Download Lecture 2

PREDICATE LOGIC
Schaum’s outline chapter 4
Rosen chapter 1
September 8, 2016
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Contents
1
Predicates and quantifiers
2
Logical equivalences for quantifiers
3
Validity and satisfiability
4
Examples
ioc.pdf
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Next section
1
Predicates and quantifiers
2
Logical equivalences for quantifiers
3
Validity and satisfiability
4
Examples
ioc.pdf
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I601: Lecture 1
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3 / 25
Motivation
Propositional logic cannot adequately express the meaning of all
statements in mathematics and in natural language.
Example
Having assumptions:
Every computer connected to the school network is functioning properly.
ELVIS is one of the computers connected to the school network.
No rules of propositional logic allow us to conclude the truth of the statement
ELVIS is functioning properly
Classical example
From the propositions
All men are mortal.
Socrates is a man.
we cannot derive that
Socrates is mortal.
ioc.pdf
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Motivation
Propositional logic cannot adequately express the meaning of all
statements in mathematics and in natural language.
Example
Having assumptions:
Every computer connected to the school network is functioning properly.
ELVIS is one of the computers connected to the school network.
No rules of propositional logic allow us to conclude the truth of the statement
ELVIS is functioning properly
Classical example
From the propositions
All men are mortal.
Socrates is a man.
we cannot derive that
Socrates is mortal.
ioc.pdf
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I601: Lecture 1
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Predicate logic
Extends propositional logic by
Individuals a, b, ... , ELVIS, ...
(Individual) variables x, y , ...
Predicates (= propositional functions) P(x), Q(x), R(x, y ), ...
Quantifiers ∀, ∃
A propositional function is a generalization of proposition:
its argument stands for en element from its domain;
its value is T or F depending on the property of its argument(s).
ioc.pdf
Jaan Penjam, email: [email protected]
I601: Lecture 1
September 8, 2016
5 / 25
Predicate logic
Extends propositional logic by
Individuals a, b, ... , ELVIS, ...
(Individual) variables x, y , ...
Predicates (= propositional functions) P(x), Q(x), R(x, y ), ...
Quantifiers ∀, ∃
A propositional function is a generalization of proposition:
its argument stands for en element from its domain;
its value is T or F depending on the property of its argument(s).
ioc.pdf
Jaan Penjam, email: [email protected]
I601: Lecture 1
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Propositional function
Formally,
for a given n ∈ {0, 1, . . .}, a predicate is a n-ary function (=a function with n arguments)
of type
P : D1 × . . . × Dn → {T , F }
U = D1 × . . . × Dn is the domain (or the universe) of the predicate P;
the statement P(x1 , . . . , xn ), becomes a proposition (representing a property of its
arguments) if either x1 ∈ D1 , . . . , xn ∈ Dn or all variables x1 , . . . , xn are bound by quantifiers
(see next slides).
Example 1
Let P(x) denote ”x > 2” and U = Z. Then
P(5) is true;
P(2) is false.
Example 2
Let R(x, y , z) denote the statement ”x + y = z” and U = Z. Then
R(1, 2, 3) is true;
R(0, 0, −5) is false.
Jaan Penjam, email: [email protected]
ioc.pdf
I601: Lecture 1
September 8, 2016
6 / 25
Propositional function
Formally,
for a given n ∈ {0, 1, . . .}, a predicate is a n-ary function (=a function with n arguments)
of type
P : D1 × . . . × Dn → {T , F }
U = D1 × . . . × Dn is the domain (or the universe) of the predicate P;
the statement P(x1 , . . . , xn ), becomes a proposition (representing a property of its
arguments) if either x1 ∈ D1 , . . . , xn ∈ Dn or all variables x1 , . . . , xn are bound by quantifiers
(see next slides).
Example 1
Let P(x) denote ”x > 2” and U = Z. Then
P(5) is true;
P(2) is false.
Example 2
Let R(x, y , z) denote the statement ”x + y = z” and U = Z. Then
R(1, 2, 3) is true;
R(0, 0, −5) is false.
Jaan Penjam, email: [email protected]
ioc.pdf
I601: Lecture 1
September 8, 2016
6 / 25
Propositional function
Formally,
for a given n ∈ {0, 1, . . .}, a predicate is a n-ary function (=a function with n arguments)
of type
P : D1 × . . . × Dn → {T , F }
U = D1 × . . . × Dn is the domain (or the universe) of the predicate P;
the statement P(x1 , . . . , xn ), becomes a proposition (representing a property of its
arguments) if either x1 ∈ D1 , . . . , xn ∈ Dn or all variables x1 , . . . , xn are bound by quantifiers
(see next slides).
