Path: dp-news.maxwell.syr.edu!spool.maxwell.syr.edu!news-spur1.maxwell.syr.edu!news.maxwell.syr.edu!newsfeed.icl.net!newsfeed.ision.net!newsfeed2.easynews.net!ision!newsfeed.freenet.de!newsfeed0.kamp.net!newsfeed.kamp.net!nx01.iad01.newshosting.com!newshosting.com!69.28.186.75.MISMATCH!hwmnpeer01.lga!hwmedia!news.highwinds-media.com!news-server.columbus.rr.com!tornado.ohiordc.rr.com.POSTED!53ab2750!not-for-mail From: J M Davitt User-Agent: Mozilla/5.0 (X11; U; Linux i686; en-US; rv:1.7.2) Gecko/20040803 X-Accept-Language: en-us, en MIME-Version: 1.0 Newsgroups: comp.databases.theory Subject: Re: Logical = relational? References: <1146765198.127635.7140@i40g2000cwc.googlegroups.com> In-Reply-To: <1146765198.127635.7140@i40g2000cwc.googlegroups.com> Content-Type: text/plain; charset=us-ascii; format=flowed Content-Transfer-Encoding: 7bit Lines: 43 Message-ID: Date: Fri, 05 May 2006 03:43:15 GMT NNTP-Posting-Host: 24.123.195.58 X-Complaints-To: abuse@rr.com X-Trace: tornado.ohiordc.rr.com 1146800595 24.123.195.58 (Thu, 04 May 2006 23:43:15 EDT) NNTP-Posting-Date: Thu, 04 May 2006 23:43:15 EDT Organization: Road Runner High Speed Online http://www.rr.com Xref: dp-news.maxwell.syr.edu comp.databases.theory:39201 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! >