The Relationship of Rationality and Logic
March 10, 2003
The two primary problems with security are not algorithms nor computation but authentication and key management. Authentication and key management involve the common factors that they involve the storage, tracking, and maintenance of rationality through logic. The problem with authentication and key management is rationality and consistency. Obviously, an authentication and key management scheme that is not useful is not productive or valuable. In order for authentication and key management to be useful, they must be both rational and consistent. This means that in order to improve the "practical" nature of authentication and key management we first must consider the less practical ideals of rationality and consistency of our logic.
Rationality and logic are both essentially comparative methodologies. The comparative relationship of rationality is that of experiential things to experiential things. Rationality is therefore an open universe of philosophy based on the artifacts of perspective experiences with respect to things. The comparative relationships of logic are idealized values related to a predetermined framework of axioms and postulates. Logic is a closed universe of philosophy, dimensionally ordered to be away from direct experience, perception, and reality of things, which are not defined through axioms and postulates.
Therefore, the primary mechanism for rationality is reasoning through analogies. Within a rational philosophy of problem solving, there are greater opportunities for detecting implied risks but greater opportunities for committing and propagating inferential driven errors.
The primary mechanism for logical reasoning takes place through a preconstructed mathematical framework of axioms and postulates. Through mathematical logic, there are opportunities for detecting explicit risks along with greater opportunities for committing and propagating procedural driven errors. The greatest advantage of applying logic through rationality is the potential for establishing a perspective that would have a reduction of logical complexity. The philosophy of reductionism is based on a rational that defines the universe in terms of parts that can be considered in isolation. The problem with this is of course it isn't entirely true. The gestalt of the whole is more than the sum of the parts.
Rationality and logic are therefore equal partners in the sense that if one or the other is allowed to take precedence over the other then there are either implied explicitness or explicit implications involved. "Implied explicitness" represents the underlying requirement for detail that becomes necessary. "Explicit implications" are additional transactions that becomes necessary. Either may result in greater degrees of complexity. Both at once can create situations that are impossible to resolve.
A "metaphysical trap" occurs during comparison of these comparison driven philosophies that lead to the belief that it is more rational to be logical and less logical to be rational. This is essentially a one way philosophical ideal and it is the reason why we tend to over emphasize the importance of the logic of mathematics. It is felt more rational to assume logical solutions are most correct, but it is illogical to assume rational solutions are most correct.
Within the logic of mathematics, consistency is often considered more important than value. The reason for this goes all the way back to the conceptual foundations of the number line, where the definitions of value and cardinality begin. If values within a number line are inconsistent, cardinality ceases to exist, except in one case. The physical universe in which we exist is not rigid. All of its dimensions expand and contract. If number lines were physical, there would then be an infinite number of possible number lines and repeatability would be in jeopardy. To constrain logic to that which is predictable is to require that consistency dictate logical values. Numbers are consistent discrimination points on a number line. Functions such as 2+2 produces 4 every time, but only if logical consistency reigns supreme.
The problem of compatibility between rationality and logic is that rationality can often be inconsistent. Just as with logic people may agree as to what is and is not rational, so too they can disagree about what is logical. For instance, while two methodologies may produce the same logical result, the rationality of procedures within a given calculation may not be the same. Moreover, with rationality, the degree to which congruence exists can never be absolute. Reason may be assigned, not necessarily calculated. Agreement always exists as a part of probability because rationality is influenced by the senses and psychological sensibility. To better understand this, it is easier to think of a concrete example.
Think of an apple. What color is it? Of course an apple could be green. But taking the context of the question into consideration, if we were to color a generalized picture of an apple, it might be more rational to consider the color red. As a society we have essentially agreed that theoretical apples are red, not just because red apples appear more frequently but because we respond to read differently. Nevertheless, on a personal level we also might have just as rationally decided on the color green. The thing is that it would also have been just as rational to consider the colors brown or black; because a rotting apple is still an apple. If we were to agree within a group to use the color red, then we still have the infinite issue of the shade of red to settle. In our world where there are many colors of apples available, there is a general societal agreement that the rational concept of an apple is red. With respect to logical agreement there is the enormous problem of absolutely agreeing on the explicit color that bars our way from absolute congruence. Reality is a perception that has been rationalized to avoid the pitfalls of logical reductionism and a rational perception can never really be logically absolute.
Rationality comes from our physical connection within our universe which is chaotic. The rule of entropy tells us that in order to establish order, we also must create far more entropy. The costs to the perfection of order are staggering. This is a universe of noise and disorder. If the human mind were to be tuned to the recognition of disorder, we might only be able to recognize our universe as a single element. Fortunately, the human mind discovers rational order with more sensitivity through patterns of consistencies rather than patterns of inconsistencies. The mind attempts to reject noise. Consistent orderly patterns allow the mind to discriminate and therefore detect and cognitively isolate the existence of inconsistencies. Rationality is fundamentally a function of consistencies.
