Introduction to Logic

Many textbooks on logic define it as a science about the forms of thinking. Such sentences are twice in error: first, logic is not a science, and second, it is not confined to thinking.

Logic is all about how people choose the ways of doing something, and hence it is a part of philosophy. Any piece of human behavior can be either logical or not, and every activity has its own logic that may be different from the logic of another activity.

However, unlike the other branches of philosophy, logic decides on acceptability of a particular act judging by the formal criteria, disregarding any social circumstances and possible consequences; this gives logic a quasi-objective appearance, as if it were independent of the people's interests and concerned only with the natural ways of things. This is why logic may look like (and be mistaken for) science.

Of course, one may study the currently known schemes of reasoning, and this would be a science analogous to, say, ethnography. Such a study would never tell a universal logical principle from mere cultural fluctuation. The traditional courses of logic enumerate the forms of syllogisms, but they never tell where all these forms are applicable - and where one should better try something else. Why the statements are build of notions? - where the different truth/verity systems come from? - how axioms and basic concepts get chosen? - to answer these and other similar questions, one needs something more general than science, the principles of making decisions on the adequacy of one's actions, reasoning included; such principles are the domain of philosophy.

Since thinking can be considered as a kind of activity, the study of its universal forms (logic) is governed by the same principles as any other logic study. However, thinking is a very special activity, due to its universality: every conscious action is mediated by thought. That is why the forms of thinking reflect the most common regularities in human activity, and hence the tendency of reducing all logic to the logic of thinking, and even worse, to the logic of a formal discourse.

Due to the ubiquity of thought, any activity at all can be considered as representing certain modes of thinking; however, some activities are more suited for that than the others - thus, theoretical science (and especially mathematics) may reveal the specificity of reason in a clearer manner because of their abstract nature allowing simple schemes to be implemented in a relatively straightforward way. On the other hand, this abstractness may cause the illusion of the subject arbitrarily designing the world at his will, without any concern for what is possible and what is not. However, any logical forms can only reflect the real position of the conscious being in the world, and the hierarchy of logical forms reproduce (in a different aspect) the hierarchy of the world. On the other hand, thinking is not pure logic, and it has to be comprehended through complementary reflections provided by many sciences, as well as the arts, or different branches of philosophy.

To start with, one might define that logic reveals universality in the forms of any activity. Primarily, there is logic of a particular activity - and one could develop it as a separate discipline. In fact, by the XX century, the humanity has accumulated enough ideas about various "special" logics: logic of deduction, logic of interrogation, logic of definition, etc. Scientology echoes with such terms as "quantum logic", "situational logic", or "temporal logic". In mathematics, numerous model of different logics have been suggested: many-valued, stochastic, fuzzy, categorial and other logical systems; however, in their research, mathematics is a science, and all mathematical models will always remain within science, with its special logic of analytical reflection. Unfortunately, most scientists are poorly educated in logic, and they cannot even correctly apply the traditional Aristotelian syllogistics, to say nothing about inductive, modal, or other schemes. Vague notions about logic outside science still hinder the development of human spirituality in general.

Syncretic, Analytical and Synthetic Logic

The first manifestation of logic is the adequacy and congruity of activities occurring in people's everyday life. 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 activity become imposed on it 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. Since such forms are relatively independent of the respective activity, their modification may be considered as a matter of convention, since the objectively existing limits of such arbitrary variations may be hidden from people's awareness.

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 is what philosophy should always desire, though it is only in praxis that synthetic logic can exist as such.

The important corollary is that logical study is applicable to any human activity, as soon as its form implies a universal component. However "irrational" people's acts may seem, they can never be completely illogical, and their logic can be revealed at a closer examination. In particular, since regular thinking is a specific activity, it can be described from the logical viewpoint.

Metaphysics, Dialectics and Diathetics

By the degree of reflectivity, Hegel distinguished three levels of logic:

1. Rational logic deals with stationary activities, where stable "standard" forms may exist for a long time. At the syncretic level, rationality appears as common sense; on the analytical level, one can find the traditional modes of reasoning enumerated by Aristotle and widely used in scientific studies from the very dawn of science; in philosophy, this way of reasoning is called metaphysical.

