11.3.1: The Logic of Not

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Bradley H. Dowden
California State University Sacramento

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In sentential logic, an inconsistent group of sentences is defined via their logical form. By definition, a sentence group is inconsistent if it implies a sentence P and also a sentence not-P. Or we could have defined "inconsistent" by saying it is some complex statement whose logical form is "P and not -P." That statement is composed of two sub-statements, the statement P and its opposite, not -P. The two sub-statements are joined by the connector and. The statement form "P and not-P" is said to be the logical form of a contradiction. Note that the statement form "not-P" does not stand for any statement that is not P; rather, it stands for any statement that negates P—that says something that must be false when P is true and that must be true when P is false. Not-P is the negation of P. This information about negation can be summarized in the following truth table for negation:

Here the capital letter T represents the possibility of the sentence at the top of its column being true, and F represents the possibility of being false. We will practice finding negations of statements. What is the negation of "What time is it?" The answer is that it has no negation because it is not a statement; it is a question. The negation of "He is on time" is "He is not on time."

Exercise \(\PageIndex{1}\)

The negation of “She’s moral” is

a. She’s immoral.
b. It’s not the case that she’s immoral.
c. She’s amoral.
d. None of the above.

Answer

Answer (d). Here are three, equivalent negations of “She’s moral:”

No, she’s not.
She’s not moral.
It’s not the case that she’s moral.

In sentential logic, there are two ways a pair of sentences can be inconsistent. They could be contradictory, or they could be contrary. They are contradictory if they are inconsistent and if, in addition, one sentence must be true while the other must be false. However, two sentences are contrary if they are inconsistent, yet could both be false. Out on the street you have to be on the alert because so many people do not recognize the difference among the words inconsistent, contradictory, and contrary. But here we will use these technical terms properly.

Sentences A and B below are contradictory, whereas sentences A and C are contrary, and sentences B and C are consistent. A is inconsistent with B. A is inconsistent with C.

A. The house is all red.
B. The house is not all red.
C. The house is all green.

A and C are contrary because they both would be false if the house were orange. Another term that occurs when people are thinking about inconsistent is “opposite.” When a squid is hiding at the bottom of the ocean in camouflage it is not signaling but hiding. A biologist might say that when a squid is in camouflage, that is the opposite of the squid’s signaling. That kind of opposite is being contrary, not being contradictory. The squid might be neither signaling nor in camouflage if it is just swimming along peacefully.

If A is inconsistent with B, is B also inconsistent with A? Yes. If A contradicts B, does B have to contradict A? Yes.

Exercise \(\PageIndex{1}\)

Create a sentence that is contrary to the claim that it's 12:26 p.m. but that does not contradict that claim.

Answer: It is noon." Both sentences could be false if it is really 2 p.m., but both cannot be true (in the same sense at the same place and time without equivocating).

Exercise \(\PageIndex{1}\)

Which statement below serves best as the negation of the statement that Lloyd Connelly is an assemblyman who lives in the capital?

a. Lloyd Connelly lives in the capital and also is an assemblyman.
b. No one from Lloyd Connelly's capital fails to be an assemblyman.
c. It isn't true that Lloyd Connelly is an assemblyman who lives in the capital.
d. Lloyd Connelly doesn't live in the capital.
e. Lloyd Connelly is not an assemblyman.

Answer: Answer (c). Answer (d) is incorrect because both the original statement and (d) could be false together. A statement and its negation cannot both be false; one of them must be true.

If you were to learn that x = 8, would it be reasonable for you to conclude that it isn't true that x is unequal to 8? Yes. The valid form of your reasoning is

One could infer the other way, too, because any statement is logically equivalent to its double negation.