CMClearMathAcademy

What Makes an Argument Valid

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

Argument validity is about guarantee. If there is any possible case where all premises are true and the conclusion is false, the argument is invalid. Learning objective: Decide whether a conclusion must follow from the premises. 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 (What Makes an Argument Valid): Worked example: Which rule or habit best matches What Makes an Argument Valid?

  1. What Makes an Argument Valid targets a specific reasoning habit.
  2. What Makes an Argument Valid focuses on decide whether a conclusion must follow from the premises.
  3. The other choices either overclaim or change the logical relationship.

Answer: What Makes an Argument Valid focuses on decide whether a conclusion must follow from the premises.

Practice problems

1. Practice case A (What Makes an Argument Valid): Name the form: If p then q. p. Therefore q.

Choices: Modus ponens · Modus tollens · Affirming the consequent · Denying the antecedent

Show solution
  1. Warm-up: First identify exactly what the question is asking: Practice case A (What Makes an Argument Valid): Name the form: If p then q. p. Therefore q.
  2. Compare each choice with the stated logical rule, and eliminate choices that change the claim's meaning.
  3. The argument uses p → q.
  4. It affirms p.
  5. Therefore q follows by modus ponens.
  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: Modus ponens

2. Practice case B (What Makes an Argument Valid): Name the form: If p then q. Not q. Therefore not p.

Choices: Modus tollens · Modus ponens · Affirming the consequent · Hypothetical syllogism

Show solution
  1. Warm-up: First identify exactly what the question is asking: Practice case B (What Makes an Argument Valid): Name the form: If p then q. Not q. Therefore not p.
  2. Compare each choice with the stated logical rule, and eliminate choices that change the claim's meaning.
  3. The argument uses p → q.
  4. It denies q.
  5. Therefore ¬p follows by modus tollens.
  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: Modus tollens

3. Practice case C (What Makes an Argument Valid): Name the form: If p then q. If q then r. Therefore if p then r.

Choices: Hypothetical syllogism · Disjunctive syllogism · Denying the antecedent · Biconditional definition

Show solution
  1. Warm-up: First identify exactly what the question is asking: Practice case C (What Makes an Argument Valid): Name the form: If p then q. If q then r. Therefore if p then r.
  2. Compare each choice with the stated logical rule, and eliminate choices that change the claim's meaning.
  3. The argument chains conditionals.
  4. p leads to q, and q leads to r.
  5. So p leads to r.
  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: Hypothetical syllogism

4. Practice case D (What Makes an Argument Valid): Name the form: p or q. Not p. Therefore q.

Choices: Disjunctive syllogism · Modus ponens · Affirming the consequent · Inverse

Show solution
  1. Warm-up: First identify exactly what the question is asking: Practice case D (What Makes an Argument Valid): Name the form: p or q. Not p. Therefore q.
  2. Compare each choice with the stated logical rule, and eliminate choices that change the claim's meaning.
  3. The argument 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: Disjunctive syllogism

5. Practice case E (What Makes an Argument Valid): Lesson focus: Which transfer task would show readiness for What Makes an Argument Valid?

Choices: Apply decide whether a conclusion must follow from the premises in a new sentence or context. · Repeat the exact worked example without changing the context. · Pick an answer before identifying the claim parts. · Use a rule from another lesson because the words sound close.

Show solution
  1. A transfer task keeps the same objective but changes the surface context.
  2. Apply decide whether a conclusion must follow from the premises in a new sentence or context.
  3. That is a better mastery signal than memorizing a practice prompt.

Answer: Apply decide whether a conclusion must follow from the premises in a new sentence or context.

Practice this interactively with instant feedback and an AI tutor.

Practice What Makes an Argument Valid Take the free placement check

More Logic lessons