# Deduction

Updated on January 26, 2010

Deduction, in the literary and philosophical use of the term, is reasoning, inference, or proof; more narrowly, in logic, it is demonstrative inference, reasoning from a more inclusive, or general, proposition (premise) to a less inclusive, or general, proposition (conclusion) contained in or subsumable under the former. An example of deduction, a syllogism, is: "No unintentional act should be punished; the killing of a pedestrian by a drunken motorist is an unintentional act; therefore, no such killing should be punished."

## Definition of Deduction

The deductive method was developed by the ancient Greek philosophers; but the Latin word deductio ("drawing out"), early used in arithmetic for the "deducting" of quantities, was first applied to the logical drawing of conclusions by the medieval Scholastics, who made the deductive method fundamental. Deduction contrasts especially with induction, recognized in Greek philosophy but strenuously advocated at the Renaissance as the "new logic" of natural science, which pursues the reverse way of inference, from members of a class to the whole class.

The supreme rule of the syllogism was that the premise cover all of any class covered by the conclusion; and although the syllogism is submerged in the more inclusive modern logic of deduction, the old criterion - that deduction is "explicative" or "analytic," bringing to light components covert or implicit in a complex - still seems appropriate. Increasingly favored, however, is another criterion - that deduction is "demonstrative," its conclusion following with certainty and necessity and not mere probability (as in induction). A deduction is hence "conclusive," which means that it cannot be counteracted by new evidence, as even the highest probability can. It is usual and convenient, therefore, to say that deduction is analytic demonstrative inference, even though some logicians deny that deduction is analytic or that induction is non-demonstrative, and even though there are "probability syllogisms," which Charles Peirce and others called deductions; for example: "Nearly all the officers signed, and he was an officer, so almost certainly he signed."

## Enrolment of Conclusions

The properties of deductive inference derive from the deductive implication, or "entailment," between premise and conclusion- the way the latter "follows from" the former. Let p and q be propositions; then, that p entails q means at least that it is logically necessary that if p (the "antecedent") is true, then q (the "consequent") must be true. It is impossible (a contradiction) that p be true and q false.

Logic tabulates many of the forms of propositions that exemplify entailment, but the relation is often obvious to unaided good sense and may be interesting regardless of its use for inference and even of whether the propositions are true. Thus one may be curious about what would be true if the antecedents were true; or, being convinced that both antecedent and consequent are true, one may appreciate that because of the entailment the antecedent explains the consequent. Even where the entailment is used for inference, moreover, this may be nondeductive; for, where p entails q and one knows q but not p, one may count q, which is not a proof of p, as some corroboration of it. Nevertheless, the definitive use of a known entailment is to validate a deductive inference.

Logicians thus often affix to statements of entailment a prescription, "the rule of deduction": Given that p entails q and that p is true, one may assert (or infer) q. This is the rule for affirming the antecedent, or modus ponens. The rule of modus tollens, or denying the consequent, is: Given that p entails q and that q is false, one may infer that p is false. Strictly this is not needed, being replaceable by a statement of entailment, namely, the fact that p implies q and that q is false entails that p is false, from which ponens yields the same result as would a direct use of tollens.

Still, some logicians regard a proliferation of rules of inference as more "natural" than catalogs of entailments, and the tollens rule well represents the way one uses an entailment to reject its antecedent because one rejects the consequent. Thus, the conclusion of the specimen syllogism about drunken driving may seem so preposterous (a reductio ad absurdum) as to suggest that at least one premise is fake, or that the apparent entailment is illusory.

## Grounds of Premises

If logic or intuition guarantees that p entails q, what guarantees p? The obvious answer is further deduction- perhaps a "chain" with "linear inference," providing premises for the premises, going back until the inquirer is satisfied, or perhaps a whole "deductive system" such as Euclid's geometry. In such a logical web or genealogical tree, "theorems" radiate along lines of entailment from a central few primary propositions, or axioms. The system provides in high degree the non-deductive values of entailments generally, but the strictly deductive ideal has been to prove all its theorems, directly or indirectly, from the axioms. The axioms, or "indemonstrables", were formerly expected to be self-evident, knowable a priori as the entailments are, so that a science thus "axiomatized" is a "deductive science" in a strong sense. Deductive systems are still exhibited and applauded, but the notion that fundamental truths are especially obvious, or even that there is only one way of axiomatizing a given subject, is today widely rejected.

Deductions normally must accept premises from nondemonstrative forms in inference and be infected with their uncertainty. Such forms include both straight induction and the "hypothetico-deductive method," which, with due precautions, confirms the antecedents of deductions by verifying their consequents through observation. Hence the saying that deduction and induction are complementary rather than opposing processes.

The 19th century idea of deductive machines has been realized in the electronic calculator, or computer. Exactly how similar its processes are to deductive reasoning strictly speaking is debatable, but as an instrument it certainly much extends the reach of human thought.

0

25

10