In the general line of the hierarchical approach, logic assumes numerous layered structures and systems reflecting both the history of its development and the modes of its application. There is no absolute form; the whole of hierarchy can be unfolded starting from any single idea. The argument about the best logic possible is hence entirely meaningful.
Historically, there were many special "logics", with all kinds of names: classical, speculative, higher-order, modal, alternative, fuzzy, constructivist, stochastic, situational, etc. Each of them represented one of the possible conversions of logic as such, without any epithets. A few hierarchical structures presented here arise each in its specific context, and they can coexist at any moment, since human activity is hierarchical, with each level demanding an appropriate logic.
Syncretic, analytical and synthetic logic
The adequacy and congruity of activities occurring in people's everyday life is the first manifestation of logic. If one acts according to the natural order of things and the current social expectations, this action is often called a “natural”, or “logical”, consequence of the objective and social situation. Internal life of a person obeys, from this point of view, its own logic; in particular, the typical routes of thought differ from one individual to another. This level of logic, where the forms of activity are not separated from the activity itself, may be called syncretic.
On the higher analytical level, the forms of one's activity become imposed on that activity as external regulations, often codified and officially accepted. For a typical example, take the traditional rules of logic studied by math students as an a priori basis of any rigor. More examples: the laws of a state, the rules of a game, editorial guidelines for the contributors to a scientific journal etc. These forms are relatively independent of the underlying activity, and their modification may seem a matter of convention, though, in fact, there are objective limitations and requirements.
The synthetic level of logic assumes that both the rules and their justification become conscious. People may intentionally change the rules for a more adequate behavior in the changing environment, so that no logical scheme is considered absolute and applicable in any situation. This means that synthetic logic does not admit any complete description, since any specification will only refer to a particular manifestation under certain conditions.
Rational, dialectical and diathetical logic
In the basis of any practical activity one finds general rationality based on the repetition of the activity's structure. Such rational logic deals with stationary activities, the "standard" forms preserved for a long time. At the syncretic level, rationality appears as common sense; on the analytical level, one can find the classical modes of reasoning described by Aristotle; in philosophy, this way of reasoning is called metaphysical. Being a necessary stage of any research, and an indispensable component of any thought, rational logic uncritically used beyond the limits of its applicability leads to biased opinions rather than knowledge and wisdom.
Dialectical logic removes metaphysical rigidity demanding that every action should be viewed in a wider context, along with its alternatives. Everything has its opposite, and the opposites are equally valid; any activity develops in struggle and mutual reflection of the opposites, and their unity can only be achieved in a higher-level activity. An example of syncretic dialectics is provided by the pragmatic attitude to the world. Analytical dialectics has been widely exercised by the ancient and medieval sophists, and this is the highest form of dialectics possible in philosophical idealism. Synthetic dialectical logic was developed in XIX-XX centuries by K. Marx, F. Engels and their followers; for political reasons, this line did not receive much public attention.
At its highest level, logic becomes aware of the universal reflectivity, when every category implicitly contains all the other categories, and the whole can be reconstructed starting from any arbitrarily selected element. Unlike dialectics, this logic does not lead to an infinite sequence of levels, the higher ones fixing the contradictions of the lower; rather, it is always aware of the whole hierarchy. Any unfolding of this hierarchy into a sequence of levels according to the dialectical schemes is considered as a particular possibility related to many others, and one arrangement of categories is as admissible as another. Still, these arrangements are not arbitrary, and the rules governing them could be called diathetics (intentional arrangement in a specific context).
Intuition, reasoning, comprehension
No logic is possible before the object and purpose of activity come to awareness. For instance, a formal definition assumes some previously formed conceptions that do not need to be defined at present. Similarly, for a formal deduction, one must be aware of the intended result, which cannot be obtained in a deductive way. This preliminary context is said to be intuitive; however, every act of insight must be logical to be practically useful.
The major part of logical research is about reasoning, discourse, formal derivation. On this level, the forms of activity are detached from the activity itself; reasoning can use them in an arbitrary manner, producing all kinds of abstract combinations. The "objectified" character of such logic simplifies its study by scientific methods, and its formal nature admits wide usage of mathematics.
There is yet another level of logic closely related to its development. Eventually, we are not satisfied with immediate results or deep knowledge—we also need comprehension, which gives us the feeling of mastery, of power, of freedom. The application of logical forms to practical tasks brings forth their interrelations, leading to the formation of higher-level logical schemes.
Adequacy, singularity, creativity
This distinction reflects the very definition of subjectivity as universal mediation. The subject takes the world as an object (nature) and reproduces it as a product (culture). Eventually, the whole world becomes thus transformed, reorganized to satisfy people's needs, assimilated in the culture. The logic of conscious activity will reproduce this fundamental hierarchy.
Indeed, one of the usual connotations of the word "logical" is "quite natural". We also speak of the "logic of things" as something to be followed in reasonable behavior. This objective logic is to organize people's acts in accordance with the already established (natural) regularities that are apparently independent of our will. This kind of behavior is considered as adequate.
On the contrary, subjective logic reflects the diversity of human ways, when the same goal is achieved quite differently by different characters. In common usage, we express it as being natural (logical) "for him", or "for her", albeit differing from our own manner. That is what we expect of the others, and of ourselves. In particular, calling somebody irrational, we mean a definite subjective logic.
Finally, the individual modes of action can be socially accepted as a common productive pattern, to become "second nature". This productive logic is creative, since it assumes a common significance of one's singularity, being useful for the others.
Inner logic, formal logic, intension
To make a thing, one needs some material to be properly shaped, that is, the form of the thing must express what it really is, its content. For instance, I take English words as my material and organize them in phrases and paragraphs to produce a description of logical hierarchy (the content of the text). These components imply three specific kinds of logic. Obviously, the choice of material will impose certain restrictions on what can be express and in which way. This inner logic can be rather restrictive; thus, there are ideas that are clearly conveyed in Russian, while it is very difficult to find an appropriate wording for them in English; conversely, some ideas are much easier to express in English. And, if I took musical notes or colors for the material, I would be driven to a quite different manner of expressing hierarchical logic. The organization of material, the form, is also associated with definite principles, formal logic. Forms are not entirely arbitrary, they depend on each other, and one has to obey some formal rules (like the grammar of the English language, or, say, the norms of coherent discourse) to produce meaning and sense. And, of course, both material and form are only needed to convey the whole, the content, and the logic of the whole is to be reflected in both inner and formal logic; in a sense, lower level logics are used to build models of intesion.
The different facets of the hierarchy of logic are not independent, since they refer to the same. In some situations one might, say, consider the triad rational → dialectical → diathetical as another expression for the triad objective → subjective → productive. However, in a different context, these two triads may represent the orthogonal dimensions of the whole, to consider objective, subjective and productive rationality along with objective rationality, dialectics and diathetics. All these schemes are mutually reflected (in Hegel's sense). Each of them can be used to unfold the hierarchy of another.
On the other hand, each hierarchy manifests itself either in a structural way, as a hierarchical structure, or in the systemic way, as a hierarchical system, with each level of the hierarchical structure becoming a separate entity interacting with the others; on the synthetic level, the hierarchy reveals itself in a sequence of development stages. Consequently, all different "logics" are present in any logic at all, occupying specific positions in the hierarchy of the whole, the special logic of an individual activity.