Example 1
Let P(x) denote ”x > 2” and U = Z. Then
P(5) is true;
P(2) is false.
Example 2
Let R(x, y , z) denote the statement ”x + y = z” and U = Z. Then
R(1, 2, 3) is true;
R(0, 0, −5) is false.
Jaan Penjam, email: [email protected]
ioc.pdf
I601: Lecture 1
September 8, 2016
6 / 25
Universal quantifier
Let U is a given domain of discourse (= universe of discourse or simply domain).
Definition
The universal quantification of P(x) is the statement
”P(x) for all values of x in the domain U .”
The notation ∀xP(x) denotes the universal quantification of P(x). Here ∀ is called the universal
quantifier. We read ∀xP(x) as ”for all x P(x)” or ”for every x P(x)”. An element for which P(x)
is false is called a counterexample of ∀xP(x).
ioc.pdf
Jaan Penjam, email: [email protected]
I601: Lecture 1
September 8, 2016
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Universal quantifier
Let U is a given domain of discourse (= universe of discourse or simply domain).
Definition
The universal quantification of P(x) is the statement
”P(x) for all values of x in the domain U .”
The notation ∀xP(x) denotes the universal quantification of P(x). Here ∀ is called the universal
quantifier. We read ∀xP(x) as ”for all x P(x)” or ”for every x P(x)”. An element for which P(x)
is false is called a counterexample of ∀xP(x).
Examples
Let P(x) be the statement ”x + 1 > x” and domain is Z. Then the quantification ∀xP(x)
is true.
Let Q(x) be the statement ”x < 2” and domain is Z. Then the quantification ∀xQ(x) is
false.
Let R(x) be the statement ”x 2 > 0” and domain is R. Then the quantification ∀xR(x) is
false. A counterexample here is x = 0.
ioc.pdf
Jaan Penjam, email: [email protected]
I601: Lecture 1
September 8, 2016
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Universal quantifier
Let U is a given domain of discourse (= universe of discourse or simply domain).
Definition
The universal quantification of P(x) is the statement
”P(x) for all values of x in the domain U .”
The notation ∀xP(x) denotes the universal quantification of P(x). Here ∀ is called the universal
quantifier. We read ∀xP(x) as ”for all x P(x)” or ”for every x P(x)”. An element for which P(x)
is false is called a counterexample of ∀xP(x).
Aanother example
Let P(x) be the statement ”x 2 > x” and domain is R. Then the quantification ∀xP(x) is
false. A counterexample is x = 12 .
However, if domain is Z, then the quantification ∀xP(x) is true.
ioc.pdf
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I601: Lecture 1
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Existential quantifier
Definition
The existential quantification of P(x) is the statement
”There exists an element x in the domain U such that P(x).”
We use the notation notation ∃xP(x) for the existential quantification of P(x). Here ∃ is called
the existential quantifier.
Remark: The meaning of ∃xP(x) changes when the domain changes. Without
specifying the domain, the statement ∃xP(x) has no meaning.
Examples
Let P(x) be the statement ”x > 3” and domain is R. Then the quantification ∃xP(x) is
true.
Let Q(x) be the statement ”x = x + 1” and domain is R. Then the quantification ∃xQ(x)
is false.
ioc.pdf
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Existential quantifier
Definition
The existential quantification of P(x) is the statement
”There exists an element x in the domain U such that P(x).”
We use the notation notation ∃xP(x) for the existential quantification of P(x). Here ∃ is called
the existential quantifier.
Remark: The meaning of ∃xP(x) changes when the domain changes. Without
specifying the domain, the statement ∃xP(x) has no meaning.
Examples
Let P(x) be the statement ”x > 3” and domain is R. Then the quantification ∃xP(x) is
true.
Let Q(x) be the statement ”x = x + 1” and domain is R. Then the quantification ∃xQ(x)
is false.
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Bound and free variables
The quantifiers are said to bind the variable x in the expressions ∀xP(x) and ∃xP(x).
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.
ioc.pdf
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I601: Lecture 1
September 8, 2016
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Bound and free variables
The quantifiers are said to bind the variable x in the expressions ∀xP(x) and ∃xP(x).
