Section 1.3 Quantifiers, Predicates, and Validity
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didn’t see the knife. Furthermore, if the knife was there on October 10, then the knife was in the
drawer and also the hammer was in the barn. But we all know that the hammer was not in the barn.
Therefore, ladies and gentlemen of the jury, my client is innocent.
Use propositional logic to prove that this is a valid argument.
S e c t I o n 1 . 3 QuantifieRS, pRediCateS, and validit y
quantifiers and Predicates
Propositional wffs have rather limited expressive power. For example, we would
consider the sentence “For every x, x > 0” to be a true statement about the
positive integers, yet it cannot be adequately symbolized using only statement
letters, parentheses, and logical connectives. It contains two new features, a
quantifier and a predicate. Quantifiers are phrases such as “for every” or “for
each” or “for some” that tell in some sense how many objects have a certain
property. The universal quantifier is symbolized by an upside down A, 4, and
is read “for all,” “for every,” “for each,” or “for any.” Thus the example sentence
can be symbolized by
(4x)(x > 0)
A quantifier and its named variable are always placed in parentheses. The second
set of parentheses shows that the quantifier acts on the enclosed expression, which
in this case is “x > 0.”
The phrase “x > 0” describes a property of the variable x, that of being positive. A property is also called a predicate; the notation P(x) is used to represent
some unspecified predicate or property that x may have. Thus, our original sentence is an example of the more general form
(4x)P(x)
The truth value of the expression (4x)(x > 0) depends on the domain of objects in which we are “interpreting” this expression, that is, the collection of objects from which x may be chosen. This collection of objects is called the domain
of interpretation. We have already agreed that if the domain of interpretation consists of the positive integers, the expression has the truth value true because every
possible value for x has the required property of being greater than zero. If the
domain of interpretation consists of all the integers, the expression has the truth
value false, because not every x has the required property. We impose the condition that the domain of interpretation contain at least one object so that we are not
talking about a trivial case.
An interpretation of the expression (4x)P(x) would consist of not only the collection of objects from which x could take its value but also the particular property
that P(x) represents in this domain. Thus an interpretation for (4x)P(x) could be
the following: The domain consists of all the books in your local library, and P(x)
is the property that x has a red cover. In this interpretation, (4x)P(x) says that every book in your local library has a red cover. The truth value of this expression,
in this interpretation, is undoubtedly false.
39
didn’t see the knife. Furthermore, if the knife was there on October 10, then the knife was in the
drawer and also the hammer was in the barn. But we all know that the hammer was not in the barn.
Therefore, ladies and gentlemen of the jury, my client is innocent.
Use propositional logic to prove that this is a valid argument.
S e c t I o n 1 . 3 QuantifieRS, pRediCateS, and validit y
quantifiers and Predicates
Propositional wffs have rather limited expressive power. For example, we would
consider the sentence “For every x, x > 0” to be a true statement about the
positive integers, yet it cannot be adequately symbolized using only statement
letters, parentheses, and logical connectives. It contains two new features, a
quantifier and a predicate. Quantifiers are phrases such as “for every” or “for
each” or “for some” that tell in some sense how many objects have a certain
property. The universal quantifier is symbolized by an upside down A, 4, and
is read “for all,” “for every,” “for each,” or “for any.” Thus the example sentence
can be symbolized by
(4x)(x > 0)
A quantifier and its named variable are always placed in parentheses. The second
set of parentheses shows that the quantifier acts on the enclosed expression, which
in this case is “x > 0.”
The phrase “x > 0” describes a property of the variable x, that of being positive. A property is also called a predicate; the notation P(x) is used to represent
some unspecified predicate or property that x may have. Thus, our original sentence is an example of the more general form
(4x)P(x)
The truth value of the expression (4x)(x > 0) depends on the domain of objects in which we are “interpreting” this expression, that is, the collection of objects from which x may be chosen. This collection of objects is called the domain
of interpretation. We have already agreed that if the domain of interpretation consists of the positive integers, the expression has the truth value true because every
possible value for x has the required property of being greater than zero. If the
domain of interpretation consists of all the integers, the expression has the truth
value false, because not every x has the required property. We impose the condition that the domain of interpretation contain at least one object so that we are not
talking about a trivial case.
An interpretation of the expression (4x)P(x) would consist of not only the collection of objects from which x could take its value but also the particular property
that P(x) represents in this domain. Thus an interpretation for (4x)P(x) could be
the following: The domain consists of all the books in your local library, and P(x)
is the property that x has a red cover. In this interpretation, (4x)P(x) says that every book in your local library has a red cover. The truth value of this expression,
in this interpretation, is undoubtedly false.