All right, perhaps I was a little too fluid in my presentation. I
will try to mathematize it appropriately, with terms carefully defined. See
below, please, for my comments clarified:
Rich AT EnglishLogicKernel DOT com
[mailto:ontolog-forum-bounces@xxxxxxxxxxxxxxxx] On Behalf Of Christopher Menzel
Sent: Monday, February 08, 2010 2:17 PM
Subject: Re: [ontolog-forum] Inconsistent Theories
On Mon, 2010-02-08 at 11:54 -0800, Rich Cooper wrote:
> Example: Let the two theories be:
> TrueS(F,x) := <expression1 of
> FalseS(F,x) := <expression2 of terminals
I don't understand what it means to say these are
theories. They look
like clauses in a BNF. I suppose a BNF is a sort
of theory, but it's a
theory that describes the grammar of a specific
language. But the left
sides here look like atomic statements in a first
order language, not a
class of expressions.
> So that there are two theories: TrueS(F,x) is the
set of Things which are
> believed (with current knowledge) to be in the
set F for terminal vector x,
> while FalseS(F,x) is the set of Things which are
believed NOT to be in F(x).
How can you possible get that from the above?
I defined it that way, where the F[k] are
Boolean membership functions. That is;
If you have a theory T[j] which has not
been fully and accurately completed; it is a Boolean function of the (here
goes) terminal linguistic symbols, whether each such terminal symbol designates
a constant, variable, function, phrase or _expression_ of the above.
T[j] is a theory of how to use each symbol
as information structured in linguistic patterns containing features F[k] that
repeat from time to time in samples.
T[j] is the jth iteration of the theory of
how to predict members of a set you are interested in. T..T[j] is the
sequence showing how the calculation got there with successive theories. Each
theory T[j] is a revision of the previous theory T[j-1] (except of course the 1st
theory T which just lets everything through).
So each function F[k](x) in the set vector
F(x) is a Boolean function that returns True if the sample x (a vector of symbols
x[i] for i in [1,m]) is a member of the set. So if we define
as one possible such Boolean member
function (x[i] designates a Turkey),
Is the nth such Boolean member function of
x where x designates a Swan, constraining the definition of theory T[j] by
defining predicates at positions 1 and n, as schematized below:
This _expression_ means that F(x) is hereby
defined to contain the vector (of size n), each element F[k] of F containing a
function F[k](x) of the terminal symbols x in sequence as experienced in a
given sample, stored in a database text cell, presently under consideration.
As hinted, x is a vector (think of the
state vector in a linear system of equations) and represents (to me at least),
a sequence of symbols which are being organized into a proper theory (our old
But to begin with, call it time t=t0, T[j]
only could think that a theory is correct as stated. At t=t1 (scale t0:t1
any way you like), T[j] has encountered some sample instances that are
predicted by T[j] to be True members of the set, but which have been found to
actually, in the real world, not do so well as TrueS in T[j]; they find
themselves better categorized as FalseS in T[j]. They are erroneous
designations which, though correct according to the current theory T[j], are
found to be False in practice.
So T[j] designates the set of correctly
encountered instances conforming with the theory T[j] versus ~T[j], which is hereby
defined as the set of correctly encountered instances that are known not to
conform with T[j] though T[j] predicts that they will so conform.
Questions, comments, condemnations,