inference the process of drawing a conclusion from premises or assumptions, or, loosely, the conclusion so drawn. An argument can be merely a number of statements of which one is designated the conclusion and the rest are designated premises. Whether the premises imply the conclusion is thus independent of anyone’s actual beliefs in either of them. Belief, however, is essential to inference. Inference occurs only if someone, owing to believing the premises, begins to believe the conclusion or continues to believe the conclusion with greater confidence than before. Because inference requires a subject who has beliefs, some requirements of (an ideally) acceptable inference do not apply to abstract arguments: one must believe the premises; one must believe that the premises support the conclusion; neither of these beliefs may be based on one’s prior belief in the conclusion. W. E. Johnson called these the epistemic conditions of inference. In a reductio ad absurdum argument that deduces a self-contradiction from certain premises, not all steps of the argument will correspond to steps of inference. No one deliberately infers a contradiction. What one infers, in such an argument, is that certain premises are inconsistent.
Acceptable inferences can fall short of being ideally acceptable according to the above requirements. Relevant beliefs are sometimes indefinite. Infants and children infer despite having no grasp of the sophisticated notion of support. One function of idealization is to set standards for that which falls short. It is possible to judge how nearly inexplicit, automatic, unreflective, lessthan-ideal inferences meet ideal requirements.
In ordinary speech, ‘infer’ often functions as a synonym of ‘imply’, as in ‘The new tax law infers that we have to calculate the value of our shrubbery’. Careful philosophical writing avoids this usage. Implication is, and inference is not, a relation between statements.
Valid deductive inference corresponds to a valid deductive argument: it is logically impossible for all the premises to be true when the conclusion is false. That is, the conjunction of all the premises and the negation of the conclusion is inconsistent. Whenever a conjunction is inconsistent, there is a valid argument for the negation of any conjunct from the other conjuncts. (Relevance logic imposes restrictions on validity to avoid this.) Whenever one argument is deductively valid, so is another argument that goes in a different direction. (1) ‘Stacy left her slippers in the kitchen’ implies (2) ‘Stacy had some slippers’. Should one acquainted with Stacy and the kitchen infer (2) from (1), or infer not-(1) from not-(2), or make neither inference? Formal logic tells us about implication and deductive validity, but it cannot tell us when or what to infer. Reasonable inference depends on comparative degrees of reasonable belief.
An inference in which every premise and every step is beyond question is a demonstrative inference. (Similarly, reasoning for which this condition holds is demonstrative reasoning.) Just as what is beyond question can vary from one situation to another, so can what counts as demonstrative. The term presumably derives from Aristotle’s Posterior Analytics. Understanding Aristotle’s views on demonstration requires understanding his general scheme for classifying inferences.
Not all inferences are deductive. In an inductive inference, one infers from an observed combination of characteristics to some similar unobserved combination. ‘Reasoning’ like ‘painting’, and ‘frosting’, and many other words, has a process–product ambiguity. Reasoning can be a process that occurs in time or it can be a result or product. A letter to the editor can both contain reasoning and be the result of reasoning. It is often unclear whether a word such as ‘statistical’ that modifies the words ‘inference’ or ‘reasoning’ applies primarily to stages in the process or to the content of the product. One view, attractive for its simplicity, is that the stages of the process of reasoning correspond closely to the parts of the product. Examples that confirm this view are scarce. Testing alternatives, discarding and reviving, revising and transposing, and so on, are as common to the process of reasoning as to other creative activities. A product seldom reflects the exact history of its production. In An Examination of Sir William Hamilton’s Philosophy, J. S. Mill says that reasoning is a source from which we derive new truths (Chapter 14). This is a useful saying so long as we remember that not all reasoning is inference. See also DEDUCTION , IMPLICATION, INDUC- TIO. D.H.S.