Re: [ontolog-forum] Inconsistent Theories

[Jawit Kien <jawit.kien@xxxxxxxxx>]

I must say,   Rich, that your more formal presentation is quite accomplished, but I'm not
sure I understand it.  Likewise, I find it surprising that no one has commented on it to date.
I will attempt to do so, but clearly, you have a lot more thought behind this than I can examine
in depth.
JK

On Mon, Feb 8, 2010 at 7:32 PM, Rich Cooper <rich@xxxxxxxxxxxxxxxxxxxxxx> wrote:

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:

Sincerely,

Rich Cooper

EnglishLogicKernel.com

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?

-chris

 

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.  

So to use John Sowa's favorite sentence:  A cat is on a mat.
I see that there are five words in alphabetical order:
1) "A"
2) "CAT"
3) "IS"
4) "MAT"
5) "ON"
 
Are these the "terminal linguistic symbols" ?

Alternately, Using the grammar:
<sentence> := <NP> <VERB> <NP>
<NP> := <ADJ> <NOUN>
<NP> := <PREP> <ADJ> <NOUN>
<VERB> := "is"
<PREP> := "on"
<NOUN> := "cat"  |  "mat"
<ADJ> := "a"
there are phrases: (again in alphabetical order)
1) "a cat"
2) "a cat is on a mat" : alternately "(1) is (4)"
3) "a mat"
4) "on a mat" : alternately "on (2)"
Are these your "terminal linguistic symbols" ?

John Sowa's analysis
yields this

(∃x:Cat)(∃y:Mat)on(x,y).

ie: ThereExists X such X is a Cat
             ThereExists Y such that Y is a Mat
                     the function "on" holds between that X and that Y.

Are any of these your "terminal linguistic symbols" ?

 

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.  

I understand a theory as a collection of axioms and theorems or alternately a set of facts and rules.
The subject of each of your theories is a boolean function of a terminal linguistic symbols?

So for our "a cat on a mat" statement is your theory applied to the sentence as a whole
ie: you have T["a cat on a mat"] ?  what does it mean to have a boolean result of a sentence?
and is it a logical statement that is the result, and you
or is the result of your T[j] the
 

T[j] is the jth iteration of the theory of how to predict members 

of a set you are interested in.  

What are you trying to predict them to be? since it is supposed to be a boolean function, are you trying to predict that they are true or false?

T[1]..T[j] is the sequence showing how the calculation got there with successive theories. 

"got there" implies you  are using a metaphor/analogy here.
Perhaps that the calculation is "moving along some path" and each "milestone" along the path is the value of T[1], T[2], T[3], up to milestone T[j].
Could you elaborate more about this movement?  What does it mean to make a "step" along the path? Since paths have surfaces, what does the surface look like? What is the "stuff" that the path consists of? What is it that is moving along the path? Does the movement only happen when the calculation completes (halts) ? Is the milestones consistent "snapshots" of the calculations?  What exactly is getting calculated?
 

Each theory T[j] is a revision of the previous theory T[j-1] (except of course the 1^st theory T[1] which just lets everything through).  

 

So now you are using a different metaphor/analogies. On the one hand, you have the T[j] being a more "refined" object that T[j-1], somehow purer, or more focused, or more accurate, or more polished, or more "good" using some measure.   The "revision" process is a succession of sub-processes, the first of which makes no "changes" at all, and each of the following ones 

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

 

Whoa. F[k](x) is a matrix/single-dimension vector of functions which take a vector of symbols ?
so F[1] is a function which is given the vector [ "a cat is on a mat" ] ?
or F[1] is a function which is given the vector ["a" , "cat" , "is" , "on" , "a" , "mat" ] ?
or what?
 

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

Okay, now you are making the F[1] (which you now say is a function and not a theory as you said before), only get a single symbol in your sample x as input. If it is a theory, wouldn't it have statements about each of the values of x[i] ? Does your := mean that you are adding this statement to the theory or does it comprise the entire Theory?

 

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

 

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

 

Why did you change from x[i] from x[7] ?  How do you know i=7 ?

I'm going to stop right now, I have a meeting, but you have quite a theory you have
worked up, but I don't quite understand it, and will send this e-mail so I can find
out if I am chasing a wild swan, wild turkey, or wild goose here.
(maybe I should be drinking that wild turkey)

JK

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?

-Rich

 

 

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