In my other notes titled “Propositions and Symbols Used in Propositional (or Symbolic) Logic” (see http://philonotes.com/index.php/2018/02/02/symbolic-logic/), I discussed the two basic types of a proposition as well as the symbols used in symbolic logic. I have also briefly discussed how propositions can be symbolized using a variable or a constant.
In these notes, I will discuss the topic “negation of statements in propositional (or symbolic) logic” or the way in which propositions or statements in symbolic logic are negated.
To begin with, we have to note that any statement used in symbolic logic can be negated. And as I have already mentioned in the previous discussion, symbolic logic uses ~ (tilde) to symbolize a negative proposition.
But how do we know that the statement is negative?
A statement is negative if it contains at least one of the following signifiers:
No
Not
It is false
It is not the case
It is not true
For example, let us consider the following statements:
- Either no students are interested in the party or it is not the case that the
administration requires the students to attend the party.
- If the company does not increase the salary of the workers, then the union will go on strike to press its various demands.
- The professor will not be absent if and only if he is not sick.
As we notice, example #1 is a compound statement, and both component statements contain the negation signs “no” and “it is not the case.” For this reason, when we symbolize the entire statement, then both component statements should be negated. Hence, if we let p stand for “No students are interested in the party” and q for “It is not the case that the administration requires the students to attend the party,” then the statement “Either no students are interested in the party or it is not the case that the administration requires the students to attend the party” can be symbolized as follows:
~ p v ~ q
In example #2, only the first component statement contains the negation sign “not.” Hence, only the first statement should be negated. Thus, if we let p stand for “The company does not increase the salary of the workers” and q for “The union will go on strike to press its various demands,” then the statement “If the company does not increase the salary of the workers, then the union will go on strike to press its various demands,” is symbolized as follows:
~ p ⊃ q
In example #3, both component statements contain a negation sign “not.” Thus, when symbolized, both component statements have to be negated. Hence, if we let p stand for “The professor will not be absent” and q for “He is not sick,” then the statement “The professor will not be absent if and only if he is not sick” is symbolized as follows:
~ p ≡ ~ q
Now, sometimes a statement can be double (or even triple) negated. In other words, the statement contains two or more negation signs. If this happens, then the statement has to be symbolized accordingly. Consider this example: “It is not true that the professor is not sick.” If we let p stand for the entire statement, then it is symbolized as follows:
~~ p
However, since a double negation implies affirmation, then the statement can also be symbolized as follows:
p
In some cases, contradictory words, such as “kind and unkind” and “mortal and immortal,” may signify negation if and only if it is clearly specified in the statement; otherwise, the statement should not be negated. Consider the following examples:
- Lulu is generous, while Lili is unkind.
- Either George is kind or Bert is unkind.
In example #1, the word “unkind” does not clearly signify negation. Thus, the statement “Lili is unkind” is not a negative statement. Let us symbolize example #1. If we let p stand for “Lulu is generous” and q for “Lili is unkind,” then the proposition “Lulu is generous, while Lili is unkind” is symbolized as follows:
p • q
However, the word “unkind” in example #2 above clearly signifies negation because of the presence of the contradictory words “kind and unkind” in the statement. Now, if we let p stand for “George is kind” and q for “Bert is unkind,” then the statement “Either George is kind or Bert is unkind” is symbolized as follows:
p v ~q
This is because the statement “Either George is kind or Bert is unkind” can also be stated in this manner: “Either George is kind or Bert is not kind.”
Rule in Negation
The negation of a true statement is false; while the negation of a false statement is true.
Obviously, the rule in negation says that if a particular statement is true, then it becomes false when negated. And if a particular statement is false, then it becomes true when negated. The truth table below illustrates this point.
Let us determine the truth-value of a negative statement by applying the rule in negation.
Consider the example below.
It is not the case that the administration requires the students to attend the party.
Again, if we let p stand for the statement “The administration requires the students to attend the party,” then the statement is symbolized as p. However, since the statement contains a negation sign “It is not the case,” then the statement is negative. Thus, the statement has to be symbolized as follows:
~ p
Now, if we assume that the statement “The administration requires the students to attend the party” is true, that is, the administration did indeed require the students to attend the party, then the statement “It is not the case that the administration requires the students to attend the party” is absolute false. To illustrate:
Please note that when we assign a truth-value to a statement, we assign it to the statement without the negation sign. Thus, if we have the statement ~ p, and if we assign, for example, True value to the statement, we assign it to p and not to ~ p.