Valid and Sound Arguments: 10 Worked Examples and a Worksheet

What valid and sound mean

Validity concerns the connection between premises and conclusion. Ask whether every premise could be true while the conclusion is false. If such a situation is possible, the argument is invalid. If none is possible under the stated logical interpretation, the argument is valid. One successful instance does not establish a guarantee.

Soundness requires validity and true premises. An invalid argument is unsound. A valid argument with a false premise is also unsound. A sound argument must have a true conclusion, but a true conclusion alone cannot show that the argument is sound. Cases 3, 4 and 8 make that distinction visible.

The words apply here to deductive arguments. We are not evaluating the strength of an inductive forecast, the soundness of an entire proof system or the truth of a single formula. Those are different tasks. For the guarantee-versus-estimate distinction, use the deductive and inductive reasoning worksheet.

Use this two-step judgment diagram

“I could not think of a counterexample” is weaker than “none is possible.” For a small propositional form, inspect every truth-table row. For other forms, use an appropriate proof or countermodel. Familiar vocabulary and confidence in the speaker are not substitutes for this check.

At the second step, seek evidence for each premise. A record may establish what happened in a fictional exercise; a real argument may require a current log, a source or clarification of a disputed principle. If the evidence is incomplete, report that limit. Soundness remains an objective property; “not established” describes what you can responsibly claim about it.

Compare validity with premise and conclusion truth

“May be true or false” describes what that combination of labels permits across arguments; it does not say every particular argument has both possibilities. The last row concerns incomplete knowledge, not a third logical truth value. If you independently verify a conclusion, that still does not verify every premise or validate the offered inference.

How to read the ten examples

For each original fictional case, write (1) whether the form is valid and why; (2) which premises are true, false or unverified in the supplied record; and (3) whether soundness is established. For an invalid form, supply a situation with every premise true and the conclusion false. A false premise is not a counterexample to validity. Use classical propositional logic, the material conditional and inclusive or. Unknown evidence does not make a claim false. Records are only for the truth check; do not add them as premises.

For these exercises, “if p then q” uses the material conditional: it is false only when p is true and q false. It does not assert causation, timing or that p actually occurred. “p or q” is inclusive: either or both may be true. Natural-language uses of “if” and “or” can require a different analysis; state the interpretation before judging their form.

Cases 1–3 deliberately hold one valid form fixed while changing the record. Cases 4–6 contrast three easily confused conditional patterns. Cases 7–8 show what inclusive “or” allows. Cases 9–10 address missing truth evidence and a conditional conclusion.

Download the ten questions without answers · Download the explained answer key. Try a question before opening its answer. Write the counterexample or inference reason, not just “valid” or “invalid.”

Ten worked examples and self-test answers

Check the conditional forms with four truth-table rows

Let p mean “N is sealed” and q mean “N is scanned.” T and F mean true and false. The “both premises” columns ask whether every premise of that form is true in the row.

For modus ponens, only the first row has both premises true; its conclusion q is true. For affirming the consequent, the first and third rows have both premises true. The third row has p false, so it supplies the needed counterexample. Checking only the first row would miss that failure.

The second row makes the conditional itself false. It therefore cannot refute either form’s validity: their premises are not all true there. This is why record B in case 2 does not disprove modus ponens. Change the form, or change the truth of an instance, and record which change you made.

Apply the method to an argument you encounter

1. Copy the exact conclusion. Preserve its scope and negation. “If checking occurs, a receipt follows” differs from “checking occurred.”
2. List the offered premises. Separate them from information you happen to know. Mark an unstated connecting premise as a proposed addition.
3. Make the interpretation explicit. Decide what “or,” “if,” “all” and key terms mean in context. Changing a word’s meaning midway changes the argument.
4. Test the guarantee. A counterexample must preserve every premise and falsify that exact conclusion. Show why each premise survives.
5. Check truth independently. Record evidence, a false premise or an unresolved question. Do not turn unavailable evidence into a negative finding.

Finding a bad argument for a conclusion does not prove the conclusion false. You may need a different argument, better evidence or a revised claim. Similarly, finding a false premise in a valid argument shows why this argument is unsound; it does not refute the inference pattern shared by other instances.

For a study-group discussion, ask one person to propose a counterexample and another to audit every premise in it. Use cases 4 and 8 to catch hidden premises, and cases 2 and 9 to distinguish a demonstrated falsehood from an evidence gap. The examples and records are original teaching constructions. No real dispatch records, professional review or measured learning outcomes are claimed.