Classical Logic

Since classical logic is one of the most developed parts of logic in general, and there are over two thousand years of literature on its numerous aspects, I cannot pretend to cover everything in this brief introduction. However, some explications are necessary, since it is in comparison with classical logic that other levels of logic can be comprehended.

What is classical?

Enumeration of the typical schemes of reasoning given by Aristotle and his school is commonly considered as the origin of logic as a special discipline. However, in Aristotle's texts, formal reasoning was never treated separately from the other aspects of being, including both physical nature and the movements of the human soul. This tradition of philosophical logic has never been interrupted in the course of many centuries, and it continues to the present time. The opposite of classical logic, sophistry, tried to reduce reasoning to mere manipulation with abstractions, and this line has got its clear expression in the modern logical positivism, identifying the schemes of reasoning with reasoning itself, formal models of logic with logic, the form of speech with its content.

Still, classical logic does not cover all the scope of philosophical logic, being concerned mainly with its structural aspects abstracted from their development. This relatively static character makes classical logic most useful in everyday life, while it follows the firmly established cultural standards; however, this inherent rigidity may lead to logical problems in more dynamic situations, where no stable norms could be observed - dialectical logic and hierarchical logic are more appropriate in such cases.

In classical logic, all the objects are supposed to never change during the discourse, so that the whole complexity of their relations could be observed "simultaneously". Of course, one does not mean the physical time here, but rather some "logical time", the order of discourse. Classical logic can be used to treat motion, and even development, - but this treatment will always be "classical", that is, accentuating static regularities within any process.

Branches of classical logic

As any logic at all, classical logic is applicable to any activity, and not only formal discourse. However, traditionally, the ideas of classical logic developed in application to analytical reasoning, which significantly influences the logical terminology, and provides the absolute majority of examples available.

Due to the universal character of classical logic, it is applicable to any sphere of human activity, and there may be separate applied disciplines treating the logic of any particular occupation. However, the universality of logic also means that such special "logics" will be all like one another, with mainly terminological difference, and hence it is enough to consider one particular object area, to get the logical tools for another. The logic of that scheme transfer also contains a static component that can be treated within classical logic.

Analytical reasoning is rather convenient for logical study due to its essentially formalized character. That is why most logical research has been centered on various formal systems expressible in some natural or artificial languages.

Within this "language-oriented" logic, one could distinguish logic of definition (formation of notions), logic of interrogation (problem formulation) and logic of discourse (currently, the most developed part). Depending on objective relations considered, and the detailed structure of the logical schemes involved, propositional logic, predicate logic, modal logic and many other special logics have been historically formed. A few modern models like multi-valued, fuzzy or categorial logic continue that line, remaining entirely within the scope of classical logic, despite all their "alternative" look.

Logical forms

Notions (concepts), statements (propositions) and inferences (arguments) make the hierarchy of fundamental logical forms in classical logic. They all are interdependent, and none of them can be reduced to the others.

1. Notion
   This level represents the activity of distinction, separating one object from another. Notions are not mere labels of things, they imply knowledge about things in their relation to each other, and hence a notion can be considered as a hierarchy of possible statements about the object.

   The notion should not be confused with a word of a natural or artificial language; quite often, there are no adequate words, and lengthy explanations and clarifications may be needed. In many cases no verbal explication can be given at all, and one has to learn notions practically, doing something under somebody's guidance.

2. Statement
   Statements are build of notions, they relate notions to each other, reflecting the objective relations in the world. Therefore, the number of possible statements is unlimited, since the world is inexhaustible, and ever new relations between notions will reflect additional objective regularities. In a statement, notions are connected in definite order, subordinated to the meaning of the statement as a whole. This integral meaning cannot be reduced to the meanings of the notions involved, and even less to a sentence of natural language or a formal construct; whole books may be sometimes needed to convey the meaning of one sentence, and some relations between notions can only be grasped in practical activity.

   However, statements are useless on themselves. They merely express ideas in a form, suitable for further production of other statements, in an inference scheme. Every statement has numerous consequences, without which the sentence has no sense; that is how one comes to the idea of the statement as a hierarchy of possible conclusions.

3. Inferences
   Inference is used to produce new statements (conclusions) from a number of other statements (premises) subordinated within a specific inference scheme. Inference schemes represent the most general regularities of the world, including both nature and culture, and they are usually applicable to many special cases. However, this high level of abstraction results in a higher vulnerability of a conclusion, which is most sensitive to minor shifts in the meanings of the notions involved; this implies that the applicability of a scheme must be substantiated for every instance of its usage.

   Like statements represent various relations between notions, inferences connect different relations to each other. Since a notion can be considered as a hierarchy of statements, an inference can also be regarded as a kind of unfolded notion.

   As with notions and logical statements, conclusions do not need to be entirely verbal - rather, they are universal schemes controlling the succession of conscious actions within a specific activity; as long as the activity (that is, its motive) remains the same, the consistency of activity can be achieved via logical conclusion.

