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.
Validity means there is no assignment with all premises true and the conclusion false. Build such a countermodel to show invalidity. Modus ponens and modus tollens preserve truth; affirming the consequent does not. A valid form can still have false premises, so validity alone is not soundness.
What you'll learn
- Decide whether a conclusion must follow from the premises
- Explain the idea in plain English before using symbols
- Use examples, non-examples, or counterexamples to check the reasoning
Worked example
Problem. Premises: p→q and p. Is the conclusion q forced?
- The first premise guarantees q whenever p holds.
- The second premise supplies p.
Answer: Yes, by modus ponens
Lesson preview: this page shows 5 practice items. The interactive activity contains 6 practice items and 6 quiz items (multiple choice). Counts include repeated prompts and variants.
Practice problems
1. Premises: p→q and p. Is the conclusion q forced?
Choices: Yes, because q is always true · Yes, by modus ponens · No; it affirms the consequent · No; it denies the antecedent
Show solution
- The first premise guarantees q whenever p holds.
- The second premise supplies p.
Answer: Yes, by modus ponens
2. Premises: p→q and q. Which assignment refutes the conclusion p?
Choices: p=false, q=true · p=true, q=false · p=false, q=false · There is no counterexample
Show solution
- Both premises are true in the selected row, yet p is false.
Answer: p=false, q=true
3. Premises: p→q and ¬q. What follows?
Choices: p · q · p ∧ q · ¬p
Show solution
- A true p would require q.
- Since q is false, p cannot be true.
Answer: ¬p
4. Premises: p∨q and ¬p. What follows?
Choices: p · p ∧ q · q · ¬q
Show solution
- The disjunction needs at least one true part.
- p is excluded, so q must hold.
Answer: q
5. All laboratory keys are metal. This object is metal. Therefore it is a laboratory key. Evaluate the argument.
Choices: Invalid because no metal keys exist · Invalid: some metal objects may not be keys · Valid: the conclusion repeats a premise · Valid: all premises might be true
Show solution
- Truth of premises does not force the conclusion.
- A metal spoon is a counterexample to the inference pattern.
Answer: Invalid: some metal objects may not be keys
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