Syllogism
A syllogism is a form of deductive reasoning used to infer conclusions from a set of premises. Recognized for its logical structure, a syllogism typically includes three components: a major premise, a minor premise, and a conclusion. This reasoning method has historical roots in ancient Greek philosophy and remains a fundamental concept in logic today. Syllogisms can be categorized into three main types: conditional, disjunctive, and categorical.
Conditional syllogisms rely on "if-then" premises, while disjunctive syllogisms present mutually exclusive options to deduce a conclusion. Categorical syllogisms, the most common, categorize subjects into groups to draw conclusions based on properties shared among them. Additionally, the enthymeme is a rhetorical variant of syllogism that often implies one of its premises rather than stating it explicitly. Overall, syllogisms are critical tools in logical reasoning, enabling the construction of arguments from smaller statements to draw valid conclusions, provided the premises are accurate.
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Syllogism
In logic, a syllogism is a type of deductive reasoning that uses a set of statements known as premises to draw a conclusion. Syllogisms can also serve as persuasive devices and can build larger arguments from smaller constituent parts. They have been used in logic since the age of antiquity, most prominently by the great philosophers of ancient Greece. The word entered English through the Latin term syllogismus, which has etymological origins in the Greek syllogizesthai, a word that is usually translated as "to infer," "to count," or "to reckon."
There are three main subcategories of syllogisms: conditional syllogisms, disjunctive syllogisms, and categorical syllogisms. All three forms use similar logic structures but employ different reasoning methods to reach conclusions. A rhetorical device known as the enthymeme is also sometimes classified as a syllogism, although enthymemes are not actually complete syllogisms in the technical sense of the term.
Background
Syllogisms are among the most commonly used forms of deductive reasoning. Deductive reasoning is also known as top-down logic, as it uses general premises to reach specific conclusions. Its counterpart, inductive reasoning, is also known as bottom-up logic, as it uses specific premises to reach general conclusions.
In their most basic form, syllogisms consist of three elements: a major premise, a minor premise, and a conclusion. The major premise is a statement that contains the predicate of the conclusion, while the minor premise is a statement that contains the subject of the conclusion. The conclusion itself follows as a logical consequence of the major premise and the minor premise and is proven to be true so long as both premises are also true. If one or both premises are false, the conclusion will also be false even though valid, sound logic was used to reach it.
A simple example of a basic syllogism with valid logic leading to a true conclusion can be given as follows:
Major premise: All people are mortal.
Minor premise: Socrates is a person.
Conclusion: Therefore, Socrates is mortal.
Similarly, a simple example of a basic syllogism with valid logic leading to a false conclusion can be given as follows:
Major premise: Every gray-haired woman is a grandmother.
Minor premise: Joan is a gray-haired woman.
Conclusion: Therefore, Joan is a grandmother.
The first example generates a true conclusion because both of its premises can be verified through observation and historical facts. The second example generates a false conclusion because its major premise is demonstrably not true, as observation and analysis of historical facts would surely uncover some gray-haired women who have no children at all and, therefore, cannot be grandmothers. However, despite the falsity of the major premise and the resultant invalidity of the conclusion, the logic used to draw the conclusion that Joan is a grandmother is otherwise perfectly valid.
Overview
The three main types of syllogisms are differentiated by the structures of their respective premises and the methods they use to draw conclusions from those premises. Conditional syllogisms follow the general form: "If A is true, then B must also be true." Because their major premises, by definition, exist only in theoretical terms, conditional syllogisms are sometimes called hypothetical syllogisms. Conditional syllogisms also tend to generate more false conclusions than any other type of syllogism, largely because their major premises exist only in theoretical terms.
Two related rules of inference, known as modus ponens and modus tollens, can be used to form conditional syllogisms. Modus ponens is a form of propositional logic that follows this form: "A implies B, and A is known to be true. Therefore, B is also true." Modus tollens, also known as "denying the consequent," follows this form: "A implies B, and A is known to be not true. Therefore, B is also not true." The first example demonstrates modus ponens, while the second demonstrates modus tollens:
Example 1
Major premise: If tomorrow is sunny, the race will be held.
Minor premise: Tomorrow will be sunny.
Conclusion: Therefore, the race will be held.
Example 2
Major premise: If someone is at the door, the dog will bark.
Minor premise: The dog did not bark.
Conclusion: Therefore, there is no one at the door.
Disjunctive syllogisms use major and minor premises that cannot both be true at the same time as a means of determining which of them is true and what the conclusion should therefore be. They follow this form: "Either A or B is true. A is not true. Therefore, B is true." An example of a disjunctive syllogism can be given as follows:
Major premise: The cat is either gray or black.
Minor premise: The cat is not gray.
Conclusion: Therefore, the cat is black.
The third and most common type of syllogism is the categorical syllogism. Categorical syllogisms frequently draw on a branch of logic known as set theory, which is used to make observations and draw conclusions about things that belong to overlapping groups. Categorical syllogisms follow this form: "All A are B. C is A. Therefore, C is also B." The earlier example about Socrates's mortality is also an example of a categorical syllogism:
Major premise: All people are mortal.
Minor premise: Socrates is a person.
Conclusion: Therefore, Socrates is mortal.
The enthymeme is sometimes considered a type of syllogism, although it technically belongs to the domain of rhetoric, not logic. Enthymemes are, in essence, implied or incomplete syllogisms that are usually used as persuasive tools. They are sometimes referred to as rhetorical syllogisms. For example, "She is an American citizen. She is entitled to due process."
Although it is not formally presented as such, the statements form a syllogism with an implied major premise:
Major premise (implied): All American citizens are entitled to due process.
Minor premise: She is an American citizen.
Conclusion: Therefore, she is entitled to due process.
Bibliography
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