
Predicate: A sentence that contains a finite
number of variables and becomes a statement
when values are substituted for the variables.
◦ Domain: the set of all values that can be substituted for
the variables
◦ EX: P(x): x^2 > x (“^” also means exponentiation in CS)
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Sets
◦
◦
◦
◦
xεA denotes that x is an element of set A
Common Sets: R(real), Z(integers), Q(rationals)
R⁺ and R⁻ indicate positive or negative elements only
{1,2,3…} indicates a set of all + integers using “…”
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The set of all elements that make a predicate
true is the truth set
◦ {x ε D | P(x)} – D is the domain of x
◦ This is read “the set of all x in D such that P(x)”
◦ EX: Q(n) is “n is a factor of 8”
 If the domain is Z⁺, the truth set is {1,2,4,8}
 If the domain is Z, {1,2,4,8,-1,-2,-4,-8}

Universal Statement
◦ “∀x ε D, Q(x)” is read “for all x in D, Q(x)”
◦ ∀ is the universal quantifier, pronounced “for all”
◦ A universal statement is true iff it is true for every x
in D and is false if just one x in D is false
 This one false x is a counterexample
◦ EX: ∀x ε R, x^2 ≥ x
 2^2 > 2, but (1/2)^2 < (1/2) so it’s false

Existential Statement
◦ “∃x ε D such that Q(x)” means “there exists an x in
D such that Q(x)”
◦ ∃ - Existential Quantifier
◦ This is true iff at least one x satisfies Q(x)
◦ EX: ∃x ε D such that x^2 = x
 1^2 = 1 and is the only value besides 0 that is true

Universal Conditional Statements
◦ “∀x, P(x) → Q(x)” means “for all x, if P(x) is true,
Q(x) follows
◦ A statement of this form can always be translated
to “∀x ε D, Q(x)” by making D the set of truth
values for P(x)
◦ The same general rule applies for Existential
statements
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Negation of Quantified Statements
◦ ~(All x in D, Q(x)) = There exists at least one x in D
such that ~Q(x)
 ~(∀x ε D, Q(x)) = ∃x ε D such that ~Q(x)
 The negation of “All Are” is logically equivalent to
“Some Are Not”
◦ ~(∃x ε D such that Q(X)) = ∀x ε D, ~Q(x)
 The negation of “Some Are” is logically equivalent to
“All Are Not”
◦ Caution: when translating to informal language,
check the placement of “not” to ensure no
ambiguity

Negations of Universal Conditional
Statements
◦ ~(∀x, P(x) → Q(x)) = ∃x such that (P(x)^~Q(x))
 ~(for all x, if P(x) then Q(x)) = there exists an x such
that P(x) is true and Q(x) is false
 This is a culmination of a lot of information. Basically ,
this means that if a universal conditional statement is
false, then there is an element that makes the
statement false.

∀, ∃, ^, v
◦ ∀x ε D, Q(x) = Q(x₁)^Q(x₂)^…^Q(xn)
 A for all statement is logically equivalent to an and
chain of all possible Q(x)’s
◦ ∃x ε D such that Q(x) = Q(x₁) v Q(x₂)v…vQ(xn)
 A there exists statement is logically equivalent to an or
chain of all possible Q(x)’s
◦ This is why the negation of for all is there exists
and vise versa

Variants of Universal Conditional Statements
◦ Given ∀x in D, P(x) → Q(x)
◦ Contrapositive: ∀x, ~Q(x) → ~P(x)
 Logically Equivalent to original
◦ Converse: ∀x, Q(x) → P(x)
 Not Logically Equivalent
◦ Inverse: ∀x, ~P(x) → ~Q(x)
 Not Logically Equivalent

When ∀ and ∃ are both used
◦ The quantifiers are co-equal and are evaluated
from left to right
 ∀x ε D, ∃y ε S such that P(x,y)
 For any x in D, a value y in S can be found to satisfy the
predicate P(x,y) (given x, choose y)
 ∃x ε D such that ∀y ε E, P(x,y)
 There exists an x in D such that any y in E satisfies P(x,y)
(choose x, then given y)

Limits
◦ “Lim(n → ∞) an = L” means that as n approaches
infinity, the value a[n] approaches L
◦ For any positive number ε, there exists an integer Z
such that whenever n > Z, a sub n sits between
L- ε and L+ ε.
◦ Symbolically: ∀ε > 0, ∃ an integer Z such that
∀integers n, if n > Z then L- ε < an < L+ ε
 This definition is logically complex, but can be easily
tested with values

Negations of Multiply Quantified Statements
◦ For any number of quantifiers, the same basic rules
can be applied
 ~(∀x in D, ∃y in E such that P(x,y) = ∃x in D such that
∀y in E, ~P(x,y)
 ~(∃x in D such that ∀y in E, P(x,y) = ∀x in D, ∃y in E
such that ~P(x,y)
 These two equivalencies were obtained using the
negations of quantified statements introduced earlier