2. Dialectical logic removes the rigidity of rationality and demands that every action should be viewed in a wider context, along with its alternatives. Everything has its opposite, which may be as valid as the original, so that the actual activity develops in the 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; this is also the highest form of dialectics possible in philosophical idealism. Synthetic dialectical logic was developed by K.Marx and F.Engels and their followers.

3. At its highest level, logic is aware of the deep reflectivity of any category, so that every one of them contains all the other categories in it and the whole may be reconstructed starting from anywhere. 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 should be considered here as a particular possibility related to many others, and one arrangement of categories is as possible as any other. Still, these arrangements are not arbitrary, and the rules governing them could be called diathetics (intentional arrangement). Hegel used to call such logic "speculative" since the only application he could find for it was systematic study of philosophical categories. However, the domain of diathetical logic is predominantly in praxis, where nothing can be done without a clear intention.

Levels of Intelligence

The presence of logic in human behavior makes it intelligent. Three levels of intelligence can be easily distinguished, following the usual scheme: syncretism, analysis, synthesis. These levels correspond to the specific branches of logic:

1. Logic of insight. This level forms the natural basis for any logicality, since no logical development is possible before the object and purpose of activity come to awareness. Thus, any formal definition assumes some previously formed conceptions which do not need to be defined at the current consideration level. Also, to start formal deduction, one has to be aware of the intended result, which cannot be obtained in a deductive way. The ability of insight is commonly known as intuition, and there can be no good scientist without it.

2. Logic of discourse. This level is commonly associated with logic, and the major part of logical research refers to discourse. Since, in discourse, the forms of activity become completely detached from the material processes underlying them, reasoning may proceed in quite an arbitrary way, producing abstract combinations of any complexity. Still, the "objectivized" character of this logic simplifies studying it using the methodology of science. The ability of linking several actions in a logical chain is known as intellect.

3. Logic of comprehension. Neither intuition, nor discourse is not enough for understanding, which requires a delicate balance of intuition and intellect characteristic of true ingenuity. The basis for such synthesis is in the practical application of logical forms, permanent interaction with the world. In other words, comprehending one logical form requires relating it to another, synthesizing them in a higher-level scheme.

It should be noted that some intelligence may be found in animal behavior too. However, this does not make animal behavior identical to human behavior, since the latter assumes more than mere intelligence: not only the form of activity should be universal, but also its contents. This can only be achieved through communication mediated by signs.

Hierarchical Integrity

The levels of logic as described above can only exist as components (or aspects) of a whole, and one level does not deny another, despite their being opposites. The dominance of one kind of logic in a particular activity means that the other logical modes will be used in the same time for self-control, not allowing too much abstraction to lead the activity away from reality. Thus, formal reasoning must be grounded on a sound intuition, and dialectical logic implies formal consistency prescribed by the classical laws; within classical logic, one can observe that basic propositional logic would govern any application of predicate calculus, or any higher-order formal system.

Regularity and Truth

The central category of logic might be called regularity. It provides the basis for any logical development, the criteria of logicality, as well as the principles of developing logic itself. Regularity in logic plays the same role as perfection in aesthetics, or ideal in ethics. On the other hand, regularity distinguishes logic from aesthetics, which primarily deals with the unique - and from ethics, which considers regularity and irregularity as two sides of the whole.

Regularity may manifest itself in many ways. However, the most popular form of regularity is known under the name of truth, which was often claimed the only goal of any logical study. Yes, truth is as attractive in science, as beauty in the arts; however, just like beauty becomes artistic when it is enhanced to the degree of perfection, truth may be considered "logical" only if it is persued in a regular way. On the opposite, consistent falsehood may lead to a different logic, which may be more appropriate under certain circumstances. The two logics (based on verification and falsification) are mutually reflected, and this is the cause of their parallel usage in science, where the results obtained within either of the two are often considered equally valid a priori, without special reduction procedures required at a higher level of logic.

Philosophical epistemology considers such categories as objective, relative and absolute truth. One may also distinguish rightness, correctness, adequacy, etc. The relations between all these categories are to be traced in logic. Another task of logic is to describe the existing logical forms and trace their origin in human activity. In general, logic is to reflect the development of logical forms as a part of the general development of the world and human praxis as its part.

