Predicate Logic

Predicate Logic is the Bases of all the Logic used in Formal Methods in Software Engineering

1. Logic & Formal Methods
Predicate Logic
Instructor: Dr H. Farooq Ahmad
Sarmad Sadik
TA: Muhammad Afzal, Maqbool
Reference: Discrete Mathematics with Examples by
Simpson

2. Topics
•Universal quantification
•Existential quantification
•Satisfaction and validity
•The negation of quantifiers
•Free and bound variables
•Substitution
•Restriction
•Uniqueness
•Equational reasoning
•Natural deduction
•One point rule
Page 2

3. Pros and cons of propositional
logic
•Propositional logic is declarative
•Propositional logic allows conjunction, disjunction,
negated information
•Propositional logic is compositional:
•meaning of B
1,1
 P
1,2
is derived from meaning of B
1,1
and of
P
1,2
•Meaning in propositional logic is context-independent
(unlike natural language, where meaning depends on context)
•Propositional logic has very limited expressive power
(unlike natural language)

4. First-order logic
•Whereas propositional logic assumes the world
contains facts,
•first-order logic (like natural language) assumes the
world contains
–Objects: people, houses, numbers, colors, baseball
games, wars, …
–Relations & functions: red, round, prime, brother
of, bigger than, part of, comes between, father of,
best friend, one more than, plus, …

5. Syntax of FOL: Basic elements
•ConstantsKingJohn, 2, NUS,...
•FunctionsSqrt, LeftLegOf,...
•Variablesx, y, a, b,...
•Connectives, , , , 
•Equality =
•Quantifiers , 

6. Predicates
•A predicate is a proposition whose truth
depends on the value of one or more
variables.
•For example, “n is a perfect square” is a
predicate whose truth depends on the value
of n.
•A function like notation is used to denote a
predicate supplied with specific variable
values. P(n)=“n is a perfect square”
•P(4) is true and P(5) is false.
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7. Universal Quantification
•Universal Quantification denoted as
•Universal Quantification allows us to capture
statements of the form “for all” or “for every”.
•For example, “every natural number is
greater than or equal to zero” can be written
formally as

0:  nNn
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8. Universal Quantification
•A quantified statement consists of three
parts: the quantifier, the quantification,
which describes the variables and the type of
variable with which the statement is
concerned and the predicate which is
normally some statement about the
quantified variables.
•Every predicate logic statement can be
considered as follows
predicate tion quantifica quantifier 
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9. Universal Quantification
nmnmnmNnm :,
nmnmnmNnNm  :;:
nmnmnmNnNm  ::
small is d d owns p small is p:;:  DogdPeoplep
Page 9

10. Exercise
•Everybody likes Jaffa cakes
•All vegetarians don’t like Jaffa cakes
•Everybody either likes Jaffa cakes or is
a vegetarian
•Either every body likes Jaffa cakes or
everybody is a vegetarian
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11. Solution
n) vegetariaisx People:x(cakes) Jaffa likes x People:x(
n) vegetariaisx cakes Jaffa likes(x People:x
cakes)) Jaffa likes(x n vegetariais(x People:x
cakes Jaffa likes x People:x




Page 11

12. Universal Quantification
•Law 1
•Example
):()):():(( qpXxqXxpXx 
strong) is p tallis pPerson:p(
strong)) is pPerson:p( tall)is pPerson:p((


Page 12

13. Universal Quantification
•Law 2
•Example
)):():(():( qXxpXxqpXx 
strong) is p tallis pPerson:p(
strong)) is pPerson:p( tall)is pPerson:p((


Page 13

14. Existential quantification
•Existential quantifier denoted by
•As universal quantification is used to
assert that a certain property holds of
every element of a set, existential
quantification is used to assert that such a
property holds of some (or at least)
elements of a set
•“Some natural numbers are divisible by 3”
may be written as

03 modn N:n 
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15. Exercise
•Some people like Jaffa cakes
•Some vegetarians don’t like Jaffa cakes
•Some people either like Jaffa cakes or
are vegetarian
•Either some people like Jaffa cakes or
some people are vegetarian
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16. Solution
n) vegetariaisx People:x(cakes) Jaffa likesx People:x(
n) vegetariais x cakes Jaffa likes(x People:x
cakes) Jaffa likes(x n vegetariaisx People:x
cakes Jaffa likesx People:x




