deduction


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deduction,

in logiclogic,
the systematic study of valid inference. A distinction is drawn between logical validity and truth. Validity merely refers to formal properties of the process of inference.
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, form of inference such that the conclusion must be true if the premises are true. For example, if we know that all men have two legs and that John is a man, it is then logical to deduce that John has two legs. Logicians contrast deduction with inductioninduction,
in logic, a form of argument in which the premises give grounds for the conclusion but do not necessitate it. Induction is contrasted with deduction, in which true premises do necessitate the conclusion.
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, in which the conclusion might be false even when the premises are true. Deduction has to do with necessity; induction has to do with probability. The famous Aristotelian syllogismsyllogism,
a mode of argument that forms the core of the body of Western logical thought. Aristotle defined syllogistic logic, and his formulations were thought to be the final word in logic; they underwent only minor revisions in the subsequent 2,200 years.
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 is one species of deductive reasoning, which was greatly extended by the development of symbolic logicsymbolic logic
or mathematical logic,
formalized system of deductive logic, employing abstract symbols for the various aspects of natural language. Symbolic logic draws on the concepts and techniques of mathematics, notably set theory, and in turn has contributed to
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.

Bibliography

See R. J. Ackermann, Modern Deductive Logic (1971); P. J. Hurley A Concise Introduction to Logic (1985).

Deduction

 

the progression from the general to the particular. More specifically, the term “deduction” denotes the process of logical inference, that is, a progression according to particular rules of logic from certain given statements (the premises) to their consequences (conclusions); the consequences can always in some sense be characterized as particular cases (examples) of the general premises. The term “deduction” is also used to denote the concrete derivation of conclusions from particular premises (as a synonym for the term “proof” in one of its meanings). More often, deduction is the generic name for the general theory of constructing correct inferences. According to this last usage those sciences—for example, mathematics, theoretical mechanics, and some branches of physics—are commonly called deductive sciences whose assumptions (or at least most of them) are obtained as the result of certain general “basic laws” (principles, postulates, axioms). The axiomatic method which is used to derive these particular propositions is frequently called the axiomatic-deductive method.

The study of deduction is the main task of logic. Sometimes logic (or in any case formal logic) is even defined as the theory of deduction, although logic is by no means the only science studying the methods of deduction. Psychology studies deduction in the process of actual individual thinking and its formation, while epistemology (the theory of knowledge) studies it as one of the basic methods (along with other methods, particularly various forms of induction) of scientific knowledge about the world.

Although the term “deduction” itself was apparently first used by Boethius, the concept of deduction as proof of some proposition by means of a syllogism may already be found in Aristotle (Prior Analytics). In the philosophy and logic of the Middle Ages and modern times there have been significant differences in views on the role of deduction among other methods of knowledge. R. Descartes, for example, opposed deduction to intuition, by which, according to him, the human mind “directly discerns” the truth. For Descartes, deduction gives the mind only “mediated” (obtained by reasoning) knowledge. (The primacy of intuition over deduction proclaimed by Descartes was revived much later and in a substantially modified and elaborated form in the concepts of what is known as intuitionism.) F. Bacon and later the other English logical inductivists (W. Whewell, J. S. Mill, A. Bain) noted correctly that a conclusion obtained by means of deduction does not contain (to use a contemporary expression) any “information” that is not contained (albeit concealed) in the premises. For this reason they regarded deduction as a “secondary” method and believed that only induction gives true knowledge. Finally, representatives of the school that arose primarily out of German philosophy (C. Wolff and G. W. Leibniz), also proceeding essentially from the idea that deduction does not produce “new” facts, reached exactly the opposite conclusion. This school of philosophy maintained that knowledge obtained by deduction is “true in all possible worlds” (or as later expressed by I. Kant, analytically true). This determines its “immutable” value (as distinguished from “factual,” or “synthetic,” truths obtained by the inductive generalizing of data from observation and experiment; these truths are valid, so to speak, “only because of a coincidence of circumstances”).

From a modern point of view the question of the relative “advantages” of deduction or induction has to a significant extent lost its meaning. F. Engels wrote that “induction and deduction belong together as necessarily as synthesis and analysis. Instead of one-sidedly lauding one to the skies at the expense of the other, we should seek to apply each of them in its own place, and that can be done only by bearing in mind that they belong together and that they supplement each other” (Dialektika prirody, 1969, pp. 195-96). However, regardless of the dialectical interrelationship of deduction and induction noted here and regardless of their applications, studying the principles of deduction is of enormous independent importance. It is precisely the investigation of these principles as such that has constituted the principal content of all formal logic from Aristotle to the present day. Moreover, intensive work is now being done on the creation of various systems of inductive logic. In inductive logic—and such is the dialectic of these at first glance polar concepts—a kind of ideal is the creation of “deductive-like” systems, that is, aggregates of the rules that can be followed to obtain conclusions which, if they are not 100 percent certain (as is knowledge obtained by means of deduction), at least have a sufficiently high degree of verisimilitude, or probability.

