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Modus Tollens

A free Logic lesson from the “Argument Validity” unit, with a worked example and practice problems including step-by-step solutions.

Modus tollens reasons backward by denying the conclusion. If q did not happen, then p could not have happened under the rule p → q. Learning objective: Use the valid pattern p → q, ¬q, therefore ¬p. 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 (Modus Tollens): Worked example: If the file was saved, then it appears in the folder. The file does not appear in the folder. What conclusion follows?

  1. Worked Example: First identify exactly what the question is asking: Example case A (Modus Tollens): Worked example: If the file was saved, then it appears in the folder. The file does not appear in the folder. What conclusion follows?
  2. For modus tollens, use p -> q together with not q to conclude not p.
  3. Modus tollens starts with p -> q.
  4. The second premise denies q.

Answer: The file was not saved.

Practice problems

1. Practice case A (Modus Tollens): Practice: If the file was saved, then it appears in the folder. The file does not appear in the folder. What conclusion follows?

Choices: The file was not saved. · The file was saved. · The folder must be full. · No conclusion follows.

Show solution
  1. Warm-up: First identify exactly what the question is asking: Practice case A (Modus Tollens): Practice: If the file was saved, then it appears in the folder. The file does not appear in the folder. What conclusion follows?
  2. For modus tollens, use p -> q together with not q to conclude not p.
  3. Modus tollens starts with p -> q.
  4. The second premise denies q.
  5. Therefore not p 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 file was not saved.

2. Practice case B (Modus Tollens): Practice: Which argument form does this use? "If it is a weekday, then the library is open. The library is not open."

Choices: Modus Tollens · Modus Ponens · Hypothetical Syllogism · Disjunctive Syllogism

Show solution
  1. Modus Tollens has the pattern p -> q; ¬q; therefore ¬p.
  2. Match the premises and conclusion to the pattern.
  3. This form is valid.

Answer: Modus Tollens

3. Practice case C (Modus Tollens): Practice: Which symbolic pattern names Modus Tollens?

Choices: p -> q; ¬q; therefore ¬p · p -> q; p; therefore q · p -> q; q -> r; therefore p -> r · p or q; ¬p; therefore q

Show solution
  1. Warm-up: First identify exactly what the question is asking: Practice case C (Modus Tollens): Practice: Which symbolic pattern names Modus Tollens?
  2. For modus tollens, use p -> q together with not q to conclude not p.
  3. Modus Tollens has a specific shape.
  4. p -> q; ¬q; therefore ¬p
  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 -> q; ¬q; therefore ¬p

4. Practice case D (Modus Tollens): Practice: Is Modus Tollens 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. Warm-up: First identify exactly what the question is asking: Practice case D (Modus Tollens): Practice: Is Modus Tollens a valid argument form?
  2. For modus tollens, use p -> q together with not q to conclude not p.
  3. Modus Tollens preserves logical force.
  4. Modus tollens starts with p -> q.
  5. The second premise denies q.
  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: Valid: the conclusion must follow from the premises.

5. Practice case E (Modus Tollens): Practice: Which sentence best describes Modus Tollens?

Choices: From p → q and ¬q, conclude ¬p. · 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. Modus Tollens has a precise shape.
  2. From p → q and ¬q, conclude ¬p.
  3. The distractors describe a different form or no real rule.

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

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