Page 16

17. Existential quantification
•Law 3
•Example
)):():(():( qXxpXxqpXx 
)))(:())(:((
))()(:(
csmallCarccfastCarc
csmallcfastCarc


Page 17

18. Existential quantification
•Law 4
•Example
)):():(():( qXxpXxqpXx 
)))(:())(:((
))()(:
csmallCarccfastCarc
csmallcfastCarc


Page 18

19. Satisfaction and validity
•The predicate n>3 can be considered neither true
nor false unless we know the value associated with
n
•A predicate p is valid if and only if it is true for all
possible values of the appropriate type. That is, if a
predicate p is associated with a variable x of type X,
then p is valid if, and only if,
•Example
trueis :pXx
true toequivalent is 0nN:n as validis 0n predicate The 
Page 19

20. Satisfaction and validity
•A predicate p is satisfiable if and only if it is
true for some values of the appropriate type.
That is, if a predicate p is associated with a
variable x of type X, then p is satisfiable if,
and only if ,
•Example
trueis :pXx
true toequivalent is
0nN:n as esatisfiabl is 0n predicate The 
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21. Satisfaction and validity
•A predicate p is unsatisfiable if, and only if, it
is false for all possible values of the
appropriate type.
•If a predicate p is associated with a variable x
of type X, then p is unsatisfiable if, and only
if,
false is :pXx
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22. Satisfaction and validity
•The analogies between valid, satisfiable and
unsatisfiable predicates, and tautologies,
contingencies and contradictions:
•valid predicates and tautologies are always
true
•satisfiable predicates and contingencies are
sometimes true and sometimes false
•unsatisfiable predicates and contradictions
are never true
Page 22

23. The negation of quantifiers
•The statement “some body like Brian” may
be expressed via predicate logic as
•To negate this expression, we may write as,
•which in natural language may be expressed
as “nobody likes Brian”
Brian likes : pPersonp 
Brian likes : pPersonp 
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24. The negation of quantifiers
•Logically saying “nobody likes Brian” is equivalent to
saying “everybody does not like Brian”.
•The negation of quantifiers behaves exactly in this
fashion, just as in natural language, “nobody likes
Brian” and “everybody does not like Brian” are
equivalent so in predicate logic
•And
•are equivalent.
Brian) likes :( pPersonp 
Brian) likes (: pPersonp 
Page 24

25. The negation of quantifiers
•Law 5
•When negation is applied to a quantified
expression it flips quantifiers as it moves
inwards(i.e negation turns all universal
quantifiers to existential quantifiers and vice
versa, and negates all predicates)
pXxpXx  :):(
pXxpXx  :):(
Page 25

26. The negation of quantifiers
•Consider the statement,
•The effect of negation of this expression can
be determined as,
),,(::: zyxpZzYyXx 
),,(::: zyxpZzYyXx 
),,(::: zyxpZzYyXx 
),,(::: zyxpZzYyXx 
),,(::: zyxpZzYyXx 
Page 26

27. Exercise
•Everyone likes everyone
•Everyone likes someone
•Someone likes everyone
•Someone likes someone
Page 27

28. Free and bound variables
•Consider two predicates
•n>5…..Eq. 1
•In equation 2, the variable n is bound, it
is existentially quantified variable.
•In equation 1, n is free as its value may
not be determined or restricted
.....Eq.25:  nNn
Page 28

29. Free and bound variables
•Example
•In this statement, y and x are two variables. y is
universally quantified and it is bound, while x is free
variable.
•In statement,
•The occurrence of x in the declaration is binding, any
occurrences of x in p are bound, and any occurrences
of any variables other than x are free
richard likes x robin) likesy People:y( 
pXx:
Page 29

30. Free and bound variables
•Example
•Exercise, Determine the scope of the universal and
existential quantifiers of the following predicate,
•The distinction between predicates and propositions can
be stated as predicates are logical statements which can
contain free variables while propositions are logical
statements which contain no such holes
3)0:(  xxyxNx
)):(:( rqYypXx 
Page 30

