Disjunctive Syllogism
A free Logic lesson from the “Argument Validity” unit, with a worked example and practice problems including step-by-step solutions.
If p or q is true and p is false, q must be true. This is useful in cases and elimination arguments. Learning objective: Use either/or reasoning when one option is ruled out. Prerequisite: No formal prerequisite. Work in this lesson starts with ordinary language, then connects the idea to symbols only after the meaning is clear. Example 1: 'If p then q; p; therefore q' is valid modus ponens. Example 2: 'If p then q; q; therefore p' is affirming the consequent and is invalid. A common misconception is to treat familiar wording as proof; instead, check exactly what the statement says and what follows from it.
What you'll learn
- Use either/or reasoning when one option is ruled out
- Explain the idea in plain English before using symbols
- Use examples, non-examples, or counterexamples to check the reasoning
Worked example
Problem. Example case A (Disjunctive Syllogism): Worked example: Either the input is valid or the form shows an error. The input is not valid. What conclusion follows?
- Disjunctive syllogism starts with an or statement.
- One option is ruled out.
- The remaining option follows.
Answer: The form shows an error.
Lesson preview: this page shows 5 practice items. The interactive activity contains 20 practice items and 10 quiz items (multiple choice). Counts include repeated prompts and variants.
Practice problems
1. Practice case A (Disjunctive Syllogism): Practice: Either the input is valid or the form shows an error. The input is not valid. What conclusion follows?
Choices: The form shows an error. · The form does not show an error. · Both options are impossible. · The input is valid.
Show solution
- Disjunctive syllogism starts with an or statement.
- One option is ruled out.
- The remaining option follows.
Answer: The form shows an error.
2. Practice case B (Disjunctive Syllogism): Practice: Which argument form does this use? "The shape is either a triangle or a circle. It is not a triangle."
Choices: Disjunctive Syllogism · Modus Ponens · Modus Tollens · Hypothetical Syllogism
Show solution
- Disjunctive Syllogism has the pattern p or q; ¬p; therefore q.
- Match the premises and conclusion to the pattern.
- This form is valid.
Answer: Disjunctive Syllogism
3. Practice case C (Disjunctive Syllogism): Practice: Which symbolic pattern names Disjunctive Syllogism?
Choices: p or q; ¬p; therefore q · p -> q; p; therefore q · p -> q; ¬q; therefore ¬p · p -> q; q -> r; therefore p -> r
Show solution
- Disjunctive Syllogism has a specific shape.
- p or q; ¬p; therefore q
- This form is valid.
Answer: p or q; ¬p; therefore q
4. Practice case D (Disjunctive Syllogism): Practice: Is Disjunctive Syllogism a valid argument form?
Choices: Valid: the conclusion must follow from the premises. · Invalid: the conclusion does not have to follow. · It depends only on whether the conclusion sounds realistic. · It is valid only when the sentences use variables.
Show solution
- Disjunctive Syllogism preserves logical force.
- Disjunctive syllogism starts with an or statement.
- One option is ruled out.
Answer: Valid: the conclusion must follow from the premises.
5. Practice case E (Disjunctive Syllogism): Practice: Which sentence best describes Disjunctive Syllogism?
Choices: From p ∨ q and ¬p, conclude q. · From p → q and p, conclude q. · It picks the conclusion that sounds most familiar. · It works only when the sentences are short.
Show solution
- Disjunctive Syllogism has a precise shape.
- From p ∨ q and ¬p, conclude q.
- The distractors describe a different form or no real rule.
Answer: From p ∨ q and ¬p, conclude q.
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