The logical operator meaning "if A is true, then B is true". Usually represented with an arrow, something like "A -> B". It is logically equivalent to (NOT A OR B). The contrapositive of an implication is equivalent to the implication.

The truth table for implication:

A  B | A -> B
-----+--------
  0 |   1
0  1 |   1
1  0 |   0
1  1 |   1
Note that the implication is always true when A is false; therefore, you cannot make any statement about B's value when A is not true.

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.

Standard logic uses the material conditional, which says that the statement "if p then q" is false only when p is true but q is false, and true otherwise. This has two interesting consequences. First, when p is false, the statement is true regardless of whether q is true or false. Second, when p and q are both true, the statement is true regardless of whether a relationship between p and q exists.

The results of the material conditional seem strange because English speakers expect such statements to be true because a relationship exists between p and q. In fact, I suggest that when an English speaker makes such statement, what they're asserting isn't that either p or q is true. What they're asserting is their expectation (i.e. that a relationship between them exists).

It seems to me that when one translates an English phrase like "if p then q" into a logical statement, one should use something more intuitive than the material conditional. Note that the counterfactual conditional is not quite enough, because that conditional is based on possible world semantics, so it's actually a slight generalization of the material conditional. One should instead use a new conditional that represents the existence of a constructive argument linking p and q. This means, at a minimum, that such an argument should not depend on the law of the excluded middle.

Im`pli*ca"tion (?), n. [L. implicatio: cf. F. implication.]

1.

The act of implicating, or the state of being implicated.

Three principal causes of firmness are. the grossness, the quiet contact, and the implication of component parts. Boyle.

2.

An implying, or that which is implied, but not expressed; an inference, or something which may fairly be understood, though not expressed in words.

Whatever things, therefore, it was asserted that the king might do, it was a necessary implication that there were other things which he could not do. Hallam.

 

© Webster 1913.

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