Download Chapter 2. The Logic of Quantified Statements

2.1.1
Chapter 2. The Logic of
Quantified Statements
• Predicates
• Quantified Statements
• Valid Arguments and Quantified Statements
2.1.2
Section 1. Predicates and
Quantified Statements I
• In Chapter 1, we studied the logic of compound
statements, but the argument reasoning in there
cannot show the validity of the following simple
argument:
All men are mortal.
Socrates is a man.
Therefore, Socrates is mortal.
2.1.3
Predicates
• To study these types of logical arguments, we
turn to predicate calculus.
• A predicate is a sentence that contains a finite
number of variables and becomes a statement
when specific values are substituted for the
variables.
• The domain of a predicate variable is the set of
all values that may be substituted in place of the
variable.
2.1.4
Predicate Notation
• If P(x) is a predicate and x has a domain D, the
truth set of P(x) is the set of all elements of D
that make P(x) true when substituted for x.
• The truth set is denoted {x ∈ D | P(x)}.
• If P(x) and Q(x) are predicates and the common
domain of x is D, then the notation P(x) ⇒ Q(x)
denotes that the truth set of P(x) is a subset of
the truth set of Q(x).
• If P(x) and Q(x) have the same truth set, we
denote this as P(x) ⇔ Q(x).
2.1.5
The Universal Quantifier
• We often find predicates involved when we are
making claims about properties that some or all
the elements of a set obey. This leads us to look
at statements using one of two quantifiers.
• The Universal Quantifier: If P(x) is a predicate
over a domain D, we say a universal statement
is one of the form “∀x ∈ D, P(x).”
• This universal statement is true provided P(x) is
true for every x in D.
• Any x ∈ D with P(x) false, is a counterexample.
2.1.6
Examples
• Example 1: Let D = {1,2,3,4,5} and let P(x) be
the predicate x2 ≥ x. Using the Method of
Exhaustion, we find that 12 ≥ 1, 22 ≥ 2, 32 ≥ 3,
42 ≥ 4, and 52 ≥ 5 are all true, hence the
universal statement ∀x ∈ {1,2,3,4,5}, x2 ≥ x is
true.
• Example 2: If we change this universal
statement to: ∀x ∈ R, x2 ≥ x, it is no longer true
since x = 1/2 is a counterexample.
2.1.7
The Existential Quantifier
• The Existential Quantifier: If P(x) is a predicate
over a domain D, we say an existential
statement is one of the form “∃x ∈ D ∋ P(x).”
• This existential statement is true provided P(x)
is true for at least one x in D, and is false if P(x)
is false for every x in D.
• From this, we see that the negation of an
existential statement is a universal statement,
and, likewise, the negation of a universal
statement is an existential one.
2.1.8
More Examples
• Consider: ∃x ∈ D ∋ x2 < 0.
• Example 1: If D = C (the Complex numbers),
then x = i yields i2 = (−1) < 0, hence the
existential statement is true.
• Example 2: If D = R, then by the properties of
R, we know that x2 ≥ 0 for all x in R, hence the
existential statement is false.
• This second example show us the negation of
∃ x ∈ R ∋ x2 < 0 is the universal statement
∀x ∈ R, x2 ≥ 0.
2.1.9
Negations of Quantifiers
• As seen in the previous example, the negation of
an existential statement is a universal statement.
• Formally, we denote:
~[∃x ∈ D ∋ P(x)] ≡ ∀x ∈ D, ~P(x).
• By the same process, we have that:
~[∀x ∈ D, P(x)] ≡ ∃x ∈ D ∋ ~P(x).
• Intuitively, the first says the opposite of at least
one thing satisfying a property is that none do,
and the opposite of all things satisfying the
property is that at least one does not.
2.1.10
Examples of Negations
• The negation of:
Some people are sad.
is
All people are not sad.
• The negation of:
All integers are rational.
is
At least one integer is irrational.
• Which of each pair is true?
2.1.11
Universal Conditional
• The statement:
∀x, if P(x), then Q(x)
is called the universal conditional.
• Many mathematical statements are universal
conditionals.
• Example: ∀x ∈ R, if x > 2 then x2 > 4 (formal)
is equivalent to: (informally)
– Every real number greater than 2 has a square
greater than 4.
– The square of any real number greater than 2 is
greater than 4.
2.1.12
Negation of Quantified Conditionals
• Since we see the properties of symbolic logic
carry over when dealing with quantified logic,
we deduce that:
~[∀x ∈ D, if P(x), then Q(x)]
is
∃x ∈ D ∋ P(x) and ~Q(x).
• Similarly, ~[∃x ∈ D ∋ if P(x), then Q(x)]
is
∀x ∈ d, P(x) and ~Q(x).
• Negate: 1. Every CS student studies CMSC203.
2. Some CS students study CMSC203.