Mathematics is a kind of implied sense of order which man has discovered through rational reasoning. Mathematics does some things well because the process of reaching certain kinds of solutions does not require the creation of more disorder. Logic as we are taught it in school does not directly occur in nature even though we obviously can see the rationality of the existence of a manifestation of the result of logic arising through nature. There may be one rock, two leaves and three raindrops, but this does not mean that the concepts one, two and three exist in nature. Because mathematical logic is expressed through inherent definition and through logical necessity isolated from physical reality, if logic is to be useful to our physical being it must be indirectly associated with rational input. To create a relationship between the one, two, and three things they must be intellectually isolated, compartmented as representative impulses within our minds. There is therefore an "order" to the representation that is expressed through logic. Physical "things" exist independently in the first order and are considered in the second order by the mind. They then are symbolically represented in the third order to be operated on or communicated.
In this model, our "order" begins independently of human beings. In the first order of "universal reality," it does not matter to the mind if a physical thing exists. (The sound of a tree falling in the forest is still a sound, even if a person is not there to hear it.) In the second order, of "perspective reality" the presence of a thing with respect to the human mind becomes important. In the second order, we are actually considering a sensory driven manifestation as a discriminatory reference with respect to our mind. The "rational" symbolic third order is yet another different kind reality. It is a manifestation the second order that has been encoded into symbolic transforms. This third order symbolic transform is no longer a momentary sensory event, but is a relationship of coded discrimination pointers to our memories.
Our conceptualizations of mathematical logic exists in a fourth order so that in order to be practical and useful, logic must be rationally fed through three orders and remain consistent.
If the first order is inconsistent, the second order will be disrupted, preventing the third order from acquiring useful symbolic content.
If the second order is inconsistent, the third order will be biased, conflicting with both perception and logic.
It is only when the first and second order produces viable rational order pointers that the fourth order can optimally be utilized.
The idea is not that the logic of the fourth order should be equal to second order perception. Biased logic overriding rationality results in increased complexities with greater potential of error.
The moment we directly associate a third order logical variable such as "x" to a "thing" in physical reality, we are implying the existence of both a rational and consistent relationship. But that is not entirely possible in part because what we are not stating through such a simple pointer is the dimension of time. Time requires a separate pointer to becomes separate "thing," which is to be separately considered and associated. If we were to choose to do this then the concept of "x" would become "xt." To express "x" as a period would require the "t1=>t2" such that "x(t1,t2)" becomes the period of second order observation. We are logically beginning to model our perceptions. But hold on, we also know through relativity that our measure of time is dependent on our perspective of velocity.
Just as with Einstein's Theory of Relativity, we must now stipulate a consistent rigidity of consequences as our "standard" concept of the universe. In other words, we are rationally stating that t1 and t2 are both comparable because of the logical requirement for consistency within our standard of second order observation. Except through the acceptance of rigidity, (or rationality) there are no simple means through which we might otherwise place value on t1 and t2 with respect to x without inducing non-repeatable error.
Precisely the same kinds of issues are involved with respect to authentication of identity. Identity (i) becomes "i1=>i2" or "x(i1,i2) with respect to time (t1,t2) or x((i1(t1),(i2(t2)). In other words, an attempt at the introduction of greater accuracy and repeatability within a stochastic environment also induces both complexity and if not maintained, also systematically induces tremendous opportunity for greater ranges of error. Highly precise solutions in logic become difficult to impossible to calculate. In fact, the principles of calculus are based as much on the maintenance of rationality as logic because in order to obtain a derivative we must choose to eliminate information (irrational noise) in rationally systematic way in order to acquire the best solution.
The point to be made is that the rigidity of logical specification, from which accuracy and repeatability is derived requires incomplete third order representation. A primary purpose of rationality is therefore to choose to feed the "rational" input so that becomes a balance between the relationships of accuracy and consistency, with respect to the usefulness of logic.
All of this brings us back to the original issue of rationality and logic within an optimum security philosophy. When we choose to govern our security processes with respect to the single ideal of logic, we are also choosing to consolidate the independent concepts of rationality and logic into a single concept and thereby inducing error within our security design. Practical logic is therefore not independent of rational consequences within the engineering of authentication and key management designs.
The general philosophy of this paper was produced independently by Roy D. Follendore III, over a period of nearly two decades for the purpose of better understanding the problems of security and cryptography. Generally supportive and related information on recent academic laboratory studies can be found in the article "Compromise Is Name Of The Game In How Brain Works, Say University Of Toronto Researchers" www.sciencedaily.com/releases/2003/03/030306075932.htm
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