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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

Why it matters: Argument validity helps learners spot reasoning errors in word problems, explanations, and public claims.

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?

  1. Worked Example: First identify exactly what the question is asking: 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?
  2. For a syllogism, follow how one premise rules in, rules out, or chains to the next claim.
  3. Disjunctive syllogism starts with an or statement.
  4. One option is ruled out.

Answer: The form shows an error.

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
  1. Warm-up: First identify exactly what the question is asking: 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?
  2. For a syllogism, follow how one premise rules in, rules out, or chains to the next claim.
  3. Disjunctive syllogism starts with an or statement.
  4. One option is ruled out.
  5. The remaining option follows.
  6. Verify the selected choice by checking that it preserves the stated logical meaning and that the other choices change the rule or claim.

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
  1. Disjunctive Syllogism has the pattern p or q; ¬p; therefore q.
  2. Match the premises and conclusion to the pattern.
  3. 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
  1. Warm-up: First identify exactly what the question is asking: Practice case C (Disjunctive Syllogism): Practice: Which symbolic pattern names Disjunctive Syllogism?
  2. For a syllogism, follow how one premise rules in, rules out, or chains to the next claim.
  3. Disjunctive Syllogism has a specific shape.
  4. p or q; ¬p; therefore q
  5. This form is valid.
  6. Verify the selected choice by checking that it preserves the stated logical meaning and that the other choices change the rule or claim.

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
  1. Disjunctive Syllogism preserves logical force.
  2. Disjunctive syllogism starts with an or statement.
  3. 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
  1. Disjunctive Syllogism has a precise shape.
  2. From p ∨ q and ¬p, conclude q.
  3. The distractors describe a different form or no real rule.

Answer: From p ∨ q and ¬p, conclude q.

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