Re: Logical = relational?

From: J M Davitt <jdavitt_at_aeneas.net>
Date: Sat, 06 May 2006 01:38:20 GMT
Message-ID: <g6T6g.32477$P2.6020@tornado.ohiordc.rr.com>

Mikito Harakiri wrote:
> J M Davitt wrote:
>

>>Mikito Harakiri wrote:
>>
>>>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
>>
>>Could you name these "five logical
>>constants/expressions" and the "five relational counterparts?"

>
>
> This snippet is from Vaughan Pratt paper "Origins of the calculus of
> Binary Relations".
>
> 0 - the least element in BA
> 1 - the greatest element in BA
> -a - complement of a
> a+b - disjunction (or since it is algebraic perspective is it actually
> sum modulo 2?)
> ab - conjunction

Whew! At least I recognized the first five -- sort of. I was thinking "false, or antitheorem" and "true, or theorem" and "negation." But what's curious is that 0 and 1 are ordered -- at least, I infer that from the words "least" and "greatest" -- and I don't see the benefit without implication. But, that's just my ignorance.

> 1' - identity/diagonal relation
> ^a - converse of a (a relation with interchanged 1st and 2nd attribute)
> a;b - composition
>
> Those are standard 8 operations that can be found in any article about
> binary relational algebra (BRA). I refer you to Pratt paper for the
> remaining 2 operations/constants (I don't quite understand them
> myself).

I took a look at a web-handy PDF... didn't get too far. I'm not sure I can find the time to learn what the converse of a relation is good for; maybe I'll wish I had, later.

Thank-you. Received on Fri May 05 2006 - 20:38:20 CDT