Foundations of Mathematics

Axiomatic System

What is deduction and induction and how are they used in mathematics?

Deduction in logic is when conclusions are drawn from premises and syllogisms (for more information on these terms, see above). In this instance, a deduction is a form of inference or reasoning such that the conclusion is true if the premises are true; or, based on general principles, particular facts and relationships are derived. Deductive logic also means the process of proving true statements (theorems) within an axiomatic system; if the system is valid, all of the derived theorems are considered valid. For example, if it is known that all dogs have four legs and Spot is a dog, we logically deduce that Spot has four legs; other examples of deductive reasoning include Aristotle’s syllogisms.

Induction is a term usually used in regards to probability, in which the conclusion can be false even when the premises are true. In contrast to deduction, the premise gives grounds for the conclusion, but does not necessitate it. Inductive logic generates “correct” conclusions based on observation or data. (But note that not all inductive logic leads to correct generalizations, making the validity of many such arguments probabilistic or “iffy” in nature.)

One can see how both these processes work in the scientific world, especially in the scientific method, in which general principles are inferred from certain facts. For example, by observation of events (induction) and from principles already developed (deduction), new hypotheses are formulated. Hypotheses are then tested by applications; and as the results satisfy the conditions of the hypotheses, laws are developed by induction. Future laws are then often developed, many of them determined by deduction.


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