monadic algebra
Let be a Boolean algebra. An existential quantifier operator on is a function such that
-
1.
,
-
2.
, where , and
-
3.
, where .
A monadic algebra is a pair , where is a Boolean algebra and is an existential quantifier operator.
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 is false iff is false. For example, suppose is a real number. Let be the statement . Then is false no matter what is. Likewise, is always false too.
-
2.
implies ; in other words, if is false, then so is . For example, let be the statement , where . By itself, is neither true nor false. However is always true.
-
3.
iff . For example, suppose again is real. Let be the statement and the statement . Then both and are true. It is easy to verify the equivalence of the two sentences in this example. Notice that, however, is false.
Remarks
-
•
One may replace condition 3. above with the following three conditions to get an equivalent definition of an existential quantifier operator:
-
(a)
-
(b)
-
(c)
From this, it is easy to see that is a closure operator on , and that and are both closed under .
-
(a)
-
•
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 , , or where 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 is a function such that
-
1.
,
-
2.
, where , and
-
3.
, where .
Every existential quantifier operator on a Boolean algebra induces a universal quantifier operator , given by
Conversely, every universal quantifier operator induces an existential quantifier by exchanging and in the definition above. This shows that the two operations are dual to one another.
References
- 1 P. Halmos, S. Givant, Logic as Algebra, The Mathematical Association of America (1998).
Title | monadic algebra |
---|---|
Canonical name | MonadicAlgebra |
Date of creation | 2013-03-22 17:48:57 |
Last modified on | 2013-03-22 17:48:57 |
Owner | CWoo (3771) |
Last modified by | CWoo (3771) |
Numerical id | 9 |
Author | CWoo (3771) |
Entry type | Definition |
Classification | msc 03G15 |
Classification | msc 06E25 |
Related topic | QuantifierAlgebra |
Related topic | PolyadicAlgebra |
Defines | existential quantifier operator |
Defines | universal quantifier operator |