Discrete Mathematics, Chapter 1.4-1.

5:
Predicate Logic

Richard Mayr

University of Edinburgh, UK

Richard Mayr (University of Edinburgh, UK) Discrete Mathematics. Chapter 1.4-1.5 1 / 23

 Outline

1 Predicates

2 Quantifiers

3 Equivalences

4 Nested Quantifiers

Richard Mayr (University of Edinburgh, UK) Discrete Mathematics. Chapter 1.4-1.5 2 / 23

 Propositional Logic is not enough

Suppose we have:
“All men are mortal.”
“Socrates is a man”.
Does it follow that “Socrates is mortal” ?
This cannot be expressed in propositional logic.
We need a language to talk about objects, their properties and their
relations.

Richard Mayr (University of Edinburgh, UK) Discrete Mathematics. Chapter 1.4-1.5 3 / 23

 Predicate Logic

Extend propositional logic by the following new features.

Variables: x, y , z, . . .
Predicates (i.e., propositional functions):
P(x), Q(x), R(y ), M(x, y ), . . . .
Quantifiers: ∀, ∃.
Propositional functions are a generalization of propositions.
Can contain variables and predicates, e.g., P(x).
Variables stand for (and can be replaced by) elements from their
domain.

Richard Mayr (University of Edinburgh, UK) Discrete Mathematics. Chapter 1.4-1.5 4 / 23

 Propositional Functions

Propositional functions become propositions (and thus have truth

values) when all their variables are either
I replaced by a value from their domain, or
I bound by a quantifier
P(x) denotes the value of propositional function P at x.
The domain is often denoted by U (the universe).
Example: Let P(x) denote “x > 5” and U be the integers. Then
I P(8) is true.
I P(5) is false.

Richard Mayr (University of Edinburgh, UK) Discrete Mathematics. Chapter 1.4-1.5 5 / 23

 Examples of Propositional Functions

Let P(x, y , z) denote that x + y = z and U be the integers for all

three variables.
I P(−4, 6, 2) is true.
I P(5, 2, 10) is false.
I P(5, x, 7) is not a proposition.
Let Q(x, y , z) denote that x − y = z and U be the integers.
I P(1, 2, 3) ∧ Q(5, 4, 1) is true.
I P(1, 2, 4) → Q(5, 4, 0) is true.
I P(1, 2, 3) → Q(5, 4, 0) is false.
I P(1, 2, 4) → Q(x, 4, 0) is not a proposition.

Richard Mayr (University of Edinburgh, UK) Discrete Mathematics. Chapter 1.4-1.5 6 / 23

 Quantifiers
We need quantifiers to formally express the meaning of the words
“all” and “some”.
The two most important quantifiers are:
I Universal quantifier, “For all”. Symbol: ∀
I Existential quantifier, “There exists”. Symbol: ∃
∀x P(x) asserts that P(x) is true for every x in the domain.
∃x P(x) asserts that P(x) is true for some x in the domain.
The quantifiers are said to bind the variable x in these
expressions.
Variables in the scope of some quantifier are called bound
variables. All other variables in the expression are called free
variables.
A propositional function that does not contain any free variables is
a proposition and has a truth value.

Richard Mayr (University of Edinburgh, UK) Discrete Mathematics. Chapter 1.4-1.5 7 / 23

 Quantifiers: Example

Consider this formula

Φ = ∀y . ((P(y ) ∧ Q(x)) ∨ ∀x. (P(x) ∨ Q(y ) ∧ P(z)))

How many free variables does this formula contain? Clicker

1 One
2 Two
3 Three
4 Four
5 None

Richard Mayr (University of Edinburgh, UK) Discrete Mathematics. Chapter 1.4-1.5 8 / 23

 Universal Quantifier

∀x P(x) is read as “For all x, P(x)” or “For every x, P(x)”.

The truth value depends not only on P, but also on the domain U.
Example: Let P(x) denote x > 0.
I If U is the integers then ∀x P(x) is false.
I If U is the positive integers then ∀x P(x) is true.

Richard Mayr (University of Edinburgh, UK) Discrete Mathematics. Chapter 1.4-1.5 9 / 23

 Existential Quantifier

∃x P(x) is read as “For some x, P(x)” or “There is an x such that,

P(x)”, or “For at least one x, P(x)”.
The truth value depends not only on P, but also on the domain U.
Example: Let P(x) denote x < 0.
I If U is the integers then ∃x P(x) is true.
I If U is the positive integers then ∃x P(x) is false.

