Disjunctive Syllogism — Definition, Formula & Examples
Disjunctive syllogism is a rule of inference stating that if you know "P or Q" is true and P is false, then Q must be true. It lets you eliminate one option from an "or" statement to conclude the other.
Given premises and , the conclusion follows as valid. Symbolically: . The rule applies symmetrically: from and , one may conclude .
Key Formula
Where:
- = First proposition in the disjunction
- = Second proposition in the disjunction
- = Negation of P (P is false)
How It Works
Start with a disjunction — a statement of the form "P or Q." Then identify that one of the disjuncts is false (you have its negation as a separate premise). Since at least one part of an "or" statement must be true, the remaining disjunct must be the true one. This reasoning works because "or" in logic means at least one side holds.
Example
Problem: Given: (1) The shape is a triangle or a quadrilateral. (2) The shape is not a triangle. What can you conclude?
Identify the disjunction: Premise 1 has the form P ∨ Q, where P = "the shape is a triangle" and Q = "the shape is a quadrilateral."
Identify the negation: Premise 2 tells us the shape is not a triangle, giving us ¬P.
Apply disjunctive syllogism: Since one part of the "or" is false, the other must be true.
Answer: The shape is a quadrilateral.
Why It Matters
Disjunctive syllogism appears frequently in two-column geometry proofs and discrete mathematics courses when you need to narrow down possibilities. It also mirrors everyday reasoning — if a number is even or odd, and you prove it is not even, you know it is odd.
Common Mistakes
Mistake: Trying to conclude ¬Q from P ∨ Q and P (affirming one disjunct to deny the other).
Correction: In an inclusive "or," both P and Q can be true simultaneously. Disjunctive syllogism only works by negating one disjunct, not by affirming one.