Logic Review: Quantifiers and Truth Tables
Equivalent Ways of Expressing Quantified Statements
Statement
Equivalent Statement
Example
All A are B.
There are no A that are not B.
All poets are writers. There are
no poets that are not writers.
Some A are B.
There exits at least one A that is a B.
Some people are lazy. At least one
person is lazy.
No A are B.
All A are not B.
No math books have pictures. All
math books do not have pictures.
Some A are not B.
Not all A are B.
Some students do not work hard.
Not all students work hard.
Negations of Quantified Statements
Statement
Negation Statement
All A are B.
Some A are not B.
Some A are B.
No A are B.
Example
All people are honest. Some
people are not honest.
Some roads are open. No roads are
open.
Truth Tables
Symbolic Statement
Negation: P
Conjunction: P Q
Disjunction: P Q
PQ
Conditional:
Biconditional: P
Conjunction (
P
T
T
F
F
):
Q
English Statement
Not P.
P and Q. P but Q. P yet Q. P nevertheless Q.
P or Q.
If P, then Q. Q if P. P is sufficient for Q.
Q is necessary for P. P only if Q. Only if Q , P.
P if and only if Q. If P, then Q and If Q, then P. P iff Q.
P is necessary and sufficient for Q.
P and Q.
Q
P Q
T
T
F
F
T
F
F
F
Conditional: PQ. If P, then Q.
P
Q
PQ
T
T
T
T
F
F
F
T
T
F
F
T
Disjunction (
P
T
T
F
F
Biconditional: P
Q
T
F
T
F
): P or Q
P Q
T
T
T
F
Q. P if and only if Q.
P
Q
T
T
F
F
T
F
T
F
P Q
T
F
F
T