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.

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)).

P | Not(P) | Not(not(P)) | Not(not(not(P))) |
---|---|---|---|

T | F | T | F |

F | T | F | T |

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 v_{i}, 1 < = i < = n, are said to be
**linearly independent**, if for all scalars c_{i},
if the sum of c_{i} v_{i} is the 0 vector then each scalar
c_{i}=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 v_{i}'s **not
being linearly independent **means

for at least one choices of scalars cElsewhere, 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_{i}, 1 < = i < = n, not[ if the sum of c_{i}v_{i}is the 0 vector, then each c_{i}=0 for all 1 < = i < = n].

for at least one choices of scalars cHowever, 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 c_{i}, 1 < = i < = n,[( the sum of c_{i}v_{i}is the 0 vector) and {not(each c_{i}=0 for all 1 < = i < = n)}].

for at least one choices of scalars cIn this situation, let j be an integer in [1,n] such that c_{i}, 1 < = i < = n, [(the sum of c_{i}v_{i}is the 0 vector) and (at least one of the c_{i}'s is not 0 for some integer i in [1,n])].

the sum of cSince -c_{i}v_{i}, for all integers i in [1,n] except i=j, is equal to -c_{j}v_{j}.

Go to an overview of logic. |