Validity and soundness
“All fish can fly. A salmon is a fish. So a salmon can fly.” The conclusion is false, yet the argument is valid, and logic has a separate word for what went wrong.
What a valid argument promises
An argument is a set of premises offered in support of a conclusion. forall x defines validity through counterexamples: a counterexample is a case in which all the premises are true and the conclusion is false, and an argument is valid if and only if there is no such case. In other words, a valid argument promises that if you accept every premise, you cannot reject the conclusion without contradicting yourself. It makes no promise about whether the premises are in fact true.
That is why the flying-salmon argument is valid. If all fish could fly and a salmon were a fish, a salmon would have to fly; there is no case where both premises hold and the conclusion fails. The trouble lies in the first premise, which is false. An argument that is valid and has only true premises is called sound. A sound argument has a true conclusion, guaranteed. The salmon argument is valid but unsound.
| Conclusion true | Conclusion false | |
|---|---|---|
| Valid | possible (sound if all premises are true) | possible, but then a premise is false |
| Invalid | possible | possible |
The table above shows the one combination validity rules out: valid, every premise true, conclusion false. Every other combination can happen. In particular, an invalid argument can have true premises and a true conclusion. “Seoul is in Korea. So water boils at 100 °C at sea level” has a true premise and a true conclusion, but the premise gives no support at all, so the argument is invalid. A true conclusion does not make an argument good.
Validity comes from form
Many valid arguments are valid because of their shape, whatever the sentences are about. Two shapes come up constantly. Modus ponens: p → q, p, therefore q. Modus tollens: p → q, ¬q, therefore ¬p. Two look-alikes are invalid, and the Stanford Encyclopedia of Philosophy lists them among the formal fallacies. Affirming the consequent: p → q, q, therefore p. Denying the antecedent: p → q, ¬p, therefore ¬q. “If the battery is dead, the car won’t start. The car won’t start. So the battery is dead” is affirming the consequent; the car might be out of fuel.
To test an argument made of letters, build one truth table with a column for each premise and one for the conclusion, then look for a counterexample row: every premise T, conclusion F. If you find one, the argument is invalid. If none exists, it is valid. For affirming the consequent, the row p false, q true is the counterexample: p → q is true, q is true, and the conclusion p is false.
| valid | No case with all premises true and the conclusion false |
| sound | Valid, and every premise is true |
| counterexample | A case where the premises are true and the conclusion false |
| modus ponens | p → q, p, so q (valid) |
| modus tollens | p → q, ¬q, so ¬p (valid) |
| affirming the consequent | p → q, q, so p (invalid) |
| denying the antecedent | p → q, ¬p, so ¬q (invalid) |
- An argument is valid when
- its conclusion is true
- there is no case where all premises are true and the conclusion false
- all its premises are true
- “All birds are green. A crow is a bird. So a crow is green.” This argument is
- valid and sound
- valid but unsound
- invalid
- A sound argument can have a false conclusion.
- True
- False
- An invalid argument can have true premises and a true conclusion.
- True
- False
- “If it snowed, the roads are white. The roads are not white. So it did not snow.” Form?
- modus ponens
- modus tollens
- affirming the consequent
- “If you study, you pass. You passed. So you studied.” Form?
- modus ponens
- affirming the consequent
- denying the antecedent
- “If it is a holiday, the bank is closed. It is not a holiday. So the bank is open.” Form?
- denying the antecedent
- modus tollens
- modus ponens
- Which row is a counterexample to “p → q, q, so p”?
- p T, q T
- p T, q F
- p F, q T
- p F, q F
- You find no counterexample row in a complete truth table. The argument is
- valid
- sound
- invalid
- A valid argument has a false conclusion. What follows?
- At least one premise is false.
- All premises are false.
- The argument is invalid after all.
- Valid: no case with true premises and a false conclusion.
- Sound: valid plus all premises true.
- Modus ponens and modus tollens are valid; affirming the consequent and denying the antecedent are not.
- Test with a truth table: look for a row with every premise T and the conclusion F.
Sources for this course (9)
- P. D. Magnus, Tim Button, Aaron Thomas-Bolduc, Richard Zach, forall x: Calgary , chapters 1 to 6 on sentences, arguments, validity, soundness, connectives and truth tables (forallx.openlogicproject.org/html/)
- Stanford Encyclopedia of Philosophy, Robin Smith, "Aristotle's Logic" (plato.stanford.edu/entries/aristotle-logic/)
- definition of deduction from Prior Analytics I.2 24b18-20, the three figures and the medieval names Barbara and Celarent
- Aristotle, On Interpretation 17a, E. M. Edghill translation (classics.mit.edu/Aristotle/interpretation.1.1.html)
- "a prayer is a sentence, but is neither true nor false"
- Irving H. Anellis, "Peirce's Truth-functional Analysis and the Origin of Truth Tables" (arxiv.org/abs/1108.2429): Peirce manuscript of 1893, Shosky's 1997 find on a 1912 Russell typescript
- Emil Post, "Introduction to a general theory of elementary propositions", American Journal of Mathematics 43 (1921)
- Wittgenstein, Tractatus Logico-Philosophicus (1921, English 1922)
- 4.31. Everyday example sentences are made up for teaching.
Checked: 2026-10-17