Disjunctive Syllogism

Either/Or reasoning: Either A or B, not A, therefore B.

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Either/Or reasoning: Either A or B, not A, therefore B.

Disjunctive Syllogism is a rule of inference that eliminates alternatives. Given that 'Either A or B' is true (at least one must hold), and knowing that A is false, we can conclude that B must be true. This is also called 'process of elimination' and is fundamental to decision-making and proof by cases.

P ∨ Q, ¬P ⊢ Q

Either the package was delivered or it was lost. It was not delivered. Therefore, it was lost.
Either the answer is 42 or 56. The answer is not 42. Therefore, the answer is 56.
Either John took the bus or he walked. John did not take the bus. Therefore, John walked.

How to recognize it

Look for an 'Either...or...' statement (disjunction)
One of the options is ruled out (negated)
The conclusion affirms the remaining option

Common mistakes

Assuming 'Either A or B' means exactly one is true (exclusive or). In logic, 'or' typically allows both to be true
Affirming one disjunct does not eliminate the other: 'Either A or B, A is true' does NOT mean B is false

Step-by-step walkthrough

Either the answer is 42 or 56. The answer is not 42. What can we conclude?

Identify the disjunction: 'Either 42 or 56'
One option is ruled out: 'Not 42'
Apply Disjunctive Syllogism: If P is false, Q must be true
Conclude: The answer is 56

Answer: The answer is 56

Practice guidance

Logical Deduction