A Story of "Or"
Let P and Q be sentences, each of which is either true or false (but neither is both; there are weird sentences that are both---"this sentence is false"). "P or Q" is a compound sentence, which is false only when both P and Q are false (and true in all other cases). Just to make it VERY official, here is a table which lists every possible combination of truth and falseness for P and Q and states whether "P or Q" is true for each possible combination. I apologize for repeating the obvious! In the tables, below, T stands for true, and F stands for false.
Consider the top row of this table. That row represents this situation: P is true and Q is true; in this situation "P or Q" is true. Ordinary English is confusing on this case. For example, if a father says "finish your term paper or you will be grounded for the week", most people would distrust a father who grounded you even though you finished your term paper! However, in the technical sciences (mathematics, engineering, physics, chemistry, etc.), there is agreement to read "or" as allowing both parts of the compound sentence to be true. Technical scientists use a different symbol, "P xor Q", to mean P is true or Q is true BUT NOT BOTH P AND Q ARE TRUE. Ordinary English uses one word for both kinds of "or"; in the sciences, we had to separate them. [Remember, computers are complete idiots and must be told everything quite literally---no leaps of the imagination!] If you have any questions about this, please email firstname.lastname@example.org with your question! It will be much appreciated.
What is the truth of "Q or P"? What difference does it make to "commute" the two sentences P and Q by putting Q ahead of P? Having both P and Q false has nothing to do with which is stated first (please forgive this review of what may seem obvious). It makes no difference, which of P and Q is stated first in an "or". "P or Q" MEANS EXACTLY THE SAME AS "Q or P"; the two compound sentences are true in exactly the same situations. Again, to make it very official, we display this sameness of truth values in the table below.
What is the truth of "P or P"? If P is true, both parts of this "or" sentence are true; by our technical understanding of "or", the compound "or" sentence is true. If P is false, both parts of this "or" sentence are false, and thus the compound "or" sentence is false. Thus, "P or P" MEANS THE SAME THING AS "P". It may be redundant, even boring, but it's true. Surprisingly, it is useful as well. Sometimes one wants to remove redundancy to make shorter but equivalent sentences. Sometimes one wants to introduce redundancy as part of a transformation into something that IS interesting. The following table is intended to make this observation quite official. Note that this table has only two rows because there are only two situations to consider: P is true or P is false.
We now repeat this table for "P or (Q or R)":
Please email email@example.com if you have questions about any of the tables above.
"Or" appears frequently in this type of expression: |x-2| > 5. This really means that (x-2 > 5) OR (x-2 < -5). To repeat, x-2 does one of two things (in this, it can't do both by axioms about real numbers): x-2 is bigger than 5 or x-2 is less than -5. It follows that x > 7 OR x < -3 (and clearly not both in the context of the real number system).
The expression z < = 4 is also an "or" sentence: it means z < 4 OR z=4. There is a theorem of real numbers that says, if a < = b and b < = a, then a=b. Note that this is an "if-then" sentence, with the "if" part being the "and" of two "or" sentences! We'll analyze this example later!
"Or" can appear in proofs in one of two very ordinary ways. Suppose that you have to prove "P or Q". A classic style of proof is to assume that "P or Q" is false, and derive from that assumption a contradiction. When assuming "P or Q" is false, we of course work with "P" being false and "Q" being false (since that is the only time "P or Q" is false). So, from both "P" and "Q" being false we get (somehow) our contradiction in this style of proof. Once we have the contradiction it is clear that "P or Q" can't be false; so "P or Q" is true and we've proved what we want.
Another use of "or" in proofs is equally common. Suppose that you wish to prove that "if P or Q, then R". One style of proof here is to assume that "P or Q" is true and somehow derive from that R. But, you can't be sure which of "P" or "Q" is the true (and both might be true). So you write TWO paragraphs. In one, you assume that P is true and somehow derive the truth of R. In the other, you assume that Q is true, and get the truth of R from that. Now, since both P and Q separately imply R, you can safely say that if P or Q, then R. A review of "if-then" sentences may help in understanding this last proof style. Here's a formal proof that the strategy of the preceding paragraph is valid: