Dissecting Disjunction

Seeing as last week's topic went over so well, I figured this week I might as well do something similar and stick to making an effort to reteach myself a concept in order to help myself in the long run, and maybe even help you guys, too, if you happen to stumble upon this post in your blogging travels.

Seeing as the midyear is approaching rather rapidly, I have decided that once again I'll be picking apart something that should be so painfully easy that I simply can't get it to settle in my mind correctly unless I write it out; Disjunction.

Disjunction states that from any statement, infer its disjunction with another statement. In symbols, this is better shown as follows:

From P (stated alone), infer P v Q

From Q (stated alone), infer Q v P

Arthur is quick to point out that this in fact is a fishy rule itself; the reason, he states, is because once the first disjunct is given, the disjunction essentially follows whatever the second disjunct is, and that disjunct can seem to be pulled right out of the atmosphere, having no real relationship to the problem at first. It cannot be reversed so that from P v Q you can infer P alone or Q alone, however. While it may seem like a useless idea, something that seems entirely irrelevant, this rule proves to be one of great use when it comes to problems like the following:

( B v F) -> ~L, F : ~L

1. ( B v F ) -> ~L         Prem
2. F                              Prem

With no place else to go, there seems to be a dead end here. This proof could not possibly be solved, there isn't enough information, right? Wrong. When disjunction is used, you can very easily pull the disjunct B from the air and add the third step to solve for ~L, as follows.

( B v F) -> ~L, F : ~L

1. ( B v F ) -> ~L         Prem
2. F                              Prem
3. B v F
4. ~L                            QED

While it may seem like cheating, this is how disjunction works. It is used in order to get to the conclusion that otherwise may be unobtainable. And while this rule may seem complex and like a sort of cheat, it is in fact a logical tool to help us through our proofs.