Aha... Found this old bit of jabbering to copy/paste.



Class Algebra for AI

Class Algebra does not acknowledge subtraction or division as valid logical operations. Nor does it recognize sums and products as independent of their constituents. Formally, union and separation are the valid operations underlying all equalities. While a sum may appear to obscure the constituents, we "know" that those constituents remain intact in some form. While a summation process may not necessarily be reversible, it's transactional history remains intact whether forgotten or not. Likewise, separation operations do not cause the portion of a sum that is no longer the focus of an equation to disappear from existence. The transactional history can always be used to reconstitute the original state of the original sum before a given operation separated it.

A+B=C
A(X)+B(Y)=C(Z)

Class axis
Z may be any of
1. An instance or entity
2. A class or concept
3. A method or process

1. Instance/Entity
If Z is an instance, the equality should be considered a definition or an identification. Those A(apples) and those B(oranges) are C[that](instance of a collection of fruit).

In this case, C is always 1, for no reference to [that] may indicate more or less than the singular instance to which it refers.

When A is 1, X is an instance. When A is greater than 1, X is a collection of similar instances whose differentiation is not significant in the context of the definition/identification. The same rules apply to B and Y.

While X and Y, may be the same kind of thing, it is not normal to refer to them separately unless they differ in some way that is significant to the definition/identification. That 1 (left leg) and that 1 (right leg) are 1 (your legs) is not an uncommon formulation, but those 2 (legs) are 1 (your legs) is a simpler formulation.

No algebraic movement of variables from side to side is permitted, but A(X) and B(Y) may be transposed.

Instance Definition/Identification Laws
C=1
A>=1
B>=1
X<>Y
A(X)<>C(Z)-B(Y)
B(Y)<>C(Z)-A(X)
A(X)+B(Y)=B(Y)+A(X)=C(Z)


2. Class/Concept
If Z is a class or concept, again C is always 1, and Z is the resultant class or concept arrived at by combining A(X) and B(Y).

The formulation...
"an (a=1) animal (x) with (+) (implied 2) wings (y) is (=) a (c=1) bird (z)"
...is a linguistic summation serving as a definition/identification of a subclass of animals.

Again standard algebraic treatment of the equality is invalid. It is nonsensical to say that because "an animal with wings is a bird", so too, "a bird without wings is an animal". The later formulation does nothing to refine, clarify, or identify a class or concept.

Further, one of x or y is typically a pre-existing class, while the other is an attribute or feature being applied to the pre-existing class in an attempt to clearly identify the resultant subclass z. It is atypical to state that "wings with an animal are a bird", but linguistically common to revise the summation words to other forms such as "wings on an animal make it a bird". These are not precisely logically equivalent, but with respect to class algebra version 1, we'll treat them as such.

Class/Concept Definition Laws
C=1
A>=1
B>=1
X<>Y
A(X)<>C(Z)-B(Y)
B(Y)<>C(Z)-A(X)
A(X)+B(Y)=B(Y)+A(X)=C(Z)

3. Method/Process
A method or process is a special form of concept in which sequence is vital to the formulation.

In the formulation...
"(a=1) turn on the shower faucet (x) then (+) (b=1) stand in the falling water (y) to (=) take a (c=1) shower (z)
...demonstrates how sequence is vital to method/process definition. One cannot stand in the falling water until the shower has been turned on.

Method/Process Definition Laws
C=1
A>=1
B>=1
X<>Y
A(X)<>C(Z)-B(Y)
B(Y)<>C(Z)-A(X)
B(Y)+A(X)<>C(Z)


--------------
c=a-b
c(z)=a(x)-b(y)

Specialized Separation as Definition
Again, Class Algebra does not recognize subtraction as a valid logical operation. In cases of abnormality or deviation from a norm, instances of classes may be identified by a feature of a class to which they belong that is lacking.

(c=1) Charlie (z) is (=) a (a=1) boy without (-) (implied b=2) ears (y)

This form of definition is useful insofar as it can be used to rapidly identify the major class to which an instance belongs, and then describe a slight deviation from a normal instance of that major class to create a subclass which is easily identifiable.

"Which of these plates is chipped?"
"Do you see that bald man (man with no hair) over there?"

Abnormality equations can actually be instructive if little is known about the normal class.

"Normal boys have 2 ears."
"Normal plates are not chipped."
"Normal men are not bald."

Because abnormality equations are just as common as typical definition equations, a good AI system will learn as much, or more, about classes from abnormality as it will from definition.

Well, I botched that. Thought the note was an introduction to the thread.

At any rate, some basics... The only valid concept (class) operations are union and separation. Mathematical operations are a convenience that can easily drop context and transactional history.

Example 1:
(1 apple + 1 orange = 2 fruits) is true
(2 fruits - 1 apple = 1 orange) may or may not be true

This suggests that a fundamental law of what I like to call "class algebra" is: a union equation requires that the items being unified are instances of or subclasses of the result.

Thanks for the efforts. I have been reading this over and over, considering and reconsidering. I had a class in symbolic logic back in 2006 and got an A, but there was the day that I spent five hours on a homework problem... Just to say, I agree with this:
"Formally, union and separation are the valid operations underlying all equalities."

That would be integration and differentiation (epistemological, not mathematical).

When you say "I feel like there is much more to add to the idea that measurement omission is the key to concept formation." you identified an interesting problem: what (ELSE) is concept formation? The only difference between atomic isotopes is indeed a measurement, so when we say "Carbon-14" that is a concept, a very complex concept, actually, but for which measurement is critical to the definition.

You say also: "I'm a bit obsessed with the idea that a simple set of rules could be added to this foundation and taught to individuals (myself, for instance) to help them quickly identify contradictions and logical fallacies."

Just the actual rules of Boolean algebra would be a great beginning for 99.9% of humanity (including many here). I mean, you could use "predicate calculus" as well to reduce terms to symbols for easy manipulation without embumbrance of emotionalisms like patriotism, family values, tradition, hope for change, etc. But basically, just knowing actual formal logic is a serious step in the right direction.

What I fear is that you are looking for an automatic or quasi-automatic way - some kind of algorithm or heuristic - to always know the truth. Nothing like that exists or can. For one thing, truth is contextual. When you remove context you risk removing meaning.

For me, these problems are LINGUISTIC. I speak a couple of languages and knew a few more and am pretty handy with almost anything human or computer. If you want to know the rules of thought, you need to know how people actually think. Language reveals that.

Just for instance - In English, we have singular and plural only. But Russian (which Ayn Rand knew) has the dual, also. 1 2 Many. That is how people thought for thousands of years and still think today, but in English, we lost the dual - you have to search for it: pair of shoes; brace of pheasants; wedding couple - and so we make generalizations about pluralities all the time.

As you noted, we say most, some, all, none, etc. with seeming abandon.