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.
ioc.pdf
Jaan Penjam, email: [email protected]
I601: Lecture 1
September 8, 2016
9 / 25
Bound and free variables
The quantifiers are said to bind the variable x in the expressions ∀xP(x) and ∃xP(x).
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.
Examples
In the statement ∃x(x + y = 1), the variable x is bound, but the variable y is free;
In the statement ∃x(P(x) ∧ Q(x)) ∨ ∀xR(x), all variables are bound
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Quantifiers as conjunctions and/or disjunctions
If the domain is finite then universal/existential quantifiers can be expressed by
conjunctions/disjunctions.
If the domain U = {1, 2, 3, 4}, then
I
∀xP(x) = P(1) ∧ P(2) ∧ P(3) ∧ P(4), and
I
∃xP(x) = P(1) ∨ P(2) ∨ P(3) ∨ P(4).
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.
ioc.pdf
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I601: Lecture 1
September 8, 2016
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Quantifiers as conjunctions and/or disjunctions
If the domain is finite then universal/existential quantifiers can be expressed by
conjunctions/disjunctions.
If the domain U = {1, 2, 3, 4}, then
I
∀xP(x) = P(1) ∧ P(2) ∧ P(3) ∧ P(4), and
I
∃xP(x) = P(1) ∨ P(2) ∨ P(3) ∨ P(4).
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.
ioc.pdf
Jaan Penjam, email: [email protected]
I601: Lecture 1
September 8, 2016
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Quantifiers as conjunctions and/or disjunctions
If the domain is finite then universal/existential quantifiers can be expressed by
conjunctions/disjunctions.
If the domain U = {1, 2, 3, 4}, then
I
∀xP(x) = P(1) ∧ P(2) ∧ P(3) ∧ P(4), and
I
∃xP(x) = P(1) ∨ P(2) ∨ P(3) ∨ P(4).
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.
Example
Let P(x) be the statement ”x 2 < 10”. Then it is true for the domain U1 = {1, 2, 3}, but it is
false for the domin U2 = {1, 2, 3, 4}.
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Examples of nested quantifiers
Statements with complex semantics require nested quantifiers.
Example 1
Statement: ”Every real number has an inverse w.r.t. addition”
Domain U = R. The property is expressed by
∀x∃y (x + y = 0)
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Examples of nested quantifiers
Statements with complex semantics require nested quantifiers.
Example 1
Statement: ”Every real number has an inverse w.r.t. addition”
Domain U = R. The property is expressed by
∀x∃y (x + y = 0)
Example 2
Statement: ”Every real number except zero has a multiplicative inverse.”
Domain U = R. The property is expressed by
∀x(x 6= 0 → ∃y (xy = 1))
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Notation: quantifiers with restricted domains
An abbreviated notation is often used to restrict the domain of a quantifier.
∀x < 0.(x 2 > 0) is another way of expressing ∀x(x < 0 → x 2 > 0).
√
√
∀x ∈ A.(0 < x 6 5) is another way of expressing ∀x(x ∈ A → (0 < x) ∧ (x 6 5)).
∃z > 0.(z 2 = 2) is another way of expressing ∃z(z > 0 ∧ z 2 = 2).
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Precedence of quantifiers
The quantifiers ∀ and ∃ have higher precedence than all logical operators
from propositional calculus
For example
∀xP(x) ∨ Q(x) means (∀xP(x)) ∨ Q(x) rather than ∀x(P(x) ∨ Q(x)).
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Next section
1
Predicates and quantifiers
2
Logical equivalences for quantifiers
3
Validity and satisfiability
4
Examples
ioc.pdf
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I601: Lecture 1
September 8, 2016
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Equivalences
Statements involving predicates and quantifiers are logically equivalent
if and only if they have the same truth value no matter which
predicates are substituted into these statements and which domain of
discourse is used for the variables in these propositional functions.
We use the notation S ≡ T to indicate that two statements S and T
involving predicates and quantifiers are logically equivalent.
Examples
∀x.¬¬S(x) ≡ ∀x.S(x)
∀x.(P(x) ∧ Q(x)) ≡ ∀x.P(x) ∧ ∀x.Q(x)
∀x.∀y .P(x, y ) ≡ ∀y .∀x.P(x, y )
∀x.∃y .P(x, y ) 6≡ ∃y .∀x.P(x, y )
ioc.pdf
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Equivalences
Statements involving predicates and quantifiers are logically equivalent
if and only if they have the same truth value no matter which
predicates are substituted into these statements and which domain of
discourse is used for the variables in these propositional functions.
