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Testing Arguments with Truth Tables

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

Truth tables test arguments by looking for a counterexample row: premises true and conclusion false. No such row means the form is valid. Learning objective: Use a truth table to test an argument form. 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 (Testing Arguments with Truth Tables): Worked example: Which rule or habit best matches Testing Arguments with Truth Tables?

  1. Testing Arguments with Truth Tables targets a specific reasoning habit.
  2. Testing Arguments with Truth Tables focuses on use a truth table to test an argument form.
  3. The other choices either overclaim or change the logical relationship.

Answer: Testing Arguments with Truth Tables focuses on use a truth table to test an argument form.

Practice problems

1. Practice case A (Testing Arguments with Truth Tables): 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 (Testing Arguments with Truth Tables): Name the form: If p then q. p. Therefore q.
  2. For symbolic logic, translate the symbols into plain English, then apply the truth condition for the connective.
  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 (Testing Arguments with Truth Tables): 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 (Testing Arguments with Truth Tables): Name the form: If p then q. Not q. Therefore not p.
  2. For symbolic logic, translate the symbols into plain English, then apply the truth condition for the connective.
  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 (Testing Arguments with Truth Tables): 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 (Testing Arguments with Truth Tables): Name the form: If p then q. If q then r. Therefore if p then r.
  2. For symbolic logic, translate the symbols into plain English, then apply the truth condition for the connective.
  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 (Testing Arguments with Truth Tables): 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 (Testing Arguments with Truth Tables): Name the form: p or q. Not p. Therefore q.
  2. For symbolic logic, translate the symbols into plain English, then apply the truth condition for the connective.
  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 (Testing Arguments with Truth Tables): Lesson focus: Which transfer task would show readiness for Testing Arguments with Truth Tables?

Choices: Apply use a truth table to test an argument form 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 use a truth table to test an argument form in a new sentence or context.
  3. That is a better mastery signal than memorizing a practice prompt.

Answer: Apply use a truth table to test an argument form in a new sentence or context.

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