Deduction

In ordinary language, the term deduction vaguely refers to a kind of activity in which we all are involved when we try to solve theoretical or practical problems: drawing conclusions from given premises. In logical theory, the term has a more precise meaning and refers to the special case in which the conclusion is indisputably justified on the basis of the premises. Deductive reasoning plays a pervasive role in philosophical and scientific practice as well as in everyday problem solving and decision making, and the ability of a “rational agent” to perform deductive inferences (no matter how complex) is an idealized assumption of some economical, political, and ethical theories.

This entry offers an informal overview of the general concept of deduction. A rigorous treatment is the subject of formal logic.

From a linguistic viewpoint, an inference consists of a list of sentences (premises) followed by words like hence, thus, or therefore and then by another sentence (the “conclusion”). Consider the following two examples:

In both examples, the premises provide some justification for asserting the conclusion. However, in Inference A, the justification is as strong as it gets: No rational agent can fail to recognize that its conclusion must certainly be true in all situations in which the premises are true. This is what we mean by saying that A is deductively sound. In contrast, it is perfectly conceivable that the premises of B are true while its conclusion is false. So B is not deductively sound and (much) harder to justify. According to some authors—most eminently Karl [Page 189]Popper in his Logic of Scientific Discovery—it cannot be justified at all. According to others—most eminently Bertrand Russell in his The Problems of Philosophy—it may be justified by a general “principle of induction” and thereby regarded as inductively sound.

In order to recognize that A is sound, we do not need to understand the meaning of “raven” and “black,” or the denotation of “Mr. Poe.” Indeed, all these words could be substituted by other words, such as whales, white, and Moby Dick, without affecting the soundness of the inference. However, substituting “all” with “some” would yield an obviously unsound inference. Hence, the soundness of A depends only on the meaning of “all.” A word whose meaning is essential for recognizing the soundness of an inference is called a logical word or logical constant. The other words are called extralogical and can be replaced by schematic letters. So the general form of A is as follows:

A schematic inference such as A* is called an inference rule. The soundness of all its instances can be immediately recognized by any agent who understands the meaning of “all.” (Indeed, some authors maintain that the meaning of a logical word can be completely defined by exhibiting simple inference rules, such as A*, showing how the logical word can be legitimately used in inference.) So, recognizing that an inference is deductively sound, in the simplest cases, is just part of our linguistic competence. This is why deductive reasoning has often been described as “analytic,” and it has been maintained, especially by logical neopositivists, that it conveys no new information. (This claim, however, is intuitively implausible, and Jaakko Hintikka has called it a true “scandal of deduction.”)

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