Today we will dedicate this section to logical negation denoted with the symbol ∼ or ¬ from the chapter on propositional logic, this operator has the property of changing the truth value of statements or propositional variables, although it is also used in open sentences, also called functions propositional.
Logical negation is very important when applying it to quantifiers as it can transform an open sentence into a statement and vice versa, but we will see this in future sections.
The current section is short and there are not many details to discuss this monadic operator. Without more to say, let’s get started.
What is logical negation?
While it is true that the negation of a statement does not make any logical connection, that is, it is not a logical connective itself, it does not stop being a compound statement after denying a simple statement.
A simple statement is intended to pass judgment, but affirmatively, if this judgment is a negation, then it would be a compound statement. Clarifying this point, let’s look at the concept of logical negation:
Definition of logical negation
In mathematics, the logical negation denoted with the symbol \( \sim \) is a logical operator that has the property of changing the validity of a statement \( p \), that is, it changes from true to false and vice versa, the negation of a statement \( p \) is written as \( \sim p \).
Here \( p \) does not make any reference to a simple or compound statement, its only function is simply to negate by emitting a truth value opposite to the validity of \( p \). Let’s see how negation works on a simple statement.
Example
Let the statements be:
- Dogs have 4 legs
- Dogs don’t have 4 legs
The two statements have something in common, one affirms and the other denies for the same subject and with opposite predicates.
Statement 2 can be written like this:
- Dogs don’t have 4 legs = \( \sim \) (Dogs have 4 legs) … \( ( \mathrm{I} ) \)
That is, statement 2 is the negation of statement 1, for practical reasons, statements 1 and 2 will be represented by \( p \) and \( q \) respectively, in this way it would look like this:
- \( p \) = Dogs have 4 legs.
- \( q \) = Dogs do not have 4 legs.
The statement \( ( \mathrm {I}) \) can be written like this:
- \( p = \sim q \)
Although \( p \) is a compound statement, the negation \( \sim \) is not a logical connector because it does not connect to another statement.
Of course statement 1 is true and statement 2 is false in case you thought I forgot.
Logical negation of simple sentences:
- \( p \) = The Tumi is gold. (In Google you can search the meaning of tumi)
\( \sim p \) = The tumi is not gold. - \( q \) = Human beings are not from Ganymede.
\( \sim q \) = Humans are from Ganymede. - \( r \) = The tiger is a feline.
\( \sim r \) = The tiger is not a feline.
Logical negation of compound statements:
- \( p \) = I am earthling and mortal.
\( \sim p \) = I am not Earthling or I am not mortal. - \( q \) = The earth is not square or circular.
\( \sim q \) = The earth is square and circular.
This type of negations for compound statements will be explained more clearly when we get to the sections of the logical disjunction.
You can make other more complex examples of the negation of other statements of another type, but this requires other logical connectives that we have not yet explained, we have already seen similar examples such as:
- My dog has legs and a tail.
Your negation would be:
- My dog has no legs or has no tail.
To explain this nomenclature, it requires the study of logical conjunction and disjunction.
Logical negation is commonly considered to be a connective but as a monadic operator, because it affects only one statement, it also takes the name of logical complement.
Negation truth table
As we have already indicated before, the only thing this logical operator does is change the validity of the statements.
If a statement is false, the negation of the statement is true, in the same way, if a statement is true, the negation of such a statement is false.
The following truth table I show the change of the validity of any statement.
\[ \begin{array}{ c | c } p & \sim p \\ \hline V & F \\ F & V \end{array} \]
Where \( V \) means that the statement is true and \( F \) means that the statement is false.
Some Logical Laws of Denial
Common property is the law of double negation (or colloquially referred to as the law of the negation of the negation), for example:
- Affirmative statement: Dogs are carnivores.
- Negated statement: \( \sim \) (dogs are carnivores) = dogs are not carnivores.
- Double negation: \( \sim \) ( \( \sim \) (dogs are carnivores) ) = \( \sim \) (dogs are not carnivores) = dogs are carnivores.