We use the notation S ≡ T to indicate that two statements S and T
involving predicates and quantifiers are logically equivalent.
Examples
∀x.¬¬S(x) ≡ ∀x.S(x)
∀x.(P(x) ∧ Q(x)) ≡ ∀x.P(x) ∧ ∀x.Q(x)
∀x.∀y .P(x, y ) ≡ ∀y .∀x.P(x, y )
∀x.∃y .P(x, y ) 6≡ ∃y .∀x.P(x, y )
ioc.pdf
Jaan Penjam, email: [email protected]
I601: Lecture 1
September 8, 2016
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Equivalences
Statements involving predicates and quantifiers are logically equivalent
if and only if they have the same truth value no matter which
predicates are substituted into these statements and which domain of
discourse is used for the variables in these propositional functions.
We use the notation S ≡ T to indicate that two statements S and T
involving predicates and quantifiers are logically equivalent.
Examples
∀x.¬¬S(x) ≡ ∀x.S(x)
∀x.(P(x) ∧ Q(x)) ≡ ∀x.P(x) ∧ ∀x.Q(x)
∀x.∀y .P(x, y ) ≡ ∀y .∀x.P(x, y )
∀x.∃y .P(x, y ) 6≡ ∃y .∀x.P(x, y )
ioc.pdf
Jaan Penjam, email: [email protected]
I601: Lecture 1
September 8, 2016
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Equivalences
Statements involving predicates and quantifiers are logically equivalent
if and only if they have the same truth value no matter which
predicates are substituted into these statements and which domain of
discourse is used for the variables in these propositional functions.
We use the notation S ≡ T to indicate that two statements S and T
involving predicates and quantifiers are logically equivalent.
Examples
∀x.¬¬S(x) ≡ ∀x.S(x)
∀x.(P(x) ∧ Q(x)) ≡ ∀x.P(x) ∧ ∀x.Q(x)
∀x.∀y .P(x, y ) ≡ ∀y .∀x.P(x, y )
∀x.∃y .P(x, y ) 6≡ ∃y .∀x.P(x, y )
ioc.pdf
Jaan Penjam, email: [email protected]
I601: Lecture 1
September 8, 2016
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Equivalences
Statements involving predicates and quantifiers are logically equivalent
if and only if they have the same truth value no matter which
predicates are substituted into these statements and which domain of
discourse is used for the variables in these propositional functions.
We use the notation S ≡ T to indicate that two statements S and T
involving predicates and quantifiers are logically equivalent.
Examples
∀x.¬¬S(x) ≡ ∀x.S(x)
∀x.(P(x) ∧ Q(x)) ≡ ∀x.P(x) ∧ ∀x.Q(x)
∀x.∀y .P(x, y ) ≡ ∀y .∀x.P(x, y )
∀x.∃y .P(x, y ) 6≡ ∃y .∀x.P(x, y )
ioc.pdf
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Quantifications of two variables.
Example where ∀x.∃y .P(x, y ) 6≡ ∃y .∀x.P(x, y )
Let P(x, y ) denote”x + y = 0”, where the domain for all variables consists of all real numbers.
∃y .∀x.P(x, y ) means the proposition
”There is a real number y such that for every real number x, x + y = 0.
Actually, no matter what value of y is chosen, there is only one value of x for which
x + y = 0. Hence, the proposition ∃y .∀x.P(x, y ) false.
∀x.∃y .P(x, y ) means the proposition
”For every real number x there is a real number y such that x + y = 0.
In fact, given a real number x, there is a real number y such that x + y = 0; namely,
y = −x. Hence, the statement is true.
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Quantifications of two variables.
The meanings of the different possible quantifications involving two
variables.
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De Morgan’s law for quantifiers
The rules for negating quantifiers are:
¬(∀x.P(x)) ≡ ∃x.¬P(x)
¬(∃x.P(x)) ≡ ∀x.¬P(x)
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Next section
1
Predicates and quantifiers
2
Logical equivalences for quantifiers
3
Validity and satisfiability
4
Examples
ioc.pdf
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Validity
An assertion in predicate calculus is logically valid (or simply valid) if it is true in every
interpretation, that is iff it is true
for all domains
for every propositional functions substituted for the predicates in the assertion
Valid assertions in predicate logic play a role similar to tautologies in propositional logic.
