Deduction

Deduction Defined

According to Aristotle (385–322 B.C.E.), deduction is inference from the general to the particular. Although he discussed both deduction and induction in his book the Organon (The Instruments), his primary focus was deduction, mainly in the form of his syllogism, which is still deemed as representative of deductive logic. The Oxford English Dictionary adopts Aristotle’s definition and defines deduction as “the process of deducing or drawing a conclusion from a principle already known or assumed; spec. in logic, inference by reasoning from generals to particulars; opposed to induction.”

Later logicians have broadened the definition of deduction and put more types of inferencing methods into this category. Two widely used logic textbooks, Irving Copi and colleagues’ Introduction to Logic (14th edition) and Patrick Hurley’s A Concise Introduction to Logic (12th edition), both define a deductive argument as one whose premises support the conclusions in the way that it is impossible (as opposed to improbable in induction) for the premises to be true and the conclusions to be false. Hurley lists five types of typical deductive arguments: arguments based on mathematics, arguments from definitions, categorical syllogism, hypothetical syllogism, and disjunctive syllogism. Arguments based on mathematical calculations, measurements, and axioms (but not statistics, which forms the basis of inductive arguments, as they infer from a sample to the population) have the validity of their conclusion fully guaranteed from the premises. An argument from a definition is deductive because the information in the conclusion is fully contained in the definition, the premise.

A syllogism is a special format of deductive argument that has two premises and a conclusion. Categorical syllogism is the classic syllogism discussed by Aristotle, in which both premises and the conclusion are either universal statements or particular statements. For instance,

• Major premise: All men are mortal.
• Minor premise: Socrates is a man.
• Conclusion: Therefore, Socrates is mortal.

Copi and colleagues and Hurley also classify the immediate inferences such as reduction and conversion of categorical propositions as deduction and develop other type of syllogisms in the deductive system, such as hypothetical syllogism and disjunctive syllogism. A hypothetical syllogism adopts the syllogism format but has a conditional statement as the first premise. For instance,

• Major premise: If it rains, the floor will be wet.
• Minor premise: It rains,
• Conclusion: Therefore, the floor must be wet.
• A disjunctive syllogism has a disjunctive statement as the first premise. For instance,
• Major premise: The flower is either red or blue.
• Minor premise: The flower is not blue.
• Conclusion: Therefore, the flower must be red.

Philosopher and logician Karl Popper (1902–1994), who believed that deduction is the only logical approach to reasoning, provided a clear definition of deduction. That is, a deductive statement is true if and only if when all the truth contained in the conclusion has been transmitted from the premises. He further thought that the definition is a tautology to the following one: A deductive inference is valid if and only if no counterexample exists. That statement served as the philosophical foundation of his famous theory of falsification.

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