Yep, it's yet another rule of inference used in propositional logic. It is written as Imp or Impl for short.

(P→Q) = (¬P∨Q)

(P therefor Q) = (either 'not P' or Q)

Or in English:
If it rains (P), then I'll get wet (Q).
is the same as...
Either it won't rain (~P), or I'll get wet (Q).

Back up to Rules of Inference

Material implication -- the "→" operator of propositional calculus -- is interesting because of how it differs from our "natural" notions of how implication should work. For example, it is easy to see that a false hypothesis P implies anything. Mathematical logicians don't care about these matters, but ("real") logicians do, since they want a theory explaining how people think.

This is unfortunate, as it leads to unintuitive consequences. First, consider a false conclusion: "if the moon is made of blue cheese, then I am a porcupine". Here I'm generating a fairly logical sequence, one which is even found in everyday speech. But consider the converse "if the moon is made of blue cheese, then Richard Nixon was president of the U.S.A.". Here I am deducing a correct (and true) conclusion from a false hypothesis. Since I'm not likely to be able to establish the hypothesis, it still doesn't look too bad.

But it means that material implication doesn't capture what we think of as implication, i.e. an ordered "logical" process. Because no amount of argument will lead you from "the moon is made of blue cheese" to "X was the president of the U.S.A."! So this implication means very little -- just that you are unable to establish "the moon is made of blue cheese".

Unfortunately, if you're looking for an implication that is a logical operator, there's no alternative. A logical operator depends only on the truth values of its arguments, and it's easy to find examples of things we'd like to call "implication" satisfying each of the lines in the truth table for P→Q:

```   P   Q P→Q
T   T   T
T   F   F
F   T   T
F   F   T
```

To generate a "real" implication, we'd want to add stuff requiring some (semantic? syntactic??) connection between P and Q. Nobody has any ideas how to do this, so we're stuck with MI. And even if you could do it, you'd lose the ability of formal logic to deal with deduction without considering the "character" of statements: you'd no longer be able to replace statements with letters, and letters with statements, knowing this does not affect truth values.

A form of implication involving only a weak connection between the propositions related. A proposition P materially implies another proposition Q if and only if either P is false or Q is true (or both). It occurs in a stronger form as logical implication, which is the form of implication used in mathematics and some formal semantics, but logical implication is too strong to be useful in most ordinary contexts. The philosopher David K. Lewis and others have attempted to develop a logic of relevant implication, but material implication is still more commonly used in philosophy.

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