**Statements With Multiple Quantifiers**

While it would be convenient if the world in general (and discrete mathematics
in particular) consisted only of simple *if-then* statements, the reality
is that much of the logic that must be contended with is made up of multiple
events strung together by various conditions and quantifiers. For example:

**You can fool some of the people all of the time.**

Written formally, this sentence can be expressed as:

people
** p** such that time

The above sentence contains *multiple quantifiers*. Furthermore,
the order in which the quantifiers appear will greatly affect the meaning
of the sentence.

Some birds sing all the time. | All birds sing sometimes. |

birds
| time
b, t sings.b |
birds
, time
b | t sings.b |

Except for the order of quantifiers, both formal sentences are the same, yet the meaning of these two sentences is very different.

Practice Exercises |

**Negating Statements With Multiple Quantifiers**

Recall that, in the last review, the
negation of a "for all" statement is a "there exists"
statement, and vice versa. Then how do you negate a statement with more than
one quantifier? Try treating the statement like an algebraic expression:
break it down into manageable, logical components and use what you have already
learned to derive the negation. For instance, using an earlier example: *You
can fool some of the people all of the time, *start by deriving a formal
logic expression.

people
** p** | time

Breaking it down into component parts and negating the sentence as a whole should give you something that looks like this:

~(people
** p** (time

Now negate the outside "there exists" quantifier (remember, the negation of a "there exists" statement is a "for all" statement) producing:

people
** p** ~(time

Do the same thing again, this time to the inside quantified statement.

people
** p**, time

Practice Exercises |

**Universal Conditional Statements: **

Using the statement form

D, if P(x) then Q(x)x

the following are examples of other forms of universal conditional statements:

Contrapositive form

D, if ~Q(x) then ~P(x)x

Converse form

D, if Q(x) then P(x)x

Inverse

D, if ~P(x) then ~Q(x)x

Like the conditional statements presented in section 1.2, a universal conditional statement is logically equivalent to its contrapositive, but not to its converse or inverse forms.

Sufficient Condition

"

, m(x) is axsufficient conditionfor n()" means "xx, if m() then n(x)".x

(If m() occurs, then n(x) will happen.)x

Necessary Condition"

, m(x) is axnecessary conditionfor n()" means "x, if ~m(x) then ~n(x)".x

(If m() does not occur, then n(x) cannot happen.)x

Only If"

, n(x)xonly ifm(x)" means ", if ~m(xx) then ~n()" or "xx, if n() then m(x)".x

(m() is a necessary condition for n(x).)x

Practice Exercises |