Up A Level
"And"
"And" of An "Or"
Contrapositive
"For All"
"If and Only If"
"If..., Then..."
"Not"
"Not" of An "And"
"Not" of An "If...Then"
"Not" of An "Or"
"Or"
"Or" of An "And"
Short Tautologies
"There exists"

# 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.

Truth Table For "Or"
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 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 ramsey@math.hawaii.edu 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.

"OR" 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 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.

Redundancy with "OR"
P P P or P
T T T
F F F
Consider, finally, an "or" sentence within an "or" sentence: "(P or Q) or R". Here P, Q, and R are sentences (each one is either true or false, but not both). The parentheses indicate that "P or Q" is compounded with a third sentence R (think of "P or Q" being evaluated first for truthfulness, and then R is taken into account). Inevitably, someone will create the sentence "P or (Q or R)" [I just did]. By our understanding, of "or", "(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".

"Or" 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 sentence 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)":

"OR" 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 "or"-ed with a true sentence. 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.

Please email ramsey@math.hawaii.edu if you have questions about any of the tables above.

Some Questions To Test Understanding.
• How many different compound sentences can be made from "(P or Q) or R" by permuting the letters P, Q, and R? How many more can be made by permuting the letters P, Q, and R and moving the parentheses to enclose the last two sentences of the compound expression? Do the compound sentences created by permutation and/or moving parentheses all mean the same thing? Try to list all the possibilities in an order in which each one is obtained from the previous one by one of two actions: commuting two sentences in an "or" or by moving parenthesis.
• Consider four sentences "or"-ed: "(P or Q) or (R or S)". List all the possible equivalent sentences obtained by moving parentheses and/or permuting the letters P, Q, R and S. How many are there? Do they all mean the same thing?
• Suppose that 99 sentences are "or"-ed in some order, with parentheses placed so that each "or" connects only two sentences (possibly compound themselves). When is this compound sentence true?

Some Uses of "Or" in Writing About Mathematics

"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:
 If(P or Q), then R means [not(P or Q)] or R, see If...then, which means [(not(P)) and (not(Q))] or R, see "not" applied to an "or" sentence, which means [(not(P)) or R] and [(not(Q)) or R], see "and" distributes over "or", which means [if P, then R] and [if Q, then R] see If...then
Because meaning here is transitive (which is the same as saying that "if and only if" is transitive), if (P or Q), then R means the same as (if P, then R) and (if Q, then R).