Example
∀x.(P(x) ∨ ¬P(x))
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Validity
An assertion in predicate calculus is logically valid (or simply valid) if it is true in every
interpretation, that is iff it is true
for all domains
for every propositional functions substituted for the predicates in the assertion
Valid assertions in predicate logic play a role similar to tautologies in propositional logic.
Example
∀x.(P(x) ∨ ¬P(x))
ioc.pdf
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Satisfiability
An assertion in predicate calculus is satisfiable iff it is true
for some domain
for some propositional functions that can be substituted for the predicates in the assertion
Valid assertions in predicate logic play a role similar to tautologies in propositional logic.
Examples
∀x.∃y .P(x, y ) is satisfiable
The domain N, and the propositional function 6 satisfy this assertion.
∀x.(P(x) ∧ ¬P(x)) is unsatisfiable
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Satisfiability
An assertion in predicate calculus is satisfiable iff it is true
for some domain
for some propositional functions that can be substituted for the predicates in the assertion
Valid assertions in predicate logic play a role similar to tautologies in propositional logic.
Examples
∀x.∃y .P(x, y ) is satisfiable
The domain N, and the propositional function 6 satisfy this assertion.
∀x.(P(x) ∧ ¬P(x)) is unsatisfiable
ioc.pdf
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Next section
1
Predicates and quantifiers
2
Logical equivalences for quantifiers
3
Validity and satisfiability
4
Examples
ioc.pdf
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Example 1: If x is irrational then so is
√
x
Let Irrational(x) denote the propositional function ”x is irrational”, and
Rational(x) = ¬Irrational(x)
Proposition
√
∀x ∈ R+ . Irrational(x) → Irrational( x)
Proof. Let x be positive real number. We will show the contrapositive, i.e.
√
∀x ∈ R+ . Rational( x) → Rational(x)
√
In other words we prove that if x is rational then so is x.
√
Assume that x is a rational number. Then,
√ by definition, there must exists
two natural numbers m and n such that x = m/n. But then x = m2 /n2 and,
since m2 and n2 are natural numbers, which by definition implies that x is a
rational number as required.
ioc.pdf
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Example 2: n is even iff n2 is even
Let Even(x) denote the propositional function ”x is even”, and Odd(x) = ¬Even(x)
Proposition
∀n ∈ Z. Even(n) ↔ Even(n2 )
Proof. Let n ∈ Z
Necessity. Let’s assume that n is even, i.e. there exists an integer k such that
n = 2k. Then n2 = (2k)2 = 4k 2 = 2(2k 2 ), and thus for ` = 2k 2 , n2 = 2` and so
is even.
Sufficiency. We will show the contrapositive,i.e. Odd(n) → Odd(n2 ). Let’s
assume that n is odd, i.e. there exists an integer k such that n = 2k + 1. Then
n2 = (2k + 1)2 = 2(2k 2 + 2k) + 1,and thus for ` = 2k 2 + k, n2 = 2` + 1 and so is
odd.
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Example 3: 4k = i 2 − j 2
Proposition
∀k ∈ N. ∃i ∈ N. ∃j ∈ N. 4k = i 2 − j 2
Proof. Let k ∈ N. Let i = k + 1 and j = k − 1.
i 2 − j 2 = (k + 1)2 − (k − 1)2 = k 2 + 2k + 1 − k 2 + 2k − 1 = 4k.
ioc.pdf
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Example 4: Either n2 ≡ 0 (mod 4) or n2 ≡ 1 (mod 4)
Remark: Formula a ≡ b (mod 4) stands for the proposition ”b is the remainder after
division of a by 4.
Proposition
∀n ∈ Z. n2 ≡ 0 (mod 4) ∨ n2 ≡ 1 (mod 4)
Proof. Let n ∈ Z. n is either even or odd. We consider each case separately.
(1). Assume n is even. Then there exists m such that n = 2m. But then
n2 = 4m2 ≡ 0 (mod 4).
(2). Assume n is odd. Then there exists m such that n = 2m + 1. But then
n2 = 4m2 + 4m + 1 = 4(m2 + m) + 1 ≡ 1 (mod 4).
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