In formal logic in the narrower sense of the term, it is entirely true—for both the system of logical rules itself and for any application of them in any area—that everything contained in any “analytic truth” (or “logical truth”) obtained by deduction is already contained in the premises from which it is derived. Each application of a rule consists in a general statement being related (applied) to some concrete (particular) situation. Some rules of logical inference fit this description in a perfectly obvious way. For example, various modifications of the so-called rule of substitution say that the property of provability (or the ability to be derived from the given system of premises) is preserved with any substitution of concrete expressions of the same type for the elements of an arbitrary formula of a given formal theory. The same is true for the common procedure of constructing axiomatic systems by means of diagrams of axioms—that is, expressions that become specific axioms after the substitution of concrete formulas of a given theory for the generic designations of the expressions.

But no matter what concrete form a particular rule may have, any application of it is always deductive in nature. The immutability, necessity, and formality of rules of logic, which do not permit exceptions, provide rich opportunities for the automation of the process of logical deduction by the use of electronic computers.

Deduction is often understood as the actual process of logical sequence. This leads to a close relationship (and sometimes even to identity) between the concept of deduction and the concepts of derivation and sequence, and this relationship is also reflected in logical terminology. For example, the phrase “theorem of deduction” is ordinarily applied to one of the most important correlations between the logical copula of implication (which is a logical form of the verbal expression “if…, then …”) and the relationship of logical sequence, or derivability: if consequence B is derived from premise A, then the implication A ⊃ B (If A, then B) is demonstrable, that is, it can be derived without any premises, from the axioms alone. (Under certain general conditions the theorem of deduction is correct for all logical systems proper and in some cases it is simply postulated for them as an initial rule.) Other logical terms linked to the concept of deduction are similar in nature. For example, statements derived from one another are called deductively equivalent, and the deductive completeness of a system (relative to some property) consists in the fact that all the expressions of a given system that have this property (for example, being true owing to a certain interpretation) are demonstrable in it.

The properties of deduction are essentially the properties of the relationship of derivability. Therefore they were primarily revealed in the course of constructing concrete logical (and logical-mathematical) formal systems (calculuses) and the general theory of such systems (the so-called proof theory). Major contributions to this study were those of Aristotle, who founded formal logic, as well as those of other classical thinkers. Leibniz advanced the idea of formal logical calculus and is correctly considered to be the father of mathematical logic. G. Boole, W. Jevons, P. S. Poretskii, and C. Peirce created the first systems of algebraic logic. G. Peano, G. Frege, and B. Russell created the first axiomatic systems of mathematical logic. Important contributions were also made by the school of contemporary researchers deriving from D. Hilbert (K. Gödel, A. Church, J. Herbrand), which also includes the founders of the theory of deduction known as the calculuses of natural deduction of the German logician G. Gentzen, the Polish logician S. Jaskowski, and the Dutch logician I. Beth.

The theory of deduction is also being worked out at the present time, and researchers in the USSR include P. S. Novikov, A. A. Markov, N. A. Shanin, and A. S. Esenin-Vol’pin.

REFERENCES

Aristotle. Analitiki pervaia i vtoraia. Moscow, 1952. (Translated from Greek.)
Descartes, R. Pravila dlia rukovodstva uma. Moscow-Leningrad, 1936. (Translated from Latin.)
Descartes, R. Rassuzhdenie o metode. Moscow, 1953.
Leibniz, G. W. Novye opyty o chelovekskom razume. Moscow-Leningrad, 1936.
Tarski, A. Vvedenie v logiku i metodologiiu deduktivnykh nauk. Moscow, 1948. (Translated from English.)
Asmus, V. F. Uchenie logiki o dokazatel’stve i oproverzhenii. Moscow, 1954.

IU. A. GASTEV

deduction

[di′dək·shən]
(mathematics)
The process of deriving a statement from certain assumed statements by applying the rules of logic.

deduction

The amount deducted from the contract sum by a change order.

deduction

1. the act or process of deducting or subtracting
2. 
a. the process of reasoning typical of mathematics and logic, whose conclusions follow necessarily from their premises
b. an argument of this type
c. the conclusion of such an argument
3. Logic a systematic method of deriving conclusions that cannot be false when the premises are true, esp one amenable to formalization and study by the science of logic
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