Adequacy, Truth, Correctness

It is implicitly assumed that the notions may be either adequate or inadequate, statements may be either true or false, and conclusions may be either correct or incorrect. This dichotomic division lies in the basis of classical logic. The adequacy of notions, the truth of statements and the correctness of conclusions cannot be established within logic, requiring inquiry into the relations between the object and the subject, the world and its reflection in human activity. Subjectively, for a logician, this looks like the subject's ability to arbitrarily construct notions, ascribe truth values, or make conventions about admissible conclusions; this arbitrariness reflects the social position of a logician, working with the forms of things abstracted from the things themselves. In reality, logic can only be verified by practical activity, and never by mere formal reasoning. Logic is an instrument for generating hypotheses, and it cannot produce "new" truths from the already established.

The dichotomies of the classical logic originate from a special, but very important activity, binary discrimination, categorization. The very idea of analytical reasoning implies distinctions made, and opposing a particular thing to the rest of the world. Since analysis is a necessary level of every activity, classical logic is universal and ubiquitous; however, since human activity cannot be reduced to analysis, logic in general is wider than classical logic.

Fundamental principles of classical logic

Logical principles express the most general, universal rules governing the formal aspects of any activity. Traditionally, three logical principles (or laws) are commonly discussed in the literature: the law of identity, the law of non-contradiction (law of excluded middle), and the law of sufficient reason. However, logical "laws" are not as restrictive as the laws of a science, and they do not determine the exact form of activity, which also depends on the specific conditions of that activity lying outside the domain of (classical) logic; that is why it would be better to speak of principles rather than laws.

1. The principle of identity
   Definiteness is a distinctive feature of classical logic. Every notion, or relation between notions, or connectivity of relations is to remain the same during the current activity, which is thus made consistent, in the classical sense. The principle of identity positions classical logic as an essentially structural approach, which connects any ideas as if they were co-existent, and never changing. Obviously, this cannot be achieved on the semantic level, since the meaning of any word or phrase depends on the context. For instance, the same term is differently defined in different sentences, and it is only in the unity of all such partial definitions that a notion can be formed. This circumstance leads to communication difficulties, since no finite text can convey the universality of a notion in full, and different people may differently restore the whole by the exposed parts. It is only in common experience and co-operation that the identity of a notion, sentence or conclusion can be maintained: as long as people's activities remain relatively uniform, they will be able to rely on classical logic to organize their social behavior.

2. The principle of distinction
   In the act of binary discrimination, a person is to decide on whether one of the two available actions should be taken in response to a specific situation; the basic form of such a decision is: "To do, or not to do?" Threshold behavior may serve as a typical model: if a certain quality of the objective situation is intensive enough, the appropriate action is to be initiated. Numerous ways of implementing this dependence lead to many models of logic; all such models refer to the same human ability manifesting itself in different environments.

   Everybody can recall situations, when the very act of choice influenced the position of the threshold, thus inducing the denial of the decision almost made. In classical logic, such situations are forbidden, and any distinctions are to be preserved intact within the same activity. That is, once the situation has been put in a particular category, it will always be in this category, and no action may lead to the opposite decision; actions implying opposite categorizations of the same situation are called contradictory, and the principle of distinction does not allow to combine them in the same activity.

3. The principle of completeness
   Any human activity actualizes itself in a hierarchy of conscious actions directed to achieving definite goals. Once the goal is chosen, one has to concentrate efforts on making it closer, which requires a clear view of the goal and rejection of the paths that do not lead to it, as demanded by the principles of identity and distinction. However, one also needs some criteria for terminating the action. Thus, one might decide to stop when the goal of the action has been achieved in full. This is only possible in classical logic based on binary discrimination, so that the any goal is thought to be fully achievable, and any person is thought to be able to distinguish the achieved goal from not yet achieved. The principle of completeness demands that every action should be completed before its results are used in another action. This makes classical logic essentially sequential, with all the benefits and deficiencies of this approach.

   In the sphere of analytical reasoning, this principle takes the form of the law of sufficient justification: a notion is considered as well-defined only if the definition is specific enough and consistent with other definitions; a statement is supposed to be true only if it can be derived from other statements that have already been justified; a conclusion is acceptable only if it based on the complete set of premises and does not go beyond the domain of discourse.

Fallacies

Within classical logic, any violation of its principles is considered as a logical error. This does not necessarily mean that the results obtained in an erroneous way are themselves erroneous; however, logical errors often have a negative effect, since they are apt to replicate in other similar situations and other logical schemes, which may sometimes result in serious damage to people's well-being. That is why it is important to know about possible logical errors (fallacies) and avoid them.

Nobody is perfect, and every person will make logical errors. Any unnoticed error will result in numerous other errors, and false conclusions, up to apparent paradoxes. The only way to stop this error propagation is to treat any formal results as mere hypotheses, rather than "proofs", and never trust them too much until their validity in their application domain has been practically established. This is a very simple idea: in you plan to do something, this does not mean that you have already done it.

It should be noted that not all fallacies are unmediated. Some people may exploit the others' poor experience with logic to persuade them into wrong actions, using intentionally introduced logical errors. This is one more argument for the necessity of mass logical education.

Fallacies should not be confused with logical paradoxes. The latter do not violate the principles of classical logic, nevertheless arriving to contradictory conclusions. Sometimes, a false paradox may be encountered, with the results being only superficially contradictory, with a hidden logical error behind the contradiction.

Paradoxes arise in the boundary situations, where the applicability of classical logic becomes problematic; one can never resolve a paradox within classical logic, and a paradox may be considered as mechanism of linking different levels of logic.