Richard Mayr (University of Edinburgh, UK) Discrete Mathematics. Chapter 1.4-1.5 10 / 23

 Uniqueness Quantifier

∃!x P(x) means that there exists one and only one x in the
domain such that P(x) is true.
∃1 x P(x) is an alternative notation for ∃!x P(x).
This is read as
I There is one and only one x such that P(x).
I There exists a unique x such that P(x).
Example: Let P(x) denote x + 1 = 0 and U are the integers.
Then ∃!x P(x) is true.
Example: Let P(x) denote x > 0 and U are the integers. Then
∃!x P(x) is false.
The uniqueness quantifier can be expressed by standard
operations. ∃!x P(x) is equivalent to
∃x (P(x) ∧ ∀y (P(y ) → y = x)).

Richard Mayr (University of Edinburgh, UK) Discrete Mathematics. Chapter 1.4-1.5 11 / 23

 Precedence of Quantifiers

Quantifiers ∀ and ∃ have higher precedence then all logical

operators.
∀x P(x) ∧ Q(x) means (∀x P(x)) ∧ Q(x). In particular, this
expression contains a free variable.
∀x (P(x) ∧ Q(x)) means something different.

Richard Mayr (University of Edinburgh, UK) Discrete Mathematics. Chapter 1.4-1.5 12 / 23

 Translating English to Logic

Translate the following sentence into predicate logic: “Every student in

this class has taken a course in Java.”
Solution:
First decide on the domain U.
Solution 1: If U is all students in this class, define a propositional
function J(x) denoting “x has taken a course in Java” and
translate as ∀x J(x).
Solution 2: But if U is all people, also define a propositional function
S(x) denoting “x is a student in this class” and translate
as ∀x (S(x) → J(x)).
Note: ∀x (S(x) ∧ J(x)) is not correct. What does it mean?

Richard Mayr (University of Edinburgh, UK) Discrete Mathematics. Chapter 1.4-1.5 13 / 23

 Equivalences in Predicate Logic

Statements involving predicates and quantifiers are logically

equivalent if and only if they have the same truth value for every
predicate substituted into these statements and for every domain
of discourse used for the variables in the expressions.
The notation S ≡ T indicates that S and T are logically equivalent.
Example: ∀x ¬¬S(x) ≡ ∀x S(x).

Richard Mayr (University of Edinburgh, UK) Discrete Mathematics. Chapter 1.4-1.5 14 / 23

 Quantifiers as Conjunctions/Disjunctions

If the domain is finite then universal/existential quantifiers can be

expressed by conjunctions/disjunctions.
If U consists of the integers 1,2, and 3, then
I ∀x P(x) ≡ P(1) ∧ P(2) ∧ P(3)
I ∃x P(x) ≡ P(1) ∨ P(2) ∨ P(3)
Even if the domains are infinite, you can still think of the
quantifiers in this fashion, but the equivalent expressions without
quantifiers will be infinitely long.

Richard Mayr (University of Edinburgh, UK) Discrete Mathematics. Chapter 1.4-1.5 15 / 23

 De Morgan’s Law for Quantifiers

The rules for negating quantifiers are:

¬∀x P(x) ≡ ∃x ¬P(x)
¬∃x P(x) ≡ ∀x ¬P(x)

Richard Mayr (University of Edinburgh, UK) Discrete Mathematics. Chapter 1.4-1.5 16 / 23

 Predicate Calculus

An assertion in predicate calculus is valid iff it is true

I for all domains
I for every propositional functions substituted for the predicates in the
assertion.
An assertion in predicate calculus is satisfiable iff it is true
I for some domain
I for some propositional functions that can be substituted for the
predicates in the assertion.
Example: ∀x (F (x) ↔ G(x)) is not valid, but satisfiable.
Example: ∀x (F (x) ↔ ¬F (x)) is unsatisfiable.

Richard Mayr (University of Edinburgh, UK) Discrete Mathematics. Chapter 1.4-1.5 17 / 23

 Nested Quantifiers

Complex meanings require nested quantifiers.

