Re: ?? Functional Dependency Question ??

From: David BL <davidbl_at_iinet.net.au>
Date: Tue, 21 Oct 2008 09:37:55 -0700 (PDT)
Message-ID: <01af9f4f-8ec1-4112-b8fa-8c3ed6f6769f_at_w39g2000prb.googlegroups.com>


On Oct 22, 12:16 am, Tegiri Nenashi <TegiriNena..._at_gmail.com> wrote:
> On Oct 21, 8:40 am, David BL <davi..._at_iinet.net.au> wrote:
>
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> > On Oct 21, 10:22 pm, paul c <toledobythe..._at_oohay.ac> wrote:
>
> > > David BL wrote:
> > > > On Oct 21, 8:15 pm, paul c <toledobythe..._at_oohay.ac> wrote:
> > > >> David BL wrote:
>
> > > >> ...
>
> > > >>> RTP: (X->A)(Y->B) -> XY->AB
> > > >>> (X->A)(Y->B) : premise
> > > >>> X->A, Y->B : conjunction elimination
> > > >>> XY : premise
> > > >>> X,Y : conjunction elimination
> > > >>> A : modus ponens on X,X->A
> > > >>> B : modus ponens on Y,Y->B
> > > >>> AB : conjunction introduction
> > > >>> XY->AB : conditional proof
> > > >>> (X->A)(Y->B) -> XY->AB : conditional proof
> > > >> I can see that the second step is eliminating a "conjunction" but the
> > > >> fourth step doesn't seem to me to involve a conjunction.
>
> > > > Step 3: Premise XY (the conjunction of X and Y).
>
> > > ...
>
> > > But XY (in the usual FD notation) is a union, not a conjunction.
>
> > > (If step 4 is valid, there must be a subtlety here that goes beyond just
> > > conjunction elimination, ie., some other notion must be involved, but so
> > > far it could be eludes me.)
>
> > Ah, Ok I understand your point now. (X->A)(Y->B) -> XY->AB is a
> > formula in the propositional calculus, which is shown to be a
> > theorem. In that context XY is definitely a logical conjunction.
>
> > In FD notation we think of X,Y as sets of attributes and as you say XY
> > is a notation for the union of those sets. However there is this
> > idea of translating an FD sentence into a formula in the propositional
> > calculus. This mapping can be done as follows
> > - sets of attributes can be mapped to proposition symbols
> > - unions of sets of attributes can be mapped to logical conjunction
> > - a functional dependency can be mapped to a logical implication
>
> > Consider that in the FD world symbol X represents a set of attributes
> > from some relation R. Let some tuple of R be given. Then as a
> > proposition we interpret X as implying that we are given or can deduce
> > (for the given tuple) the values of all the attributes associated with
> > X.
>
> This perspective of FD theory into propositional calculus is new to
> me. So could you please be more specific? Sure propositional calculus
> has some axioms that have no interpretation in FD world, e.g.
>
> A -> (B -> A)
>
> or it has?

I would say wffs of FD theory map to a strict subset of wffs of the propositional calculus.

The idea is not mine. Im trying to make sense of Bob Badours post where he suggested a simple connection between FD and logical implication. Received on Tue Oct 21 2008 - 18:37:55 CEST

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