Implication is perhaps the most common tool of human reasoning. Curiously, it is virtually unused in computer logic. By that I mean that most computer languages do not have a built-in operator for implication but need to express it in terms of not a or b.
In non-computer languages (such as English), implication is often expressed in terms of if a then b, or a, therefore b, or, of course, a implies b.
Some important considerations about implication are:
- If a implies b and b implies c, then a implies c.
- If a implies b, then not b implies not a.
- Just because a implies b does not mean b implies a.
- Just because a implies b does not mean not a implies not b.
The first rule is pretty much the basis of most of philosophy. It typically presents a whole chain of implications: Because of a, there is b. Because of b, there is c. Because of c, there is d. Etc... Because of y, there is z. Therefore, because of a, there is z.
An example of the second rule might be something like: A city is a town with 100,000 or more inhabitants. The population of Gammaville is 10,000, which is less than 100,000. Therefore, Gammaville is not a city. (Note, this is a made-up example, I am not sure of the exact definition of a city.)
The third rule is quite important because people often make the mistake of thinking that if a implies b, then b implies a. Unfortunately, sometimes it does. But that does not mean it always does. An example to illustrate that a implying b does not necessary have b imply a could be: It snows on Christmas. In here, Christmas is a, snow is b. In other words, if it is Christmas it snows. But just because it snows does not mean it is Christmas. (Again, this is just an example, I know it does not always snow on Christmas, and certainly not in the Southern Hemisphere.)
The fourth rule is actually a combination of the second and third rules: It snows on Christmas does not mean that it does not snow when it is not Christmas.
One thing that seems to be the source of much confusion is the fact that false implies anything. In other words, if a is false, then "a implies b" is true regardless of whether b is true or false. This is a philosophical point: Just because we come to a correct conclusion based on a faulty premise does not invalidate the conclusion. That is, the truth remains the truth regardless of how it was discovered. But, at the same time, a faulty premise can, and often does, lead to a faulty conclusion. Hence, false implies anything.