Classical Logic

Classical Logic

Classical logic is probably the most developed part of logic in general, and its numerous aspects are widely discussed in the literature. Still, the origin of logical rules and the overall organization of classical logic are yet poorly comprehended, and this hinders the development of the other levels of logic, since classical logic forms a natural basis for their formation, and they cannot be comprehended without relating them to classical logic. Unfortunately, the invention of symbolic logic has distracted the efforts of logicians from the general logical ideas to their special models, so that the study of the fundamental principles has been replaced with the enumeration of the possible formal schemes. However, classical logic, like any logic at all, is not merely formal; it necessarily comprises some ontological and ethical aspects. The exaggeration of the formal issues has estranged logic from ordinary life, limiting it to formal science and engineering. The lack of logic in human activity may support swindlers and profiteers, but it is incompatible with the development of consciousness and reason.

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 books, formal reasoning was never treated as separate 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 proceeds following the already established cultural norms; however, this inherent rigidity may lead to logical problems in the periods of change, of instability, in crisis situations, or upon encountering a very different culture. The new standards have not yet formed, and one needs a different logic to determine the directions of development; in such cases, dialectical or diathetical logic would be more appropriate.

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, this is not necessarily the simultaneity in the physical sense, but rather in some "logical time", the order of discourse. Classical logic can certainly be applied to motion, and even to development; but this treatment will always be "classical", that is, accentuating static regularities within any process or development phase.

Branches of classical logic

Like logic in general, classical logic is applicable to any activity at all, and not only to formal discourse. However, traditionally, the ideas of classical logic have mainly been developed for the needs of analytical reasoning, which has significantly influenced the terminology; most examples of classical logic are being taken from the domain of discourse, presenting the figures of thought rather than the schemes of activity.

Due to the universal character of classical logic, various applied disciplines treating the logic of any particular occupation can be constructed. However, the universality of logic also means that such special "logics" will be all alike, 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 explicit character and the possibility of immediate implementation of any formal scheme. Most logical research has been centered on various formal systems expressible in some natural or artificial language.

Within this "language-oriented" logic, one could distinguish the logic of definition (the formation of notions and logical rules, the logic of meaning), the logic of interrogation (the techniques for outlining the problem area, the logic of sense), and logic of discourse (including the logic of inference as its most developed component). Depending on objective relations covered and the character of the logical schemes involved, such special forms of classical logic as propositional logic, predicate logic, modal logic and many others have historically formed. Various multi-valued, categorial, fuzzy or stochastic logics continue that line, remaining entirely within the scope of classical logic, however "alternative" they might look.

Logical forms

Notions (concepts), statements (propositions) and inferences (arguments) make the commonly known hierarchy of fundamental logical forms in classical logic. They all are interdependent, and none of them can be reduced to the others. Also, one is free to unfold the hierarchy in a different dimension, considering various non-traditional categorial structures.

The level of notion 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. Notions cannot be simply denoted; quite often, the lack of adequate words results in lengthy explanations and clarifications. In many cases there is no verbal explication at all, and one has to learn notions practically, doing something under somebody's guidance.

Statements are built 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 we will discover ever new relations between notions revealing 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.

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 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 (and its motive in particular) remains the same, the consistency of activity can be achieved and inspected through logical conclusion.

Adequacy, Truth, Correctness

Within classical logic, it is implicitly assumed that the notions can be either adequate or inadequate, statements can be either true or false, and conclusions can be either valid or incorrect. This dichotomy lies in the basis of classical logic. The adequacy of notions, the truth of statements and the validity of conclusions cannot be established within logic, since it concerns the relations between the object and the subject, the world and its reflection in human activity. Subjectively, for a logician, the applicability of classical logic to practical activity 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, always operating with the forms of things abstracted from the things themselves. In reality, logic can only be verified by action, and never by mere formal reasoning. Logic is only 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, or categorization. The very idea of analytical reasoning implies making sharp distinctions, 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

The basic ideas of classical logic express the most general, universal rules governing the formal aspects of any activity. Traditionally, three logical laws have been widely discussed in the literature: the law of identity, the law of non-contradiction (also known as the law of excluded middle), and the law of sufficient justification. 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 logical principles rather than laws.

The principle of identity

Definiteness is a distinctive feature of classical logic. Every notion or a relation between notions, or mutual dependence of such relations, is to remain the same during the current activity, which is thus made consistent, in the classical sense. In classical logic, ideas merely co-exist; they are being defined once and forever, never subject to any change. The same holds for the possible relations between ideas. That is, the principle of identity positions classical logic as an essentially structural approach. Obviously, such a static picture cannot be achieved on the semantic level, since the sense of any word or phrase essentially depends on the context. For instance, a term can be introduced in many ways, with numerous formal descriptions, while the notion is only defined as the unity of such partial definitions. This circumstance is a source of communication difficulties, since no finite text can convey the universality of a notion in full and different people can differently restore the whole from 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. However, when the society is split to antagonistic classes or exclusive estates, the identity of a notion can only be maintained within the same social group.

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 can 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 that 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 combining them in the same activity.

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. In classical logic based on binary discrimination 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 valid only if it based on the complete set of premises and does not get beyond the domain of discourse. In the strictest sense, in formal logic, this principle is formulated as the law of excluded middle: any statement is either true or false (and hence its negation is true), and there is nothing in between; this formulation reveals the inherent insufficiency of classical logic.


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 a serious damage to people's well-being. That is why it is important to know about possible logical errors (fallacies) and avoid them.

There are different classifications of fallacies depending on the adopted conception of classical logic as such; neither of them can be exhaustive, as the other positions of the hierarchy that require appropriate consideration. Thus, among the commonly considered, one could distinguish fallacies of relevance, of ambiguity, and of presumption. Fallacies of relevance refer to the arguments relying on premises that aren't relevant to the discussion (for instance, irrelevant appeals). Ambiguity arises in an argument when one connotation of a word is implicitly replaced by another. Fallacies of presumption mean using false premises to derive any desirable conclusion (for instance, false dilemmas and circular arguments). All such arguments (or acts) violate the principle of identity. Other classes of fallacies arise from violating the principles of distinction or completeness.

Nobody is perfect, and every person will make logical errors. Any unnoticed error will result in a sequence of induced 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: if you are planning 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 are different from mere delusion. When people do not know something well enough, they may assert something wrong about it, but this is not a logical fallacy, despite its ability to propagate through a sequence of syllogisms. Only when a false statement is intentionally used in an argument, a logical error occurs.

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 hence a paradox could be considered as mechanism of linking different levels of logic.

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