Inference: Deductive and Inductive

Implication versus Inference

Fundamental to an understanding of deductive reasoning is a distinction between implication and inference. Implication is a logical relationship; inference is a cognitive act. Statements imply; people infer. A implies B if it is impossible for A to be true if B is false. People are said to make an inference when they justify one claim (conclusion) by appeal to others (premises). Either implications exist or they do not, independently of whether inferences are made that relate to them. Inferences either are made or are not made; they can be valid or invalid, but they are inferences in either case. Failure to keep the distinction in mind can cause confusion. People are sometimes said to imply when they make statements with the intention that their hearers will see the implications of those statements and make the corresponding inferences, but to be precise in the use of language one would have to say not that people imply but that they make statements that imply.

Aristotle and the Syllogism

The preeminent name in the history of deductive reasoning is that of Aristotle, whose codification of implicative relationships provided the foundation for the work of many generations of logicians and epistemologists. Aristotle analyzed the various ways in which valid inferences can be drawn with the structure referred to as a categorical syllogism, which is a form of argument involving three assertions, the third of which (the conclusion) follows from the first two (the major and minor premises). A syllogism is said to be valid if, and only if, the conclusion follows from (is implied by) the premises. Aristotle identified many valid forms and related them to each other in terms of certain properties such as figure and mood. Figure relates to the positions of the middle term—the term that is common to both premises—and mood to the types of premises involved. An explanation of the system Aristotle used to classify syllogistic forms can be found in any introductory text on first-order predicate logic.

That deductive reasoning makes explicit only knowledge already contained implicitly in the premises from which the deductions are made prompts the question: Of what practical use is deduction? One answer is that what is implicit in premises is not always apparent until it has been made explicit. It is not the case that deductive reasoning never produces surprises for those who use it. A mathematical theorem is a conclusion of a deductive argument. No theorem contains information that is not implicit in the axioms of the system from which it was derived. For many theorems, the original derivation (proof) is a cognitively demanding and time-consuming process, but once the theorem has been derived, it is available for use without further ado. If it were necessary to derive each theorem from basic principles every time it was used, mathematics would be a much less productive enterprise.

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