§3.1: Statements, Negations, and Quantified Statements
-We now begin our study on logic. We will focus on the more “mathematical” side of logic, but part of your work in this
chapter will include using Logic in the English language.
-Logic is built on there being two possibilities: 1. TRUE
(there is NO middle ground)
Def: A statement is a sentence that is either TRUE or FALSE, but not both simultaneously.
Ex. Determine if the following are statements and why or why not?
a. Lake City is the capital of Florida.
[This is a statement, although it is false]
b. Lake City is the best city in Florida.
[This is not a statement, since it is an opinion]
c. Go to Lake City today.
[This is not a statement, since it is a command]
d. Do you think Lake City is the capital of Florida? [This is not a statement, since it is a question]
Since we often refer to statements and do “operations” on multiple statements, writing the English form can be clumsy.
We can write statements by using symbols:
p: Lake City is the capital of Florida. (We use the letter p to represent this statement in symbolic form.)
Let us consider the statement Lake City is the capital of Florida. Is it true or false? [I hope you realize this is FALSE]
If we change the meaning of statement from false to true, or true to false, we are NEGATING the statement.
How could we form the negation of the statement “Lake City is the capital of Florida”? By changing it from a FALSE
statement to a TRUE statement.
Give the negation of the above statement:_____Lake City is NOT the capital of Florida.____________________
Def: Symbolically, the
character is used to represent negation in logic. It is spoken as “not” What set operation is
similar to this? ______This is similar to the complement operator, (the ’)
So our example: ~p:_ Lake City is not the capital of Florida._______________________________________
Ex. Let q: One succeeds. Give the symbolic form of “One does not succeed.”_ Answer: _~q
Ex. Let s: Humans and bananas do not share approximately 60% of the same DNA structure. Give the English
representation of ~s. Is the negation true or false?
The English representation of ~s would be “Humans and bananas do share approximately 60% of the same DNA
structure.” It turns out this statement is true. For other strange statements, check out exercises 25-28 in §3.1
Def: A quantified statement is one which contains quantifiers.
Def: Quantifiers are words like all, some, and no (or none). You can think of these words as those that put a
quantity on an object.