The typical expression of deductive inference is the syllogism. The essential feature of deduction is the necessary character of the connexion between the antecedent or premises and the consequent or conclusion. Granted the truth of the antecedent judgments, the consequent must follow; and the firmness of our assent to the latter is conditioned by that of our assent to the former. The antecedent contains the ground or reason which is the motive of our assent to the consequent; the latter, therefore, cannot have greater firmness or certainty than the former. This relation of necessary sequence constitutes the formal aspect of deduction. It can be realized most clearly when the argument is expressed symbolically, either in the hypothetical form:
or in the categorial form,
The material aspect of the deductive argument is the truth or falsity of the judgments which constitute it. If these be certain and evident the deduction is called demonstration , the Aristotelian apodeixis . Since the conclusion is necessarily implied in the premises, these must contain some abstract, general principle, of which the conclusion is a special application; otherwise the conclusion could not be necessarily derived from them; and all mediate inferences must be deductive, at least in this sense, that they involve the recognition of some universal truth and do not proceed directly from particular to particular without the intervention of the universal.
When, starting from general principles, we advance by a series of deductive steps to the discovery and proof of new truths, we employ the deductive or synthetic method. But how do we become certain of those principles which form our starting-points?
This ascent from the particular to the general is called induction, or the inductive or analytic method. Comparatively little attention was paid to this method during the Middle Ages. Apparatus for the accurate observation and exact measurement of natural phenomena was needed to give the first real impetus to the cultivation of the physical, natural, or inductive sciences. In these departments of research the mind approaches reality from the side of the concrete and particular and ascends to the abstract and general, while in deduction it descends from the general to the particular. But although the mind moves in opposite directions in both methods, nevertheless the reasoning or inference proper, employed in induction, is in no sense different from deductive reasoning, for it too implies and is based on abstract, necessary truths.
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