42
Formal Logic
following it. In (4), the scope of (E x) is S(x) and the scope of (4y) is T( y); parentheses or brackets can be eliminated when the scope is clear.
If a variable occurs somewhere in a wff where it is not part of a quantifier and
is not within the scope of a quantifier involving that variable, it is called a free
variable. For example, y is a free variable in
(4x)[Q(x, y) S (E y)R(x, y)]
because of the first occurrence of y, which is neither the variable of a quantifier nor
within the scope of a quantifier using y. A wff with free variables may not have a
truth value at all in a given interpretation. For example, in the interpretation where
the domain is all of the integers, the predicate P(x) means “x > 0”, and 5 means (of
course) the integer 5, the wff
P( y) ` P(5)
has no truth value because we don’t know which element of the domain y refers to.
Some elements of the domain are positive and others are not. The wff
P( y) ~ P(5)
is true in this interpretation even though we don’t know what y refers to because
P(5) is true. In both of these wffs y is a free variable.
eXAMPLe 21
In the wff
(4x)(E y)[S(x, y) ` L( y, a)]
the scope of (E y) is all of S(x, y) ` L( y, a). The scope of (4x) is (E y)[S(x, y) ` L( y, a)].
Consider the interpretation where the domain consists of all the cities in the United
States, S(x, y) is the property “x and y are in the same state,” L( y, z) is the property
“y’s name begins with the same letter as z’s name,” and a is assigned the value Albuquerque. So the interpretation of the entire wff is that for any city x there is a city y in
the same state that begins with the letter A. The wff is true in this interpretation. (At
least it is true if every state has a city beginning with the letter A.)
Translation
Many English language statements can be expressed as predicate wffs. For example, “Every parrot is ugly,” is really saying, “For any thing, if it is a parrot,
pRaCtiCe 17 What is the truth value of the wff
(E x)(A(x) ` (4y)[B(x, y) S C( y)])
in the interpretation where the domain consists of all integers, A(x) is “x > 0,” B(x, y) is “x > y,” and
C( y) is “y ≤ 0”? Construct another interpretation with the same domain in which the statement has the
opposite truth value.
Formal Logic
following it. In (4), the scope of (E x) is S(x) and the scope of (4y) is T( y); parentheses or brackets can be eliminated when the scope is clear.
If a variable occurs somewhere in a wff where it is not part of a quantifier and
is not within the scope of a quantifier involving that variable, it is called a free
variable. For example, y is a free variable in
(4x)[Q(x, y) S (E y)R(x, y)]
because of the first occurrence of y, which is neither the variable of a quantifier nor
within the scope of a quantifier using y. A wff with free variables may not have a
truth value at all in a given interpretation. For example, in the interpretation where
the domain is all of the integers, the predicate P(x) means “x > 0”, and 5 means (of
course) the integer 5, the wff
P( y) ` P(5)
has no truth value because we don’t know which element of the domain y refers to.
Some elements of the domain are positive and others are not. The wff
P( y) ~ P(5)
is true in this interpretation even though we don’t know what y refers to because
P(5) is true. In both of these wffs y is a free variable.
eXAMPLe 21
In the wff
(4x)(E y)[S(x, y) ` L( y, a)]
the scope of (E y) is all of S(x, y) ` L( y, a). The scope of (4x) is (E y)[S(x, y) ` L( y, a)].
Consider the interpretation where the domain consists of all the cities in the United
States, S(x, y) is the property “x and y are in the same state,” L( y, z) is the property
“y’s name begins with the same letter as z’s name,” and a is assigned the value Albuquerque. So the interpretation of the entire wff is that for any city x there is a city y in
the same state that begins with the letter A. The wff is true in this interpretation. (At
least it is true if every state has a city beginning with the letter A.)
Translation
Many English language statements can be expressed as predicate wffs. For example, “Every parrot is ugly,” is really saying, “For any thing, if it is a parrot,
pRaCtiCe 17 What is the truth value of the wff
(E x)(A(x) ` (4y)[B(x, y) S C( y)])
in the interpretation where the domain consists of all integers, A(x) is “x > 0,” B(x, y) is “x > y,” and
C( y) is “y ≤ 0”? Construct another interpretation with the same domain in which the statement has the
opposite truth value.