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Re: [ontolog-forum] Inconsistent Theories

To: <cmenzel@xxxxxxxx>, "'[ontolog-forum] '" <ontolog-forum@xxxxxxxxxxxxxxxx>
From: "Rich Cooper" <rich@xxxxxxxxxxxxxxxxxxxxxx>
Date: Mon, 8 Feb 2010 17:32:33 -0800
Message-id: <20100209013235.1B590138D22@xxxxxxxxxxxxxxxxx>

Hi Chris,


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 Cooper


Rich AT EnglishLogicKernel DOT com


-----Original Message-----
From: ontolog-forum-bounces@xxxxxxxxxxxxxxxx [mailto:ontolog-forum-bounces@xxxxxxxxxxxxxxxx] On Behalf Of Christopher Menzel
Sent: Monday, February 08, 2010 2:17 PM
To: [ontolog-forum]
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 terminals x>;

> FalseS(F,x)  := <expression2 of terminals x>;


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[1]..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[1] 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


F[1] := IsA(x[i],Turkey)


as one possible such Boolean member function (x[i] designates a Turkey), and


F[n] := IsA(x[7],Swan)


Is the nth such Boolean member function of x where x[7] designates a Swan, constraining the definition of theory T[j] by defining predicates at positions 1 and n, as schematized below:


F(x) := [IsA(x[i],Turkey), .., IsA(x[7],Swan)]


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 friend T[j]).  


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, qualifications?




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