Logical Degeneration

In classical logic, we evaluate the truth of any statement by assigning it a logical value true or false. From the viewpoint of human activity, this is a special case of binary discrimination, which is one of the most common operations in human (and animal) perception. Thus, for an animal, when something happens in the world, it is of vital importance to decide whether the event can be potentially dangerous, or potentially awarding, to take an appropriate action. Humans have more opportunities for decision, but, in real life, we do not evaluate anything just for the sake of evaluation; binary discrimination is always a preliminary stage for deciding what to do next.

In a more general case, we need to filter the situation through an n-slot scheme, selecting one of the predefined labels for a certain characteristic of an event, thus bringing it under the corresponding category. The operation of categorization lies in very core of human culture, and in the foundation of human thought. In particular, we often use categorization in the form of measurement.

In the stationary case, when the characteristic time of decision is much less than the typical duration of an event, n-slot categorization can be reduced to a sequence of acts of binary discrimination. This reduction does not necessarily reflect to logical organization of events, but it can model the real attempts to cope with a difficult problem step by step, subdividing it into tractable parts.

Unfortunately, the world does care much for our categorization schemes and moves on its own, so that many events cannot objectively be brought under a single category. For instance, several positions in the scheme are equally valid to characterize an event. In this case our categorization happens to be degenerate, and one could replace several slots (which are called adjacent in respect to this particular kind of degeneration) with a single slot, a higher level category. In more complex cases, all the positions in the scheme can describe the same event, and we have to introduce yet another scale to evaluate the degree of correspondence. The event is then represented as a distribution over the original categorization space, which thus becomes hierarchically organized.

The folded scheme can be unfolded back, provided the single entity occupying multiple positions is split into its separate aspects, or roles, depending on the positions occupied. That is, instead of folding our categorization scheme, we try to remove degeneration by reorganizing the objective situation itself considering it in more detail. Thus folding of categorical hierarchies corresponds to unfolding the hierarchy of activity, and the other way round.

Let us illustrate it with a simple example. The statement "His wife is better than his boss" is a typical case of binary discrimination. Given two possibilities ("better" and "no better"), we need to apply one of them to a scheme containing two logical positions ("wife" and "boss"). We can successfully apply this scheme to many individuals, and admit its universality. However, the scheme becomes degenerate, when one's wife is also one's boss. As a result, the statement cannot be said to be either true or false. One solution is to admit yet another logical value (higher level category) "neither true nor false" and fold our evaluation to a single slot. Conversely, the possibility of discrimination can be saved if we reformulated the original statement as "She is better as his wife, than as his boss", effectively removing degeneration.

Of course, all the above is valid within the range of the scheme's applicability. For instance, we cannot apply it to the individuals who have either no wife (which is quite possible) or no boss (which sounds a little bit fantastic). Also, there are many ways of degeneration, each requiring its own resolution. Thus, what is "better" in one respect can be "no better" in another. And the very idea of categorization becomes out of place when the positions or aspects of the situation change too fast to fix any momentary values; a different logic is needed in this case.

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