Thursday, July 30, 2015

PROPOSITIONAL CALCULUS OF NON-CAUSAL INFERENCE

The eye is towards future technology. The emphasis in this piece is on technological relevance, specifically to the Google engine. I will begin by explaining the key aspects of the logic, and continue with some observations about how to apply best the logic to a search program, or the nearest compatible concept.

Basically a few notes explaining the role of categories / non-causal inference in formulating logic for computers.

ON COHERENT SYSTEMATICS

The diagram represents the boundary of all possible knowledge, due to the conceptual limit on definability produced by using polar opposites. Conceptually, the only term excluded is zero. Zero may also be included if it has an opposite, with the stipulation that the entire system is ‘over-unity’.

ON SUBJECT AND CONTEXT

In a categorical system, the difference between a subject and a context becomes unclear. For, the system is highly atomistic. The only term representing consciousness is ‘consciousness’. And the only term representing knowledge is ‘knowledge’. And same for any other word. Otherwise, it is left up to the system to clarify the meaning. The meaning is thus implicitly correspondent, but the mode of relation is either coherent or incoherent.

A subject according to the formalism may be defined most simply as the first category, and a second subject as the second category, and so on.

Even more formally, the SUBJECT AXIS is the first and third category in a quadra, while the CONTEXT AXIS is the second and fourth category in a quadra. However, the order of the categories is determined to be arbitrary, a problem that is resolved by positioning opposites diagonally across from one another, producing an order that is identical from top or bottom of the diagram (although alternating in either case in binary fashion, rather than in a full permutation). However, technically the set order of deductions is arbitrary. For formal purposes, the set order of deductions is by reference to the first and second order of categories, etc.

In larger category sets, pairs of opposites may be treated arbitrarily as context or subjects, with no effect on their content, but with emphasis on preserving the most appropriate set order of the categories according to the person doing the analysis. Separate orders of categories produce separate deductions, although the resulting deductions will be formally consistent due to non-contradiction.

ON STATES AND QUALITIES

States may be seen in pairs with qualities, to clarify grammatical meaning. This also has the advantage of creating a familiar relationship between truth (states) and the senses (qualities).

In each conjunction in the logics mentioned later, one part of the conjunction represents a quality, and the other a state. The alternative (introducing all states are all qualities) is difficult, but still workable.

Thus, typically, half of statements are qualities, and half are states in quadra. The ratio remains ½ for larger diagrams, although the alternation occurs differently in diagrams that are not divisible by 4 (sub) categories.

The use of qualities and states implies a quality-state duality, with non-quality and non-state concepts excluded.

However, there is strength to this position, because of two factors:

(1) Most words can be phrased either as qualities or states, that is, as nouns or adjectives.

(2) Additional qualification may be had through the specific words being used. In other words, the formal use of nouns and adjectives does not place a great limitation on the types of meanings considered within the system.

A notation: other types of word-types can be used, provided that they have corresponding opposites, and particularly if they are categorically equivalent to nouns and adjectives.

On the other hand,

If all four quadra are occupied by a noun or an adjective (following the rule of using opposites and positioning them diagonally but otherwise unrestrained in content), then this produces greater difficulty of interpretation, but the interpretation is still valid.

So, as a rule, universal qualities and states = objective limit of difficulty.

[THE NEXT SECTIONS SHOW DIAGRAMS AND THE NORMAL AND TECHNOLOGY APPLICATIONS OF THE LOGIC].

￼A Diagram of the Next Set Level [A Sixteen-Category Diagram].

￼Other variations.

VALID FORMULAE

The most typical valid formula is AB : CD or AD : CB. For larger diagrams it is more complex. However, opposite values always relate in opposite positions.

NORMAL USE OF THE LOGIC

A range of inferences might include such as the following:

Simple inferences:

(1) What is the context of a subject (A,C) = (B,D)
(2) What is the opposite of (A,B,C,D) = (C,D,A,B)
(3) What is the deduction of (A,B,C,D) = (AB:CD and AD:CB)
(4) What is the opposite deduction of (AB:CD or AD:CB) = (AD:CB or AB:CD)

Complex inferences:

(1) What is the limit on (AB or CD)? = (CD or AB)
(2) What is the next option for Set X? The evolution of every category of X, or the application of the best against the worst properties of X, or the relationship of AB to CD, or CD to AB.
(3) How do we get better logic? By getting a better relationship.
(4) What do we do if there is no solution? Find a different context for the subject.

APPLICATIONS

Typical applications involve absolute knowledge statements, quantum logic, systemic variables, and high-level category parsing.

ABSOLUTE KNOWLEDGE STATEMENTS

These take the exact form as the deduction, but can also be worked up to larger sentences (see my videos on 16- and 64- category deductions, and my book titled How to Write Aphorisms).

