Deduction

Deduction, in logic, a process of reasoning in which reasons are given in support of a claim. The reasons, or justifications, are called the premises of the claim, and the claim they purport to justify is called the conclusion. In a correct, or valid, deduction the premises support the conclusion in such a way that it would be impossible for the premises to be true and for the conclusion to be false. In this, deduction differs sharply from induction, a process of drawing a conclusion in which the truth of the premises does not guarantee the truth of the conclusion.

The actual truth or falsity of the premises and the conclusion is not at issue in determining whether an argument is a valid deduction. In the following argument, for instance, two premises are offered in support of a conclusion:

The form of an argument determines whether it is a valid deduction. In general, arguments that display the form “All P’s are Q’s; t is P (or a P). Therefore, t is Q (or a Q)” are valid, as are arguments that display the form “If A then B; it is not the case that B. Therefore, it is not the case that A.” The following example displays the latter form:

The study of different forms of valid argument is the fundamental subject of deductive logic. These forms of argument are used in any discipline to establish conclusions on the basis of claims. In mathematics, propositions are established by a process of deductive reasoning, while in the empirical sciences, such as physics or chemistry, propositions are established by deduction as well as induction.

The first person to discuss deduction was the ancient Greek philosopher Aristotle, who proposed a number of argument forms called syllogisms, the form of argument used in our first example. Soon after Aristotle, members of a school of philosophy known as Stoicism continued to develop deductive techniques of reasoning. Aristotle was interested in determining the deductive relations among general and particular assertions—for example, assertions containing the expression “all” (as in our first example) and those containing the expression “some.” He was also interested in the negations of these assertions. The Stoics focused on the relations among complete sentences that hold by virtue of particles such as “if … then,””it is not the case that” (as in our second example), “or,””and,” and so forth. Thus the Stoics are the originators of sentential logic (so called because its basic units are whole sentences), whereas Aristotle can be considered the originator of predicatelogic (so called because in predicate logic it is possible to distinguish between the subject and the predicate of a sentence).

In the late 19th and early 20th centuries the German logicians Gottlob Frege and David Hilbert argued independently that deductively valid argument forms should not be couched in a natural language—the language we speak and write in—because natural languages are full of ambiguities and redundancies. For instance, consider the English sentence “Every event has a cause.” It can mean either that one cause brings about every event, wherein A causes B, C, D, and so on, or that individual events each have their own, possibly different, cause, wherein X causes Y, Z causes W, and so on. The problem is that the structure of the English language does not tell us which one of the two readings is the correct one. This has important logical consequences. If the first reading is what is intended by the sentence, it follows that there is something akin to what some philosophers have called the primary cause, but if the second reading is what is intended, then there may well be no primary cause.

To avoid these problems, Frege and Hilbert proposed that the study of logic be carried out using formalized languages. These artificial languages are specifically designed so that their assertions reveal precisely the properties that are logically relevant—that is, those properties that determine the deductive validity of an argument. Written in a formalized language, two unambiguous sentences remove the ambiguity of the sentence, “Every event has a cause.” The first possibility is represented by the sentence , which can be read as 'there is a thing x, such that, for every y, x causes y.' This would correspond with the first interpretation mentioned above. The second possible meaning is represented by , which can be read as 'for every thing y, there is a thing x such that x causes y.' This would correspond with the second interpretation mentioned above. Following Frege and Hilbert, contemporary deductive logic is conceived as the study of formalized languages and formal systems of deduction.

Although the examples in this article are simple, the process of deductive reasoning can be extremely complex. Conclusions are obtained from a step-by-step process in which each step establishes a new assertion that is the result of an application of one of the valid argument forms either to the premises or to previously established assertions. Thus the different valid argument forms can be conceived as rules of derivation that permit the construction of complex deductive arguments. No matter how long or complex the argument, if every step is the result of the application of a rule, the argument is deductively valid: If the premises are true, the conclusion has to be true as well.