The Order of Quantifiers
◦ By reversing the order of quantifiers ∀ and ∃, it is
possible to reverse the truth values of the
statement (this is not always so)
◦ If two of the same quantifiers are next to each other
and are reversed, the meaning is left unchanged
 ∀reals x and ∀reals y, x + y = y + x. This is true both
ways
 ∀ real numbers x and y,… is the exact same statement

The Language of First-Order Logic
◦ The collection of all symbols for quantifiers,
connectives, variables, etc. is not just a bunch of
garble. It’s a language through which most
situations can be described. There are many
dialects for many different situations but the same
general rule applies to each: you can’t learn it if you
don’t use it.
◦ It’s important that you understand the symbols
used in logic because they are the golden standard
in mathematics; be it a proof, a solution or some
other kind of process, the language of logic will be
used to describe it.

Did You Know?
◦ An entire programming language called Prolog
(programming in logic) was developed for logical
manipulation
◦ Using symbolic manipulation, Prolog can solve all
your logical problems for you if you know how to
use it
◦ This course is not language specific, but it may be
worth looking into

Rule of Universal Instantiation
◦ If something is true for everything in a domain,
then it is true for any particular thing in the domain
◦ If something is true for a type or class of objects,
then it is true for any particular object of the types
or class
 This lets you apply general rules and theorems to
specific objects without changing or having to prove
the theorem used

Universal Modus Ponens
◦ ∀x, if P(x) then Q(x): P(a) for a particular a :: Q(a)
◦ If x satisfies P(x) then it satisfies Q(x). It is known
that “a” satisfies P(a) so Q(a) must be true
 This is a valid – and very useful – argument form

Universal Modus Tollens
◦ ∀x, if P(x) then Q(x):~Q(a) for a particular a :: ~P(a)
◦ If x satisfies P(x) then it satisfies Q(x). It is known
that “a” fails Q(a) so it must also fail P(a)

Proving Validity of Arguments with Quantified
Statements
◦ An argument form is valid regardless of the
predicates used iff the conclusion is true when the
premises are all true
◦ An argument is valid iff its argument form is valid
◦ A form can be proven true using other forms,
theorems, and diagrams like truth tables
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The Use of Diagrams
◦ Picture each domain in an argument as a disk
 If a particular domain exists within a larger domain
then place the particular disk into the larger disk
 EX: all integers are real numbers. Draw a disk for reals and
then draw an integer disk inside the reals disk
 If something is not on the larger disk, it can’t be on
the smaller one either
 If a number is not a real number, then it cannot be an
integer
 If a number is an integer, then it is also a real number

More on Diagrams
◦ One must be careful to ensure
that the diagrams are fit
together properly. Both the
converse and inverse errors can
be miss-interpreted
diagrammatically to say that the
form is valid, when in fact the
form is abstract. In general,
diagrams are helpful when used
to represent argument forms.

Quantified Converse Error
◦ ∀x, if P(x) then Q(x): Q(a) for a particular a:: P(a)
 This is invalid
 EX: P(x) = “x is an integer”, Q(x) = “x is a real number”
 If Q(x) is true, P(x) doesn’t necessarily follow because x
could be .5, which is real, but is not an integer

Quantified Inverse Error
◦ ∀x, if P(x) then Q(x): ~P(a) for a particular a :: ~Q(a)
 Invalid form
 EX: P(x) = “x is an integer”, Q(x) = “x is a real number”
 If x is .5, P(x) is false but Q(x) is still true. The truth of Q(x)
didn’t follow from the truth of P(x)

Obtaining Additional Argument Forms
◦ In order to transform an established argument form
into its universal equivalent, combine each
statement in the form with Universal Instantiation
◦ EX: Universal Transitivity
 ∀x, P(x)→Q(x): ∀x, Q(x)→R(x) :: ∀x, P(x)→R(x)
 Anything that makes P(x) true makes Q(x) true,
anything that makes Q(x) true makes R(x) true,
anything that makes P(x) true must make R(x) true

Abduction
◦ Sometimes, converse and inverse errors are useful
 ∀x, P(x)→Q(x): Q(x) :: P(x) is a converse error, but is
not totally useless. It can be used to search for a
logically correct statement
 Say that a doctor knows two things:
 If someone has chicken pox, they will have red bumps
 A patient complains that he has red bumps
 It is not valid to say that the patient has chicken pox, but
the doctor knows that he can perform a test that will
prove chicken pox is true
 Thanks to the converse error, the doctor was able to
narrow his search. This comes in handy when designing
expert systems and artificial intelligence

In the world of mathematics, proving your
statements is an essential skill. Don’t believe
us?
◦ Say an engineer is designing a new type of engine
for a commercial plane. According to his
calculations, the engine will not malfunction under
standard conditions. In testing, this fact may seem
to be true, but if his calculations were not proven
valid, there will always be a lingering uncertainty.
◦ When people’s lives depend on your correctness,
you have to be absolutely sure that you are in fact
correct