A simple case is that Functional Systems [are] Non-Functional Non-Systems. If we know what non-systems are, e.g. by the opposites of the terms defining what system means, then we suddenly have a definition of what is non-functional!

A more complex case using a similar logic is: Arbitrary Infinite Mathematical Dysfunctions [are / involve] Ambiguous Finite Irrational Functions. This can be proven because the terms are by definition non-contradictory.

The deductions are ‘true’ if the word-parts of the deduction exist, but are still valid even if one word part does not exist, because then presumably its opposite exists to a greater degree. Thus, the end-product of two meaningless parts tends to simply result in a simpler form of conjunction, rather than total obletion of significance.

The worst case is meaninglessness, which can exist in two degrees in the case of quadra (1) A half-universal meaninglessness, which means that the context or subject axis, but not both, are in no way real at all---this is unlikely, or (2) If both context and subject axis are utterly without application---even less likely.

The other major case of problem at this level is if both opposite terms must be treated as the same, which I argue is essentially a result of a lack of discretion about selecting technical words. It may be a good precaution to not be a total relativist, which does not seem like too stringent of a requirement. After all, the system accommodates the possibility that the terms are meaningless in some sense, even if it does not accommodate that the terms are meaningless in all senses. So a relativist needs to argue that the terms do not exist at all---which is not even a relative claim. To the relativist, the system looks relativistic, but he should know better.

In general, there is a kind of proactive approach that is necessary, where contexts of application may be related to SOME AREA OF RESEARCH. Quantification---e.g. quantification of subject or context axes---may be necessary to produce specific applications, but with experience, certain arrangements will prove to have repeat significance. And a parser can store information about the types of contexts and subjects which typically have relevance for users with particular interests. The emphasis may be on using the best, most traditional examples over new radical ‘attempts’ at reasoning. However, the logic itself is no less radical because of this.

QUANTUM LOGIC

Quantum logic may be typified as meaning the use of a categorical deduction in a variablistic sense. This can be interpreted in terms of corresponding conjunctions.

So, for example, if you have a finite function, the critical and constructive thing to do is to look for infinite dysfunctions. This can be taken as critical advice, or also as the limit of investigation for the concept, unless the concept is ‘evolved’.

This is because, either you have to accept that ‘finite functions are infinite dysfunctions’ or you have to switch the terminology, and determine that ‘finite dysfunctions are infinite functions’. Numerous conclusions could be drawn here, but the point is that this kind of quantum selection involving the opposite terms of the original set---what I call ‘paroxysm’ --- is the best way for tapping the terms for unseen data.

And, the terms will be less limited in the case of larger conjunctions, as shown in the earlier section.

If you are uncertain about how well-founded the logic is, try placing anything representing function in place of the ‘function’ operator, and then placing the opposite of that term in the ‘dysfunction’ operator.

The result, with massive knowledge, is always that there is an irrefutable correspondence between the functional and dysfunctional component, just as there is a correspondence between function and dysfunction in general. It is only when dysfunction is rejected that function becomes incoherent, and it is only when function is rejected that dysfunction becomes incoherent.

To justify this, consider that there is an element of irrationality. Another way to justify it is to notice that universal functions have finite dysfunctions, and universal dysfunctions have finite functions. By this logic, the conclusion I drew is irrefutable, so long as any term is quantifiable.

SYSTEMIC VARIABLES

Now, the same sorts of tricks apply to large systems. It has nothing to do with the size of the system.

If one variable applies to a complex set of data, then you can predict that if the data were truly opposite, an opposite variable would apply.

It is then only up to you to determine some very clever variables, and then you can determine very clever opposite variables which would apply if your data was opposite.

You can also conclude that if the data was not opposite, say, but LESS LOGICAL, then the opposite variable would apply in a MORE LOGICAL WAY.

HIGH-LEVEL CATEGORY-PARSING

You can already see some of the relations between simple and complex inferences, and between universal and less universal and somewhat meaningless and completely meaningless statements, as well as some use of irrationalism or quantification to show variablistic relations.

Now we can use these types of relationships to produce hiearchies. Here are some examples:

The Neutral Boolean Operators: [Is, As, Just As, When, So, And As Such] can be used to arrange large bodies of logical relations in a 4X exponential conjunction hierarchy, for example, in a very large set of conjunctions:

{Is, As, Is, Just As, Is, As, Is, When,
Is, As, Is, Just As, Is, As, Is, So,
Is, As, Is, Just As, Is, As, Is, When,
Is, As, Is, Just As, Is, As, Is, And As Such,
Is, As, Is, Just As, Is, As, Is, When,
Is, As, Is, Just As, Is, As, Is, So,
Is, As, Is, Just As, Is, As, Is, When,
Is, As, Is, Just As, Is, As, Is}

TO BE COMPLETED LATER…