2
Formal Logic
computer circuitry) is a direct analog of the statement logic of this chapter. We
will study circuit logic in Chapter 8.
S e c t I o n 1 . 1 StatementS, SymboliC RepReSentation,
and tautologieS
Formal logic can represent the statements we use in English to communicate facts
or information. A statement (or proposition) is a sentence that is either true or
false.
eXAMPLe 1
Consider the following:
a. Ten is less than seven.
b. Cheyenne is the capital of Wyoming.
c. She is very talented.
d. There are life forms on other planets in the universe.
Sentence (a) is a statement because it is false. Sentence (b) is a statement because
it is true. Sentence (c) is neither true nor false because “she” is not specified; therefore (c) is not a statement. Sentence (d) is a statement because it is either true or
false; we do not have to be able to decide which.
Connectives and Truth Values
In English, simple statements are combined with connecting words like and to
make more interesting compound statements. The truth value of a compound
statement depends on the truth values of its components and which connecting
words are used. If we combine the two true statement, “Elephants are big,” and,
“Baseballs are round,” we would consider the resulting statement, “Elephants are
big and baseballs are round,” to be true. In this book, as in many logic books,
capital letters near the beginning of the alphabet, such as A, B, and C, are used to
represent statements and are called statement letters; the symbol ` is a logical
connective representing and. We agree, then, that if A is true and B is true, A ` B
(read “A and B”) should be considered true.
pRaCtiCe 1
1
a. If A is true and B is false, what truth value would you assign to A ` B?
b. If A is false and B is true, what truth value would you assign to A ` B?
c. If A and B are both false, what truth value would you assign to A ` B?
The expression A ` B is called the conjunction of A and B, and A and B are
called the conjuncts of this expression. Table 1.1 summarizes the truth value of
A ` B for all possible truth values of the conjuncts A and B. Each row of the table
1 Answers to practice problems are in the back of the text.
Formal Logic
computer circuitry) is a direct analog of the statement logic of this chapter. We
will study circuit logic in Chapter 8.
S e c t I o n 1 . 1 StatementS, SymboliC RepReSentation,
and tautologieS
Formal logic can represent the statements we use in English to communicate facts
or information. A statement (or proposition) is a sentence that is either true or
false.
eXAMPLe 1
Consider the following:
a. Ten is less than seven.
b. Cheyenne is the capital of Wyoming.
c. She is very talented.
d. There are life forms on other planets in the universe.
Sentence (a) is a statement because it is false. Sentence (b) is a statement because
it is true. Sentence (c) is neither true nor false because “she” is not specified; therefore (c) is not a statement. Sentence (d) is a statement because it is either true or
false; we do not have to be able to decide which.
Connectives and Truth Values
In English, simple statements are combined with connecting words like and to
make more interesting compound statements. The truth value of a compound
statement depends on the truth values of its components and which connecting
words are used. If we combine the two true statement, “Elephants are big,” and,
“Baseballs are round,” we would consider the resulting statement, “Elephants are
big and baseballs are round,” to be true. In this book, as in many logic books,
capital letters near the beginning of the alphabet, such as A, B, and C, are used to
represent statements and are called statement letters; the symbol ` is a logical
connective representing and. We agree, then, that if A is true and B is true, A ` B
(read “A and B”) should be considered true.
pRaCtiCe 1
1
a. If A is true and B is false, what truth value would you assign to A ` B?
b. If A is false and B is true, what truth value would you assign to A ` B?
c. If A and B are both false, what truth value would you assign to A ` B?
The expression A ` B is called the conjunction of A and B, and A and B are
called the conjuncts of this expression. Table 1.1 summarizes the truth value of
A ` B for all possible truth values of the conjuncts A and B. Each row of the table
1 Answers to practice problems are in the back of the text.