Inference
In argumentation, inference is a reasoning process expressed by an argument. In mathematics, inference is the reasoning process expressed by computation, calculation, and measurement. In general, inference is a rational movement from one concept to another. It is most easily understood by reference to a hypothetical, or "if-then" proposition. For example, the proposition, "If it is raining, then it is wet," illustrates an inference made from the concept of rain to the concept of wetness. We infer that it is wet from the assertion that it is raining. Another way of expressing inferential reasoning is by the logical rule of inference called implication. Without the ability to make inferences, rational processes would be disconnected.
The process of inference is the derivation of a proposition (the conclusion) from one or more propositions (the premises) in an argument. Depending upon the validity of the inference, there is good reason to assert that the premise or premises do indeed support the conclusion. The proposition, "Larry is sleeping," can be inferred from the propositions, "If Larry is at home then he is sleeping" and "Larry is at home," with good reason because the relationships between the terms in the premises provide inferential support for the terms in the last proposition. This means that, if the premises are true, the conclusion must be true on pain of contradicting the premises.
Similarly in mathematics, inferences are made based on the relationships between numbers. For example, we can infer the number twelve when we add together the numbers five and seven. By themselves the latter numbers do not imply the number twelve, but the process of addition allows us to combine five and seven to conclude twelve.
In logic there are two basic forms of inference. One is deduction and the other is induction. Deductive inferences are necessary because, as the Larry example shows, if the premises are true then the conclusion may not be false without contradicting the premises. Inductive inferences, on the other hand, are only probable. In an inductive argument, the truth of the premises means that it is improbable for the conclusion to be false. If the conclusion is false, however, while the premises are true, there is no contradiction between premises and conclusion. For example, "Larry will be at home today," follows with probability from "Larry has been at home for the last few days." The conclusion is not inferred with necessity, however, because it is possible that Larry is not at home today.
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