The predicate is that portion of a statement that gives information about the
subject. In logic, the predicate can be represented through the use of *predicate
symbols* and *predicate variables.* Consider the following:

**Mary is studying for a bachelor's degree at GSU.**

Let **P** be the predicate symbol for *is studying for a bachelor's
degree at GSU, *and let ** x** be the predicate variable, which
takes its value from the set of all students. Then the above sentence is
represented symbolically as

The above statement can be further refined. Let ** y** also
be a predicate variable, taking its value from the set of all colleges,
and let R stand for

The predicate is a sentence containing a specific number of variables,
and becomes a statement when specific values are substituted in place of
the predicate variables. The values are taken from the **domain **of
the predicate variables: the domain of ** x** is the set of all
students, and the domain of

A set can be defined by simply enclosing the set elements in brackets. For
instance, {2, 3, 5, 7} is a set of prime numbers less than 10. When using symbols,
sets are typically represented by upper case letters, and set elements by lower
case. Let **T** be a set of prime numbers less than 10, and ** x**
be an element of

When elements are substituted for variables in a predicate, the result is either
true or false. Let **P** be the set of all prime numbers, let ** x**
be an element of

{** x P
| R(x)**}

which is read as "the set of all ** x** in

Using and

Given predicates **P( x)** and

**P( x)**

**P( x)**

Quantifiers refer to given quantities, such as "some" or "all",
indicating the number of elements for which a predicate is true. The symbol
is translated as "for
all", "given any", "for each", or "for every",
and is known as the *universal quantifier*. The symbol is
the *existential quantifier*, and means variously "for some",
"there exists", "there is a", or "for at least
one".

A **universal statement** is a statement that
is true if, and only if, it is true for every predicate variable within a given
domain. Consider the following example:

Let **B** be the set of all species of non-extinct birds, and ** b**
be a predicate variable such that

An **existential statement** is a statement
that is true if there is at least one variable within the variable's domain
for which the statement is true.

Again, using the above defined set of birds and the predicate **R( b)**,
the existential statement is written as "

Practice Exercises |

**Negating Quantified
Statements**

If a **universal statement** is a statement
that is true if, and only if, it is true for every predicate variable within
a given domain (as stated above), then logically it is false if there exists
even one instance which makes it false. As discussed before, the statement "*All
birds fly*." is false. But its negation is not "*No birds fly*."
(also patently false). Instead, the correct negation will be "*Some birds
do not fly*." Symbolically:

~(All birds fly) | Some birds do not fly. | |

or | ||

~(b B | Q(b)) | b B | ~Q(b) |

The negation of a universal statement ("all are") is logically equivalent to an existential statement ("Some are not").

Similarly, an **existential statement** is
false only if **all** elements within its domain are false. The negation
of "Some birds are bigger than elephants" is "No birds
are bigger than elephants."

~(Some birds are bigger than elephants) | No birds are bigger than elephants | |

or | ||

~(b B | Q(b)) | b B | ~Q(b) |

The negation of an existential statement ("some are") is logically equivalent to a universal statement ("all are not").

Practice Exercises |

**Universal Conditional Statements**

Remember conditional statements?
A *universal conditional statement* is simply a universal statement with
a condition, and is symbolically represented as:

** x**,
if

Understanding universal conditional statements will further your understanding
of *if-then *(conditional) statements, in particular, why the** truth
table of an if-then** statement is constructed as it is. Recall that
a conditional statement is false only if its hypothesis is true and the result
is false, and apply this to an earlier example of a conditional statement:

**If someone kicks me, then I will yell "ouch!"**

Obviously, if someone kicks you and you yell "ouch!", the above conditional statement is true. But suppose you drop a hammer on your foot and yell ouch? Or nothing happens to you and you therefore say nothing? Neither of these situations makes the above statement false, so it becomes true by default. The only thing that makes this statement false is if someone kicks you and you do not yell "ouch!".

Putting this statement into the form of a universal conditional statement yields the following:

**For all things that can happen to me, if someone
kicks me, then I will yell "ouch!"**

If you consider that the only instance which makes this false is a true hypothesis (someone kicks you) and a false conclusion (you don't yell ouch), it should become apparent that this statement is therefore true for everything else. Particularly, it is always true when the hypothesis is false.

Practice Exercises |

**Negating Universal
Conditional Statements**

To understand the negation of a universal conditional statement, first
review *if-then* (conditional) negations and negations of universal
("for all") statements:

The negation of a **conditional** (if-then) statement is logically equivalent
to an *and* statement.

~(pq) p ~q

The negation of a universal statement is logically equivalent to an existential statement.

~(** x**,
P(

Substituting the conditional statement into the universal statement yields the following result:

~(** x**,
P(

Practice Exercises |