Alternation

Let P and Q be statements, each of which is either true or false (but neither is both; there are weird statements that are both---"this statement is false"). "P or Q" is a compound statement, called an alternation, 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.

Truth Table For Alternation
P Q P or Q
T T T
T F T
F T T
F F F

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 Unit", 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 alternation as allowing both parts of the compound statement 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 alternation; 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!]

What is the truth of "Q or P"? What difference does it make to "commute" the two statements 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 alternation. "P or Q" MEANS EXACTLY THE SAME AS "Q or P"; the two compound statements are true in exactly the same situations. Again, to make it very official, we display this sameness of truth values in the table below.

Alternation Is Commutative
P Q P or Q Q or P
T T T T
T F T T
F T T T
F F F F

What is the truth of "P or P"? If P is true, both alterns of this alternation statement are true; by our technical understanding of alternation, the compound "or" statement is true. If P is false, both alterns of this alternation statement are false, and thus the compound "or" statement 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 statements. 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.

Redundancy with Alternation
P P P or P
T T T
F F F
Consider, finally, an alternation statement within an alternation statement: "(P or Q) or R". Here P, Q, and R are statements (each one is either true or false, but not both). The parentheses indicate that "P or Q" is compounded with a third statement R (think of "P or Q" being evaluated first for truthfulness, and then R is taken into account). Inevitably, someone will create the statement "P or (Q or R)" [I just did]. By our understanding, of alternation, "(P or Q) or R" should mean the same as "P or (Q or R)"; in fact both are false exactly when all three of P, Q, and R are false (and true in all other situations). There are 8 situations to consider, T or F for each of P, Q, and R in all possible ways. These are listed in the table below, which lists the truth values of "(P or Q) or R".

Alternation Is Associative, part 1
P Q R P or Q (P or Q) or R
T T T T T
T T F T T
T F T T T
T F F T T
F T T T T
F T F T T
F F T F T
F F F F F
Let us review one row of this table, say the fifth. There, P is false, Q is true and R is true. Thus "P or Q" is true; from that, it follows that "(P or Q) or R" is true because R is being "or"-ed with a statement which is true. Note that "(P or Q) or R" is false only in the last situation, when all of P, Q, and R are false.

We now repeat this table for "P or (Q or R)":

Alternation Is Associative, part 2
P Q R Q or R P or (Q or R)
T T T T T
T T F T T
T F T T T
T F F F T
F T T T T
F T F T T
F F T T T
F F F F F
Again, let us review the fifth row of this table. Because Q and R are true there, "Q or R" is true; this makes "P or (Q or R)" be true because "P" is being alternated with a true statement. Again, note that "P or (Q or R)" is false only in the last situation when all three of P, Q and R are false.

Questions To Test Your Understanding.