Some thoughts on negation

Negation might seem a little tricky at first. Let's take a look at one: if the food is good, then the service is excellent.

Let

A: The food is good

B: The service is excellent

Then this sentence is an example of

${A\rightarrow{B}}$

We discover in the exercises that

${A\rightarrow{B}}$ is equivalent to $A'\lor{B}$ and so the negation of ${A\rightarrow{B}}$ can be written ${A\wedge{B'}}$ using De Morgan, or

The food is good, but the service is not excellent.

How, might you ask, does one arrive at this?

In order to negate the implication, we begin by negating the one case in which the implication is false: if A is true, and B is false, then the negation must be true. Hence our target wff should be true when both A and B' are true. Consider

${A\wedge{B'}}$ which in every other case is false, just as it should be!

A look at the truth table also suggests this approach:

A	B	${A\rightarrow{B}}$	?
T	T	T	F
T	F	F	T
F	T	T	F
F	F	T	F

Only one true, and the rest false. This is similar to the truth table for conjunction making A and B' true simultaneously....

Hence $A \land B'$ is one possible negation:

A	B	${A\rightarrow{B}}$	${A\wedge{B'}}$
T	T	T	F
T	F	F	T
F	T	T	F
F	F	T	F

Website maintained by Andy Long. Comments appreciated.

longa@nku.edu