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Denying the Antecedent

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

Denying the antecedent assumes the hypothesis is the only path to the conclusion. The conclusion may still happen another way. Learning objective: Recognize why p → q, ¬p, therefore ¬q is invalid. 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 (Denying the Antecedent): Worked example: If a student studies flashcards, then the student reviews vocabulary. The student does not study flashcards. What is the correct logical assessment?

  1. Denying the antecedent uses p -> q and not p.
  2. q might still happen for another reason.
  3. Therefore not q does not have to follow.

Answer: Invalid: the student might review vocabulary another way.

Practice problems

1. Practice case A (Denying the Antecedent): Practice: If a student studies flashcards, then the student reviews vocabulary. The student does not study flashcards. What is the correct logical assessment?

Choices: Invalid: the student might review vocabulary another way. · Valid: the student cannot review vocabulary. · Valid: this is modus ponens. · Invalid only because vocabulary is not math.

Show solution
  1. Denying the antecedent uses p -> q and not p.
  2. q might still happen for another reason.
  3. Therefore not q does not have to follow.

Answer: Invalid: the student might review vocabulary another way.

2. Practice case B (Denying the Antecedent): Practice: Which argument form does this use? "If a figure is a square, then it has four sides. This figure is not a square."

Choices: Denying the Antecedent · Modus Ponens · Modus Tollens · Hypothetical Syllogism

Show solution
  1. Denying the Antecedent has the pattern p -> q; ¬p; therefore ¬q.
  2. Match the premises and conclusion to the pattern.
  3. This form is a common invalid look-alike.

Answer: Denying the Antecedent

3. Practice case C (Denying the Antecedent): Practice: Which symbolic pattern names Denying the Antecedent?

Choices: p -> 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 (Denying the Antecedent): Practice: Which symbolic pattern names Denying the Antecedent?
  2. Compare each choice with the stated logical rule, and eliminate choices that change the claim's meaning.
  3. Denying the Antecedent has a specific shape.
  4. p -> q; ¬p; therefore ¬q
  5. This form is a common invalid look-alike.
  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 (Denying the Antecedent): Practice: Is Denying the Antecedent a valid argument form?

Choices: Invalid: the conclusion does not have to follow. · Valid: the conclusion must follow from the premises. · It depends only on whether the conclusion sounds realistic. · It is valid only when the sentences use variables.

Show solution
  1. Denying the Antecedent is a tempting but invalid form.
  2. Denying the antecedent uses p -> q and not p.
  3. q might still happen for another reason.

Answer: Invalid: the conclusion does not have to follow.

5. Practice case E (Denying the Antecedent): Practice: Why is Denying the Antecedent invalid?

Choices: q can still happen another way, so ¬p does not force ¬q. · Because the sentences are too long. · Because it uses symbols. · Because the conclusion sounds false.

Show solution
  1. Invalid means the conclusion can fail while premises hold.
  2. q can still happen another way, so ¬p does not force ¬q.
  3. So the conclusion is not guaranteed.

Answer: q can still happen another way, so ¬p does not force ¬q.

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