Categories and Categorial Schemes

The basic elements of art are images, or representations. Complex images get constructed from simpler images, which are as syncretic. The images of art can be produced, but cannot be defined or formalized; they may only refer to other images in a syncretic way, through allusion.

Science works with notions, which can be defined and related to other notions analytically, through their place in a relatively rigid hierarchical structure. Notions are not immediately linked to perception or action, and hence they can be combined in an formal way producing new notions, which may be quite abstract.

The basic structural element of philosophy is a category, which could be considered as the hierarchical synthesis of an image and a notion. Categories are used in categorial schemes, and cannot exist on themselves, without any reference to a scheme. Categories and schemes represent the typical (universal) ways of action, and hence they are not as dependent on perception as the images of art, while being less abstract than the notions of science.

Every logical scheme can be treated as a structure, a system, or a hierarchy:

1. Structurally, a logical scheme contains a number of logical positions linked with logical connectives. The structural aspect of a logical scheme is commonly used for definition. Every logical position is characterized by a unique collection of properties, and the process of categorization (which is the basis of analytic creativity in general) relates an empirically distinguished object to a position in some logical scheme. Inversely, the only definition an object can be given comes from its relations to the other objects, which is reflected in an appropriate logical scheme.

2. As a system, the same logical scheme may describe a number of possible inferences. The systemic aspect of a logical scheme implies splitting it into a number of substructures, so that any one of them may be used to infer the rest of the scheme. The inferences that restore less missing parts may have a higher grade of certainty. However, no inference can be absolutely trustable, since the same substructures may participate in different schemes, either of which may be used for inference, producing a different result. Hence, logical inference can only generate hypotheses, which are yet to be practically tested.

3. From the hierarchical viewpoint, a scheme represents the levels and ways of (logical) development. The scheme may be represented as a number of interrelated structures or interacting systems, forming a higher-order integrity. There may be more than two levels, and the resulting hierarchical structure or hierarchical system will represent one of the possible paths of development, from simpler to the more complex schemes. In logical hierarchies, higher levels may be considered as more general than the lower levels, thatis, to be treated as the levels of generality. However, development may follow different paths, and the same whole can be made of different parts.

Hierarchy of Schemes

Logical schemes can include many components and be rather complicated; for example, modern mathematics can operate as a scheme, or any special science as a whole. However, there are a few simple schemes that can, in many cases, be considered as fundamental, so that the more extensive schemes would originate from these simple categorial constructions.

The monad. A single category connected to itself. The only logical position of the monad is virtually identical to the only connective. Still, monads are not as trivial as it may seem, since it allows further unfolding into any other scheme, involving different sequences of generation rules. The structural aspect of the monad corresponds to the well-known principle of identity, reflecting the constancy of the subject of consideration (which, for instance, may be a definition of the science studying it). On the systemic level, the monad implies the component's transformation into itself, which may, for example, describe the normal development of a science that does not lead to contradictions requiring an extension of the limits of study. Hierarchically, the monad reflects the integrity of activity in the course of development: new levels may form, but they will be the levels of the same hierarchy.

The dyad. Two categories linked to each other. The dyad represents the opposition of the components, their difference and mutual dependence. There are two complementary unfoldings of the dyad:

[1] ® [2] ("primary", or "material" form), and

[2] Þ [1] ("secondary", or "ideal" form).

Thus, the primary form of the dyad may mean causation, while the secondary form might mean implication. The interaction between [1] and [2] may be regarded as a cycle of mutual determination: [1] produces [2], which, in its turn, produces [1'], then [2'] etc.

The triad. Three categories linked together, every one of them representing the unity of the other two. The triad implies two complementary cycles:

the "primary", or "material" cycle:		... ® [1] ® [2] ® [3] ® [1'] ® ...
the "secondary", or "ideal" cycle:		... Þ [3] Þ [2] Þ [1] Þ [3'] Þ ...

Of course, one might also conceive the schemes of the order four and higher, though, in practical usage, such schemes will most probably reduce to a number of lower-order unfoldings.