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Chapter 2 Arguments and Proofs

An argument may be described as a group of statements, one of which is claimed to follow from the others. Arguments have structure. In mathematics, the statement which is supposedly validated by the others is called the conclusion; those statements which are claimed to provide justification for the conclusion are called the hypotheses.

The type of reasoning used in arguments is traditionally divided into two basic types, deductive and inductive. It is often said that deductive reasoning involves moving from the general to the specific, whereas inductive reasoning involves moving from specific observations to claims of general principles. However, this description is a generalization that is not always the case. The major distinction might better be described in terms of whether or not the conclusion must always follow from the hypotheses. In a deductive argument it is claimed that the conclusion must be true if the hypotheses are true; that is, it is impossible for the conclusion to fail if the hypotheses hold true.

In contrast, an inductive argument involves the claim that the conclusion probably follows from the hypotheses. Deductive arguments do not become “more valid” by adding hypotheses, whereas inductive arguments may become stronger or weaker by adding hypotheses.