Sentences and Statements

Logic and mathematical proof can be studies just like algebra. In fact, much of symbolic logic is just that. A declarative sentence which is true or false, but not both, is called a statement . The following are statements:

• Babe Ruth hit 714 home runs.
• Jack Nicklaus has won 20 major golf titles.
• 2+3=6
• The 25,000th digit of is 7.

The following are not statements:

• He is a golfer.
• Why is a duck?
• x+1=0
• x-y=a

The sentence

He is a golfer.

cannot be judged true or false because we do not know who He is. If the word He is replaced by Bobby Jones forming the sentence

Bobby Jones is a golfer.

the sentence becomes a true statement. Similarly, if x in the sentence

is replaced by 2, forming the sentence

the sentence then becomes a false statement.

The letter x is a variable  in the sentence x+1=0. A letter or other symbol that can represent various elements of a universal set is called a variable. We can make a sentence a statement by replacing its variables by elements of the universal set or by attaching phrases such as For every or There exists to the sentence. For example,

is not a statement, but each of the following is a statement:

• 1<3
• 5<3
• For every real number x, x<3.
• There exists a real number x, such that x<3.

Replacements for variables of a sentence are always chosen from some universal set. Any replacement which makes a sentence true is called a solution. The set of all solutions is called the solution set of the sentence.