement $x<1$ and $\psi(x)$ the statement $x>1$ . Then both $\exists x \psi(x)$ and $\exists x \varphi(x)$ are true. It is easy to verify the equivalence of the two sentences in this example. Notice that, however, $\exists x (\varphi(x) \wedge \psi(x))$ is false.

Remarks

• One may replace condition 3. above with the following three conditions to get an equivalent definition of an existential quantifier operator:
  1. $\exists(\exists(a)) = \exists(a)$
  2. $\exists (a\vee b)=\exists(a) \vee \exists(b)$
  3. $\exists ((\exists a)')=(\exists a)'$
  From this, it is easy to see that $\exists$ is a closure operator on $B$ , and that $\exists a$ and $(\exists a)'$ are both closed under $\exists$ .
• Like the Lindenbaum algebra of propositional logic, monadic algebra is an attempt at converting first order logic into an algebra so that a logical question may be turned into an algebraic one. However, the existential quantifier operator in a monadic algebra corresponds to existential quantifier applied to formulas with only one variable (hence the name monadic). Formulas with multiple variables, such as $x^2+y^2=1$ , $x\le y+z$ , or $x_i=x_{i+1}+x_{i+2}$ where $i=0,1,2,\ldots$ require further generalizations to what is known as a polyadic algebra. The notions of monadic and polyadic algebras were introduced by Paul Halmos.

Dual to the notion of an existential quantifier is that of a universal quantifier. Likewise, there is a dual of an existential quantifier operator on a Boolean algebra, a universal quantifier operator. Formally, a universal quantifier operator on a Boolean algebra $B$ is a function $\forall : B\to B$ such that

1. $\forall (1) = 1$ ,
2. $\forall (a) \le a$ , where $a\in B$ , and
3. $\forall(a \vee \forall(b)) = \forall(a) \vee \forall(b)$ , where $a,b\in B$ .

Every existential quantifier operator $\exists$ on a Boolean algebra $B$ induces a universal quantifier operator $\forall$ , given by $$\forall (a) := (\exists (a'))'.$$ Conversely, every universal quantifier operator induces an existential quantifier by exchanging $\forall$ and $\exists$ in the definition above. This shows that the two operations are dual to one another.

Bibliography

1

P. Halmos, S. Givant, Logic as Algebra, The Mathematical Association of America (1998).