Logic is a branch of philosophy that deals with the rules of correct reasoning. Most work in the field of logic deals with a form of reasoning called an argument. An argument consists of a set of statements called premises, followed by another statement called the conclusion. If the premises support the conclusion, the argument is correct. If the premises do not support the conclusion, the argument is incorrect.
There are two types of arguments, deductive and inductive. A deductive argument is valid when the conclusion must be true if the premises are true. When the conclusion does not necessarily follow from the premises, a deductive argument is invalid. In an inductive argument, the conclusion is more or less probably true on the basis of the premises. Because the conclusion does not follow necessarily from the premises, an inductive argument is not usually deductively valid. An inductive argument may be correct or incorrect. This article deals mainly with deductive reasoning. For more information on inductive reasoning, see Inductive method .
Logic tells us whether a deductive argument is valid or invalid. The validity of such an argument depends on the form of the argument, not on the truth of its premises. As a result, an argument that depends on false premises could be valid, and an argument based on true premises could be invalid.
The categorical syllogism
is the most common form of argument in traditional deductive logic. The ancient Greek philosopher Aristotle was one of the first scholars to carry out a systematic study of the categorical syllogism.
A syllogism consists of two premises and a conclusion. A categorical syllogism is one in which every statement has one of the four forms: (1) All A are B. (2) No A are B. (3) Some A are B. (4) Some A are not B. The letters A and B, or any other letters that might be used, are terms that represent various classes of things, such as numbers, people, yellow objects, unpleasant sounds, or brown cows. The following argument is an example of a valid categorical syllogism: “All mammals are warm-blooded. All brown cows are mammals. Therefore, all brown cows are warm-blooded.” The form of this syllogism is: “All A are B. All C are A. Therefore, all C are B.”
The following categorical syllogism is invalid: “No stars are planets. Some satellites are not planets. Therefore, some satellites are not stars.” This syllogism has the following form: “No A are B. Some C are not B. Therefore, some C are not A.” We can determine that this syllogism is invalid by comparing it with another syllogism that has the same form and yields a false conclusion. Such a syllogism would be: “No precious stones are cheap things (true). Some diamonds are not cheap things (true). Therefore, some diamonds are not precious stones (false).” This syllogism fails to meet the requirement that the conclusion must be true if the premises are true. Therefore, the syllogism must be invalid.
The rules of syllogisms
enable us to test a categorical syllogism without considering similar examples or examining the argument’s structure in detail. These rules are based on certain features that occur in all valid syllogisms and distinguish them from invalid ones. For example, one rule states that no valid syllogism has two negative premises. There are two negative premises in this syllogism: “No stars are planets. Some satellites are not planets. Therefore, some satellites are not stars.” Thus, we know that this syllogism cannot be valid.
There are other rules for constructing valid syllogisms. (1) The syllogism must have exactly three terms. For example, consider this invalid syllogism: “All laws are made by Congress. v = at is a law of falling bodies. Therefore, Congress made v = at. ” The term law is unclear. It can refer to a physical law, such as the law of falling bodies, or to legislative law. As a result, this syllogism has four terms instead of three–and is invalid. (2) Two positive premises must yield a positive conclusion. (3) A negative premise and a positive premise must yield a negative conclusion. (4) The term that occurs in both premises must be modified by the words all or none at least once. (5) A term that is modified by all or none in the conclusion must be modified by all or none in one of the premises.
Modern logic
extends far beyond the work of Aristotle. Modern logicians (scholars who study logic) have developed theories and techniques to deal with deductive arguments other than categorical syllogisms. Notable modern logicians include the British mathematicians George Boole and Alfred North Whitehead and the British philosopher Bertrand Russell. These logicians, unlike traditional ones, have used mathematical methods, as well as techniques that involve symbols.
Today, logic is used mainly to test the validity of arguments. It also has important uses in working with such devices as computers and electric switching circuits.
To test an argument, a logician first analyzes its statements and expresses them as symbols. In many cases, a letter or other character in an argument stands for a whole word or phrase. For example, logicians would write the sentence “Socrates is wise” as “Ws.” The sentence “Every Greek is wise” would be written as a formula: “(x) (Gx→Wx).” The → means if___, then___. Next, the logician uses rules of derivation, also called inference rules, to determine what new formulas may be derived from the original premises. For example, one rule enables the statement “Q” to be derived from the statements “P” and “(P→Q).” Thus, the statement “The picnic is canceled” may be derived from “It is raining” and “If it is raining, then the picnic is canceled.” The logician continues to derive formulas until a conclusion has been reached.
Special uses of logic.
Special branches of logic guide much reasoning in science, law, and certain other fields. Various branches of logic guide reasoning involved in obligations, promises, commands, questions, preferences, and beliefs.
Much of the reasoning that people do in everyday life is nondeductive–that is, it produces probable conclusions rather than definite ones. For example, physicians use nondeductive reasoning in diagnosing the probable cause of a patient’s symptoms. Legal scholars often use nondeductive methods to determine what law governs a particular case.
