Validity vs. Truth
A free Logic lesson from the “Foundations of Logical Thinking” unit, with a worked example and practice problems including step-by-step solutions.
Truth belongs to individual statements. Validity belongs to arguments: if the premises were true, would the conclusion have to be true? Learning objective: Explain the difference between a true claim and a valid argument structure. 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
- Explain the difference between a true claim and a valid argument structure
- Explain the idea in plain English before using symbols
- Use examples, non-examples, or counterexamples to check the reasoning
Worked example
Problem. Example case A (Validity vs. Truth): Worked example: Which rule or habit best matches Validity vs. Truth?
- Validity vs. Truth targets a specific reasoning habit.
- Validity vs. Truth focuses on explain the difference between a true claim and a valid argument structure.
- The other choices either overclaim or change the logical relationship.
Answer: Validity vs. Truth focuses on explain the difference between a true claim and a valid argument structure.
Practice problems
1. Practice case A (Validity vs. Truth): Name the form: If p then q. p. Therefore q.
Choices: Modus ponens · Modus tollens · Affirming the consequent · Denying the antecedent
Show solution
- Warm-up: First identify exactly what the question is asking: Practice case A (Validity vs. Truth): Name the form: If p then q. p. Therefore q.
- Compare each choice with the stated logical rule, and eliminate choices that change the claim's meaning.
- The argument uses p → q.
- It affirms p.
- Therefore q follows by modus ponens.
- 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 (Validity vs. Truth): Name the form: If p then q. Not q. Therefore not p.
Choices: Modus tollens · Modus ponens · Affirming the consequent · Hypothetical syllogism
Show solution
- Warm-up: First identify exactly what the question is asking: Practice case B (Validity vs. Truth): Name the form: If p then q. Not q. Therefore not p.
- Compare each choice with the stated logical rule, and eliminate choices that change the claim's meaning.
- The argument uses p → q.
- It denies q.
- Therefore ¬p follows by modus tollens.
- 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 (Validity vs. Truth): 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
- Warm-up: First identify exactly what the question is asking: Practice case C (Validity vs. Truth): Name the form: If p then q. If q then r. Therefore if p then r.
- Compare each choice with the stated logical rule, and eliminate choices that change the claim's meaning.
- The argument chains conditionals.
- p leads to q, and q leads to r.
- So p leads to r.
- 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 (Validity vs. Truth): Name the form: p or q. Not p. Therefore q.
Choices: Disjunctive syllogism · Modus ponens · Affirming the consequent · Inverse
Show solution
- Warm-up: First identify exactly what the question is asking: Practice case D (Validity vs. Truth): Name the form: p or q. Not p. Therefore q.
- Compare each choice with the stated logical rule, and eliminate choices that change the claim's meaning.
- The argument starts with an or statement.
- One option is ruled out.
- The remaining option follows.
- 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 (Validity vs. Truth): Lesson focus: Which transfer task would show readiness for Validity vs. Truth?
Choices: Apply explain the difference between a true claim and a valid argument structure 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
- A transfer task keeps the same objective but changes the surface context.
- Apply explain the difference between a true claim and a valid argument structure in a new sentence or context.
- That is a better mastery signal than memorizing a practice prompt.
Answer: Apply explain the difference between a true claim and a valid argument structure in a new sentence or context.
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