Many arguments turn on an if-then statement, called a conditional: "If P, then Q." P is the antecedent, Q is the consequent.

Two valid forms

Modus ponens (affirming the antecedent):

• If P, then Q.
• P.
• Therefore, Q.

If it is raining, the ground is wet. It is raining. So the ground is wet.

Modus tollens (denying the consequent):

• If P, then Q.
• Not Q.
• Therefore, not P.

If it is raining, the ground is wet. The ground is not wet. So it is not raining.

Two invalid imposters

These look similar but do not work.

Affirming the consequent:

• If P, then Q.
• Q.
• Therefore, P. (invalid)

If it is raining, the ground is wet. The ground is wet. So it is raining. Not necessarily: a sprinkler could have wet it.

Denying the antecedent:

• If P, then Q.
• Not P.
• Therefore, not Q. (invalid)

If it is raining, the ground is wet. It is not raining. So the ground is not wet. Again, the sprinkler.

A quick test

A conditional only promises that P is enough for Q. It doesn't say P is the only way to get Q. Whenever there could be another route to Q, reasoning backward from Q to P, or from not-P to not-Q, fails.