Re: Logical = relational?
Date: Fri, 05 May 2006 03:43:15 GMT
Message-ID: <nRz6g.20552$YI5.12301_at_tornado.ohiordc.rr.com>
Mikito Harakiri wrote:
> 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
Would you do me a favor, please? Could you name these "five logical constants/expressions" and the "five relational counterparts?"
Thank-you.
> 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 Fri May 05 2006 - 05:43:15 CEST