Logical = relational?
Date: 4 May 2006 10:53:18 -0700
Message-ID: <1146765198.127635.7140_at_i40g2000cwc.googlegroups.com>
This looks like a silly question for the folks with database background. Sure they are the same concept. Let's put the things into wider perspective, however.
The simplest form of logic is propositional calculus. It has been algebraized by J Boole in the form of boolean algebra. Boolean algebra is isomorphic to the field of unary relations.
It was de Morgan who moved to the next step and established the calculus of binary relations in 1860. Pierce turned out to the subject in 1870, and found most of the interesting equational laws of relation algebra. The subject fell into neglect between 1900 and 1940, to be revived by Tarski. He laid out algebraic axioms that hold in any field of binary relations hoping to find a first order characterization of fields of binary relations the same way boolean algebra axioms characterise the field of unary relations. It turned out that Tarski axioms were unsufficient, moreover it has been proved that no finite system of axioms would suffice.
In modern notation Tarski algebra includes five logical
constants/operations:
0, 1, -a, a+b, ab
and five relational counterparts:
0', 1', ^a, a^+b, a;b
In the algebra of binary relations the logical product operation (which can be interpreted as set intersection) is different from relation composition operator ";"!
The next natural step is to move into the field of n-ary relations. The first attempt was made by Tarski with introduction of cylindric algebras. The relation dimension moved from 2 to n, but remained fixed.
E Codd expanded the idea to manipulate relations of mixed dimensions. The greatest contribution of Codd, however was unifying relational and logical views. The relational join is both set intersection, and relational composition! Received on Thu May 04 2006 - 19:53:18 CEST