1.1.7: More forms of implication
- Page ID
The implication ¬q → ¬p is called the contrapositive of p → q. An implication is logic- ally equivalent to its contrapositive. The contrapositive of “If this is Tuesday, then we are in Belgium” is “If we aren’t in Belgium, then this isn’t Tuesday”. These two sentences assert exactly the same thing.
Note that p → q is not logically equivalent to q → p. The implication q → p is called the converse of p → q. The converse of “If this is Tuesday, then we are in Belgium” is “If we are in Belgium, then this is Tuesday”. Note that it is possible for either one of these statements to be true while the other is false. In English, I might express the fact that both statements are true by saying “If this is Tuesday, then we are in Belgium, and conversely”. In logic, this would be expressed with a proposition of the form (p → q) ∧ (q → p).
Similarly p → q is not logically equivalent to ¬p → ¬q. The implication ¬p → ¬qis called the inverse of p → q. Although this mistake is commonly made in English, for instance people often assume that when I say: “If it is morning, I drink some coffee”, I also mean that when it is not morning I do not drink coffee. But my original statement does not tell you anything about what I do when it is not morning.
The biconditional operator is closely related to the conditional operator. In fact, p ↔q is logically equivalent to (p → q) ∧ (q → p). The proposition p ↔ q is usually read as “p if and only if q”. (The “p if q” part represents q → p, while “p only if q” is another
way of asserting that p → q.) It could also be expressed as “if p then q, and conversely”. Occasionally in English, ‘if... then’ is used when what is really meant is ‘if and only if’. For example, if a parent tells a child, “If you are good, Sinterklaas will bring you toys”, the parent probably really means to say “Sinterklaas will bring you toys if and only if you are good”. (The parent would probably not respond well to the child’s perfectly logical plea “But you never said what would happen if I wasn’t good!”)