Disjunctive Syllogism | Definition & Examples

A disjunctive syllogism is an argument with two premises and a conclusion that describes an either–or relationship. The conclusion is derived through a process of elimination when one of the two options is negated.

Disjunctive syllogism example
  • A shape is either a hexagon or an octagon.
  • The shape is not a hexagon.
  • Therefore, the shape is an octagon.

This argument fits the structure of a disjunctive syllogism because it presents a choice between two options (hexagon or octagon), negates one (hexagon), and concludes by affirming the other (octagon).

Disjunctive syllogisms are typically used in formal logic, but mathematics, computer programming, and other disciplines often use the same pattern of reasoning expressed in different ways.

What is a disjunctive syllogism?

A disjunctive syllogism is an argument in formal logic that draws its conclusion from a disjunction, or an “either–or” relationship. These syllogisms are distinguished by their focus on two options, at least one of which must be true.

The first premise of a disjunctive syllogism introduces the two alternatives (e.g., “The cat is either on the bed or under the bed”). The second premise negates one of the options either explicitly or implicitly (e.g., “The cat is not under the bed”). The conclusion that the other option is true can then be deduced with logical certainty (e.g., “Therefore, the cat is on the bed”).

In formal logic, the following formula is used to express disjunctive syllogisms:

  • Either P or Q.
  • Not P.
  • Therefore, Q.

The Latin name for disjunctive syllogisms is modus tollendo ponens, meaning “method of affirming by negating,” because the only valid way to form a conclusion using disjunction is by eliminating or negating one of the options. Disjunctive syllogisms cannot be affirmative.

Disjunctive syllogisms always express deductive reasoning, and they always consist of two premises and a conclusion. These are defining traits of syllogisms in general, including the other two main categories: hypothetical syllogisms and categorical syllogisms.

Note
In symbolic logic, disjunctive syllogisms are expressed as follows:

  • P∨Q
  • ¬𝑃
  • ∴𝑄

The symbol “∨” (called a “logical disjunction”) is used to express the “inclusive or” concept explained above (meaning that either or both might be true—but not neither). The symbol “¬” means “not,” and the symbol “∴” means “therefore.”

Disjunctive syllogism examples

Examples of disjunctive syllogisms are typically found in discussions of formal logic and other philosophical domains. Philosophers such as Immanuel Kant have explored the role of disjunctive syllogisms within the broader structures of logical reasoning to understand how human knowledge is organized and decisions are made.

Disjunctive syllogism example in logic
  • Human actions either involve free will or are solely determined by external factors.
  • Human actions are not solely determined by external factors.
  • Therefore, human actions involve free will.

Disjunctive syllogisms are used in disciplines like logic and moral philosophy, where they are used to analyze scenarios with two options. Syllogisms help simplify complex arguments and provide a clear framework for deductions.

In mathematics, the reasoning behind disjunctive syllogisms is essential, though it’s typically expressed in different notation.

Disjunctive syllogism example in mathematics
  • A number is either even (divisible by 2) or odd (not divisible by 2).
  • The real number y is not divisible by two.
  • Therefore, the real number y is odd.

One of the ways disjunctive syllogistic reasoning can be used in mathematics is to analyze and infer the properties of numbers.

The reasoning process involved in disjunctive syllogisms similarly plays a crucial role in computer programming.

Disjunctive syllogism example in computer programming
  • A variable can hold either a numerical value or a textual value.
  • Variable x cannot be numerical because it cannot be multiplied.
  • Therefore, variable x is a textual value.

In a programming language like Python, disjunctive syllogistic reasoning is used to make decisions based on different conditions or types of data stored in variables. For example, this reasoning process can be used to determine the appropriate actions to take based on whether a variable contains numerical or textual data.

Invalid disjunctive syllogisms

Disjunctive syllogisms are always valid when formed correctly. However, an attempt at forming a disjunctive syllogism can result in an invalid structure, forming a formal logical fallacy (or non sequitur fallacy).

Invalid attempts at forming a disjunctive syllogism are most likely to result in one of the following fallacies:

  • Affirming a disjunct involves wrongly concluding that because one option in a disjunction (either–or statement) is true, the other option must be false. This attempt at a disjunctive syllogism is invalid because affirming one option does not preclude the truth of the other (i.e., both could be true).
    • Example: “The flag is either blue or red. The flag is red. Therefore, the flag is not blue.”
  • Denying a disjunct involves wrongly concluding that because one option in a disjunction (either–or statement) is false, the other option must also be false. This attempt at a disjunctive syllogism is invalid because negating one option does not imply that the other is also false.
    • Example: “The flag is either blue or red. The flag is not red. Therefore, the flag is not blue.”

Frequently asked questions about disjunctive syllogisms

How can you prove the validity of a disjunctive syllogism using a truth table?

In symbolic logic, the validity of a disjunctive syllogism can be proved using a truth table. This table expresses all truth values (i.e., true or false, expressed as T or F) of the premises and conclusion under all possible conditions.

P Q PQ
(“Either P or Q.”)
¬P
(“Not P.”)
Conclusion
(“Therefore, Q”)
T

T

F

F

T

F

T

F

T

T

T

F

F

F

T

T

T

F

T

F

This truth table demonstrates that disjunctive syllogisms are valid by showing that when both premises are true (which occurs in row three) the conclusion is also true.

What is an example of a disjunctive syllogism in the media?

An example of a disjunctive syllogism in media would be the narrator of a science documentary explaining, “Either the observed celestial object is a comet, or it is an asteroid. It has a tail, which comets have but asteroids do not; therefore, it is a comet.”

Note: Examples of “either–or” arguments seen in the media typically aren’t syllogisms. Arguments found in media discourse are typically examples of inductive reasoning. (When inductive arguments present exaggerated binary options and ignore nuance, they exemplify the either-or fallacy or the false dilemma fallacy.)

 

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Magedah Shabo

Magedah is the author of Rhetoric, Logic, & Argumentation and Techniques of Propaganda and Persuasion. She began her career in the educational publishing industry and has over 15 years of experience as a writer and editor. Her books have been used in high school and university classrooms across the US, including courses at Harvard and Johns Hopkins. She has taught ESL from elementary through college levels.