Conditionals and the contrapositive

“If it rains, the match is cancelled” can be broken in only one way: it rains and the match goes ahead.

If p, then q

A conditional joins two statements with if ... then. In p → q, read “if p, then q”, the part after if is the antecedent and the part after then is the consequent. Think of a conditional as a promise. A parent says, “If you score 90 on the test, I will buy you a bicycle.” When has the parent broken the promise? Only when the child scores 90 and no bicycle arrives. If the child scores 90 and gets the bicycle, the promise was kept. If the child scores 80, the promise said nothing about that case, so whether a bicycle turns up or not, nobody can say the parent lied.

That gives the truth table of the conditional used in logic, often called the material conditional: p → q is false in exactly one row, where p is true and q is false, and true in the other three. The two rows with a false antecedent feel odd at first. “If the Moon is made of cheese, then 2 + 2 = 4” comes out true. Logic accepts this because the sentence is never caught out: there is no situation where its antecedent holds and its consequent fails. Everyday “if” sometimes carries more, such as a cause or a link in meaning, but the material conditional keeps only the part that can be checked.

Converse, inverse and contrapositive

From one conditional you can build three relatives. Start with “If an animal is a whale, then it is a mammal” (p → q). The converse swaps the parts: “If an animal is a mammal, then it is a whale” (q → p). The inverse negates both parts: “If an animal is not a whale, then it is not a mammal” (¬p → ¬q). The contrapositive swaps and negates: “If an animal is not a mammal, then it is not a whale” (¬q → ¬p).

Only the contrapositive is guaranteed to have the same truth value as the original. The original is true, and so is its contrapositive: anything that is not a mammal cannot be a whale. The converse and the inverse are both false, because a dog is a mammal and not a whale. Notice also that the converse and the inverse are contrapositives of each other, so they always agree with each other. When you need to prove p → q and a direct proof is awkward, proving ¬q → ¬p instead is a standard move in mathematics.

Only if, unless, necessary and sufficient

English hides conditionals in other words. forall x points out that “p only if q” is symbolised as p → q, not q → p. “You may board only if you have a ticket” says that boarding requires a ticket: board → ticket. It does not promise that a ticket gets you on the plane. In the conditional p → q, p is called a sufficient condition for q, and q a necessary condition for p. Having a ticket is necessary for boarding; it is not sufficient, because you also need a passport and a seat. “q if and only if p” claims both directions and is written p ↔ q, the biconditional.