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in the broad sense, that on the basis of which an inference or conclusion is drawn. Premises may be facts or judgments of facts, principles, axioms, postulates, or any events or propositions that are raw data from which some information that is new to us can be extracted directly or through reasoning. In this sense we may speak equally of premises of induction and premises of deduction.
In the narrow sense, premises proper in formally deductive logical constructs are propositions to which is applied some rule of inference or formulas symbolizing the propositions and comprising statements of the rules of inference in the investigator’s language. The concept of logical corollary is symmetrical to the concept of premise. These concepts are generally relative: a proposition may be a premise in one application of a rule of inference and a corollary in another. In logical formalisms of the axiomatic type, the premises of the first steps of deduction are stated in advance in the form of axioms and thus play the role of absolute premises, or prerequisites: the deductive procedure must necessarily begin with them. In natural calculi, in which reasoning follows the principle of assumptions that was known even in antiquity, there are no absolute premises.
Whatever their character, premises are a necessary condition for logical argumentation or proof. Here the question of the nonextraneous character of premises turns out to be essential. A premise that is extraneous to a given argument may always be replaced by the contradictory premise without damage to the argument. A law of logic that may be called the law of the extraneous premise corresponds to the rule
(A & B ⊃ C) & (A ⊃ C)) ⊃ (A & ┐ B ⊃ C)
The fundamental task of logic is to investigate the corollaries of given premises and to find nonextraneous premises corresponding to given consequences. Within the limits of the formalism of the algebra of propositions, these problems have an exhaustive solution.
M. M. NOVOSELOV