Textbook
An implication $P\Rightarrow Q$ is false only when $P$ is true and $Q$ is false. It is true in every other combination. The condition $P$ is sufficient for $Q$, while $Q$ is necessary for $P$. A necessary condition is required but need not guarantee the result.
For an integer $n$, divisibility by 4 implies evenness. The reverse fails: 6 is even but not divisible by 4. Implication expresses a logical relationship, not necessarily a temporal cause. A false premise does not prove a false conclusion.
The divisibility proof concerns an arbitrary integer $n=4k$. Then $n=2(2k)$, where $2k$ is an integer, so $n$ is even. This reasoning does not assume every even number is divisible by 4. A counterexample to the original implication would require something else: a number divisible by 4 that is not even.
Definitions and further reading: Mathematics for Computer Science.
In plain language
Implication resembles a guarantee that a purchased ticket allows entry: it fails only when a ticket was bought and entry is denied. It says nothing about a visitor without a ticket, who might enter with an invitation. Likewise, divisibility by 4 guarantees evenness even though an even number need not be divisible by 4. The check concerns one guarantee rather than the cinema’s entire operating policy or the ancestry of every two.
Example
The example checks the illustrative values 4, 6, and 3. None has a true premise and false conclusion. The value 6 also refutes the reversed implication.
Implication: What “If, Then” Promises
Value 4
Divisible by 4 and even: the promise holds.
Step 1 of 3: Value 4
Read all steps
Value 4
Divisible by 4 and even: the promise holds.
- State
- P: truecurrentQ: truecurrent
Value 6
The premise is false and conclusion true; the reverse fails.
- State
- P: falsecurrentQ: truecurrent
Value 3
Both conditions are false; the original promise is not violated.
- State
- P: falsecurrentQ: falsecurrent
For 6, the original rule’s premise is false, so it is not a counterexample to that rule. For the reversed rule, however, “even” is true and “divisible by 4” is false. The same number changes roles when the arrow is reversed. The values 4, 6, and 3 display allowed combinations; the proof using $4k$ explains why the forbidden combination cannot occur for any integer.
Related concepts: Equivalence.