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Modus Ponens

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

Modus ponens is the direct use of an if-then rule. If the rule holds and the hypothesis is true, the conclusion follows. Learning objective: Use the valid pattern p → q, p, therefore q. 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 Ponens): Worked example: If a shape is a square, then it has four sides. The shape is a square. What conclusion follows?

  1. Worked Example: First identify exactly what the question is asking: Example case A (Modus Ponens): Worked example: If a shape is a square, then it has four sides. The shape is a square. What conclusion follows?
  2. For modus ponens, use p -> q together with p to conclude q.
  3. Modus ponens starts with p -> q.
  4. The second premise gives p.

Answer: It has four sides.

Practice problems

1. Practice case A (Modus Ponens): Practice: If a shape is a square, then it has four sides. The shape is a square. What conclusion follows?

Choices: It has four sides. · It is not a square. · It may have five sides. · No conclusion follows.

Show solution
  1. Warm-up: First identify exactly what the question is asking: Practice case A (Modus Ponens): Practice: If a shape is a square, then it has four sides. The shape is a square. What conclusion follows?
  2. For modus ponens, use p -> q together with p to conclude q.
  3. Modus ponens starts with p -> q.
  4. The second premise gives p.
  5. Therefore q 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: It has four sides.

2. Practice case B (Modus Ponens): Practice: Which argument form does this use? "If a number ends in 0, then it is even. The number ends in 0."

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

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

Answer: Modus Ponens

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

Choices: p -> q; p; therefore q · p -> q; ¬q; therefore ¬p · 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 Ponens): Practice: Which symbolic pattern names Modus Ponens?
  2. For modus ponens, use p -> q together with p to conclude q.
  3. Modus Ponens has a specific shape.
  4. p -> 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 -> q; p; therefore q

4. Practice case D (Modus Ponens): Practice: Is Modus Ponens 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 Ponens): Practice: Is Modus Ponens a valid argument form?
  2. For modus ponens, use p -> q together with p to conclude q.
  3. Modus Ponens preserves logical force.
  4. Modus ponens starts with p -> q.
  5. The second premise gives p.
  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 Ponens): Practice: Which sentence best describes Modus Ponens?

Choices: From p → q and p, conclude q. · From p → q and ¬q, conclude ¬p. · It picks the conclusion that sounds most familiar. · It works only when the sentences are short.

Show solution
  1. Modus Ponens 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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