“Every real number has an inverse w.r.t. addition.”
Let the domain U be the real numbers. Then the property is
expressed by
∀x ∃y (x + y = 0)
“Every real number except zero has a multiplicative inverse.”
Let the domain U be the real numbers. Then the property is
expressed by
∀x (x 6= 0 → ∃y (x ∗ y = 1))

Richard Mayr (University of Edinburgh, UK) Discrete Mathematics. Chapter 1.4-1.5 18 / 23

 Thinking of Nested Quantification
Nested Loops
I To see if ∀x ∀y P(x, y ) is true, loop through the values of x:
I At each step, loop through the values for y .
I If for some pair of x and y , P(x, y ) is false, then ∀x ∀y P(x, y ) is
false and both the outer and inner loop terminate.
∀x ∀y P(x, y ) is true if the outer loop ends after stepping through
each x.
I To see if ∀x ∃y P(x, y ) is true, loop through the values of x:
I At each step, loop through the values for y .
I The inner loop ends when a pair x and y is found such that P(x, y )
is true.
I If no y is found such that P(x, y ) is true the outer loop terminates
as ∀x ∃y P(x, y ) has been shown to be false.
∀x ∃y P(x, y ) is true if the outer loop ends after stepping through
each x.
If the domains of the variables are infinite, then this process can
not actually be carried out.
Richard Mayr (University of Edinburgh, UK) Discrete Mathematics. Chapter 1.4-1.5 19 / 23
 Order of Quantifiers

Quantifiers can be grouped into blocks

∀x∀y . . . ∀z ∃a∃b . . . ∃c ∀u∀v . . . ∀w . . . . . .

Quantifiers can be swapped inside a block, but not between blocks.

Let P(x, y ) denote x + y = y + x and U be the real numbers.
Then ∀x ∀y P(x, y ) is equivalent to ∀y ∀x P(x, y ).
Let Q(x, y ) denote x + y = 0 and U be the real numbers. Then
∀x ∃y P(x, y ) is true, but ∃y ∀x P(x, y ) is false.

Richard Mayr (University of Edinburgh, UK) Discrete Mathematics. Chapter 1.4-1.5 20 / 23

 Quantifications of Two Variables

Statement When True? When False

P(x,y) is true for There is a pair x, y
every pair x,y. for which P(x,y) is
false.
For every x there is There is an x such
a y for which P(x,y) that P(x,y) is false
is true. for every y.
There is an x for For every x there is
which P(x,y) is true a y for which P(x,y)
for every y. is false.
There is a pair x, y P(x,y) is false for
for which P(x,y) is every pair x,y
true.

Richard Mayr (University of Edinburgh, UK) Discrete Mathematics. Chapter 1.4-1.5 21 / 23

 Negating Nested Quantifiers
Let P(x, f ) denote that person x has taken flight f .
Let Q(f , a) denote that flight f is operated by airline a.
Formulate: “There is no person who has taken a flight on every airline
in the world.”
¬∃x ∀a ∃f (P(x, f ) ∧ Q(f , a))
Now use De Morgan’s Laws to move the negation as far inwards as
possible.

¬∃x ∀a ∃f (P(x, f ) ∧ Q(f , a))

∀x ¬∀a ∃f (P(x, f ) ∧ Q(f , a)) by De Morgan’s for ∃
∀x ∃a ¬∃f (P(x, f ) ∧ Q(f , a)) by De Morgan’s for ∀
∀x ∃a ∀f ¬(P(x, f ) ∧ Q(f , a)) by De Morgan’s for ∃
∀x ∃a ∀f (¬P(x, f ) ∨ ¬Q(f , a)) by De Morgan’s for ∧

Can you translate the result back into English?

“For every person there is an airline such that for all flights, this person
has not taken that flight or that flight is not operated by this airline.”
Richard Mayr (University of Edinburgh, UK) Discrete Mathematics. Chapter 1.4-1.5 22 / 23
 An Example from Calculus

Express that the limit of a real-valued function f at point a is L.

lim f (x) = L
x→a

In predicate logic

∀ ∃δ ∀x (|x − a| < δ → |f (x) − L| < )

where the domain of  and δ are the positive real numbers and the
domain of x are all real numbers.
Now express its negation, i.e., that limx→a f (x) 6= L.

∃ ∀δ ∃x (|x − a| < δ ∧ |f (x) − L| ≥ )

Richard Mayr (University of Edinburgh, UK) Discrete Mathematics. Chapter 1.4-1.5 23 / 23