As you can see, the double negation of a statement returns the same statement, symbolically it is represented like this:
\[ \sim ( \sim p ) = p \]
The symbolic representation of the truth values for a statement \( p \) where we are assuming it to be true along with its negation and double negation is:
- \( \mathrm{V} (p) = V \)
- \( \mathrm{V} ( \sim p ) = F \)
- \( \mathrm{V} ( \sim ( \sim p ) ) = V \)
This is for the case where \(p \) is false, we would have:
- \( \mathrm{V} (p) = F \)
- \( \mathrm{V} ( \sim p ) = V \)
- \( \mathrm{V} ( \sim ( \sim p ) ) = F \)
There are other logical laws such as Morgan’s laws that we will not mention in this section because they are related to other logical connectives that we have not explained yet, but they are widely used to find relationships between logical connectives that we will already deal with in later sections of the logic course.
Examples
All the examples of double negation have the following logical form \( \sim ( \sim p ) = p \), let’s see:
- It is not true that my dog does not have a tail = my dog has a tail.
- It is not true that the earth is not round = the earth is round. (I know it’s fake, don’t worry)
- It is false that cats do not eat meat = cats eat meat.
Therefore, it is clear that the truth value of a double negation of a statement turns out to be the same statement.
Denial of categorical propositions
Negation can have other applications such as categorical propositions, there are special words that we add to the subject of an open sentence to transform it into statements, this type of statement is called categorical propositions and can be justified in a general or particular way to the subject according to the predicate that we assign to it.
Categorical propositions are statements that affirm or deny a quantity of the subject of a statement, that is, the predicate affirms or denies for all or some of a set within a category (in this case, the subject). Category propositions have the following form:
- All S are P
- No S are P
- Some S are P
- Some S are not P
And this is where the logical negation comes in, the negation of the category proposition of 1 is 2 and the negation of 3 is 4 and vice versa, that is:
- The negation of “all S are P” is “no S are P”.
- The negation of “no S are P” is “all S are P”.
- The negation of “some S are P” is “some S are not P”.
- The negation of “some S are not P” is “some S are P”.
However, the negation of “some S are P” should be “not all S are P” and that it is completely different from “some S are not P”, the explanation is made in the quantifiers section later.
But near the end of this section I make a small sketch of this point.
Denial as a function of truth
Another way to create opposition to these statements verbally speaking is with the phrase “it is not true that” without the need to introduce the adverb “no” between the subject and predicate.
This word is enough to transform a simple statement into compounds as mentioned above, although it is true that the negation of a compound statement can generate another compound statement, it can also give us a simple statement.
Let the following proposition be:
- \( p \): The door is not made of wood. (compound)
Their denial would be:
- \( \sim p \): It is not true that the door is not made of wood. (compound)
Its equivalent would be:
- \( \sim p \): The door is wooden. (simple)
When a logical operator operates on a statement with certain possible truth values, they also return a specific truth value, these logical operators are called truth functions.
In fact, all the logical connectives that we will introduce in later sections can also be treated as truth functions.
For the case of logical negation, we must be careful when we deny a statement if we are dealing with quantifiers such as “Some are” or “some are not”, since one of them is the conclusion of the other.
The correct way to use negation:
Let’s look at the following example:
- \( p \) = Some dogs have tails.
This is a categorical proposition, and it is obvious that some dogs have tails, but there must be other dogs that do not have them, we obtain the following statement:
- \( q \) = Some dogs do not have a tail.
If you look closely, the negation of \( p \) does not really contradict \( q \) since one of them has been deduced from the other and more than an apparent contradiction, really one of them is the conclusion of the other.
As I said in the previous section, the adverb “no” has some drawbacks when denying this type of statement. The correct way to deny the statement “some dogs have tails” is:
- It is not true that some dogs have tails
And it is the same as saying:
- No dog has a tail.
We will see these corrections in a section on quantifiers in the elementary set theory course for later, for now, only these small details must be taken into account.
Summary:
You can do many things with logical negation, but this will be seen when we combine with the rest of the logical connectives, in the following table I present a summary of what has been exposed so far.
| Connective | Notation | Application | Meaning | Example | True table |
| Negation | ~, ¬, | ~ p | not | Sergio is not skinny | \[ \begin{array}{ c | c } p & \sim p \\ \hline V & F \\ F & V \end{array} \] |
Thus ended this section, in the next, we will dedicate ourselves to the logical conjunction. And that would be it, see you in the next section, have a good day, greetings.