Valid Arguments

Modus ponens, modus tollens, and the fallacies that mimic them.

The idea

An argument puts forward some statements, its premises, in support of another, its conclusion. Logic does not ask whether the premises are in fact true; it asks whether the conclusion would have to be true if they were.

An argument is valid when every assignment of truth values that makes all of its premises true also makes its conclusion true. Validity is a property of the argument's form alone. A valid argument can have false premises and a false conclusion, and an argument with true premises and a true conclusion can still be invalid, when the conclusion happens to be true without the premises forcing it.

Because only the form matters, a truth table settles validity. Give each premise and the conclusion a column, and inspect only the rows in which every premise reads $T$. If the conclusion reads $T$ in all of them, the form is valid; a single row with all premises true and the conclusion false shows it is not.

A few valid forms recur often enough to carry names, and each has an invalid look-alike, so we check an argument by table rather than judge it by resemblance.

Ways to work on it

• Walkthrough. Modus ponens, modus tollens, and affirming the consequent.
• Practice. Name the valid conclusion or spot the fallacy.
• Hardest. Disprove validity with a counterexample row.

Not sure where to start? Take the ten-question placement test.