﻿ "Not"

NOT

The term "not" reverses the truth of a sentence, P. If P is true, not(P) is false. If P is false, not(P) is true. The table below formalizes this; T stands for true and F stands for false.

Truth Table For "Not"
P Not(P)
T F
F T

In technical sciences such as mathematics, but not in ordinary writing, it is OK to have more than one "not": not(not(P)), not(not(not(P))), etc. The table below gives the truth values for these. Not(not(P)) is evaluated by finding the truth values of not(P) first, and then reversing them. Likewise, one evaluates not(not(not(P))) by reversing the truth values of not(not(P)).

Truth Table For Repeated "Not"'s
P Not(P) Not(not(P)) Not(not(not(P)))
T F T F
F T F T
As this table shows, "not(not(P))" means exactly the same thing as "P" and "not(not(not(P)))" means the same as "not(P)". There is a general rule that holds here (which is proved by the principle of induction): an odd number of repeated "not"'s means the same as just one; an even number of repeated "not"'s means the same as none.

Here is an example of double negatives from high school geometry. Two lines in the Euclidean plane intersect if they have at least one point in common; they are parallel if they have no point in common. What does it mean that two lines are NOT parallel? That means, not(they have no point in common). Notice the double negative "not" and "no". They "cancel": two lines are NOT parallel means that they have at least one point in common (that is, they intersect). There is more happening in this example than just the cancelling of two negatives (a for-all statement is being converted to a there-exist statement---a discussion saved for elsewhere).

Here is an example from linear algebra that bewilders many students. Some vectors vi, 1 < = i < = n, are said to be linearly independent, if for all scalars ci, if the sum of ci vi is the 0 vector then each scalar ci=0 for all 1 < = i < = n. Note that the prefix "in" on independent usually means "not" dependent. It should be no surprise that linear algebra textbooks often define linearly dependent to be not linearly independent. That's a classic example of a double negative!

Elsewhere, you may learn that not[(for all x) (P(x))], where P(x) is some sentence about x, has the same meaning as (there is at least one x) [not(P(x))] (see "for all"). So, in the example of the previous paragraph, the vi's not being linearly independent means

for at least one choices of scalars ci, 1 < = i < = n, not[ if the sum of ci vi is the 0 vector, then each ci=0 for all 1 < = i < = n].
Elsewhere, perhaps at the "not(if P, then Q)" page, you may learn that "not(if P, then Q)" means the same as "P and (not(Q))". If we use that above in our analysis of not being linearly independent, we discover that the vi's not being linearly independent means
for at least one choices of scalars ci, 1 < = i < = n,[( the sum of ci vi is the 0 vector) and {not(each ci=0 for all 1 < = i < = n)}].
However, the second sentence in the "and" just above is again the "not" of a "for all" sentence, and is equivalent to "there is at least one integer i in [1,n] for which ci is not 0". Thus we discover that the vi's not being linearly independent means
for at least one choices of scalars ci, 1 < = i < = n, [(the sum of ci vi is the 0 vector) and (at least one of the ci's is not 0 for some integer i in [1,n])].
In this situation, let j be an integer in [1,n] such that cj is not 0. In the equation, the sum of ci vi is the 0 vector, move the term cjvj to the other side: one obtains
the sum of ci vi, for all integers i in [1,n] except i=j, is equal to -cjvj.
Since -cj is not equal to zero, we may divide both sides of the equation by it:
the sum of [ci/(-cj)] vi, for all integers i in [1,n] except i=j, is equal to vj.
We have just proved that, when the vi's are not linearly independent, one of them can be written as a linear combination of the others. The converse is also true, and is left to the reader.