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 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.
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.
In mathematics, the reasoning behind disjunctive syllogisms is essential, though it’s typically expressed in different notation.
The reasoning process involved in disjunctive syllogisms similarly plays a crucial role in computer programming.
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 P ∨ Q
(“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 eitheror fallacy or the false dilemma fallacy.)