31. Substitution
•Consider statement,
•we may replace the name n with any term t,
provided that t is of the same type as n. This process
is called substitution: the expression p[t/n] denotes
the fact that the term t is being substituted for the
variable name n in the predicate p.
•Example, we may denote the substitution of 3 for n
in the predicate n>5 by
5:  nNn
]/3[5nn
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32. Substitution
•Also, we may denote the substitution of 3+4
for n in n>5 by
n>5[3+4/n]
•Substitution is only for free variables
Page 32

33. Substitution
•More than one substitution in case of more than one free
variable
•Consider predicate,
•Here x and y are both free variables. We may substitute
nigel for x as follows,
•Now substituting ken for y,
sad isy happy isx 
sad isy happy is nigel
/x]sad)[nigel isy happy is(x


sad isken happy is nigel
]sad)[ken/y isy happy is (nigel


Page 33

34. Substitution
•Similarly, in one step, overall substitution may be
denoted as,
•These two substitutions take place sequentially,
first nigel is substituted for x, then ken is
substituted for y. However, for simultaneous
substitutions,
/x][ken/y]sad)[nigel isy happy is(x 
ken/y]/x,sad)[nigel isy happy is(x 
Page 34

35. Substitution
•To illustrate the difference between sequential and
simultaneous substitution, consider the example,
•In former there is only one occurrence of y and in latter
there are two occurrences of y
783
]/8,/)[73(


y
yxyyx
true
yyy
yxyyx




7838
]/8[73
]/8][/)[73(
Page 35

36. Restriction
•Filters role in predicate logic
•Considering an example, all natural numbers
which are prime numbers and are greater
than 2 are odd, may be written as,
•Using a filter, statement may be written as,
•Form of restricted predicates as,
odd isn 2)n prime is(n N:n 
odd isn 2n prime isn |N:n 
predicate n restrictio |tion quantifica quantifier 
Page 36

37. Restriction
•Restriction can also be used in existential
quantification, “there has been a female
British Prime Minister”, may be written as
Minister PrimeBritish a waspfemale is p|People:p 
predicatenrestrictio|tionquantifica universal 
predicate n restrictio tion quantifica universal 
odd isn 2n prime isn |N:n 
odd isn 2)nprime is(n N:n 
Page 37

38. Restriction
•Law 6
•Law 7
t)pX:x(t)p|X:( x
):()|:( tpXxtpXx 
Page 38

39. Exercise
•Represent the following statement in the
form,
•Everyone who likes cheese likes spam
•There is a person who likes cheese that
likes spam
•Everyone who likes cheese also likes spam
and likes bananas
predicate n restrictio |tion quantifica quantifier 
Page 39

40. Solution
bananas likes pspam likes pcheese likes p|People:p
spam likes pcheese likes p|People:p
spam likes pcheese likes p|People:p



Page 40

41. Uniqueness
•Existential quantifier, allows us to represent
statements such as “there is at least one x,
such that…”.
•In order to be more specific, consider, “there
is exactly one x, such that…”
•Consider an example statement,
it is the case that the predicate x+1=1 is true
for exactly one natural number:0
11xN:x 
Page 41

42. •The operator which allow us to state that “
there is exactly one natural number, x, is
denoted as
•As an example,
1

11:
1
 xNx
Uniqueness
richest a is cCountry:c
1

Page 42

43. Uniqueness
•Assuming the statement
•It is sometimes useful to pinpoint exactly which
element of X for which predicate p holds. Also, it
is convenient to perform some operation on this
value.
•The µ operator allows us to do exactly this, the
statement (µx:X | p) is read as “ the unique x
from set X such that p holds of x”
•Example, (µx:N | x+1=1) where µ is associated
with value 0
pXx:
1
Page 43

44. Uniqueness
•When a unique element of the relevant type does possess
the relevant property, then we are free to construct
statements as given previously
•On the other hand, µ expressions that are not associated
with a unique element generate an undefined value.
•As an example, the µ expressions
(µn:N | n is even)
and
(µc:Country | population (c) > 10,000,000)
are both undefined
Page 44

45. Uniqueness
•We can incorporate terms into set
comprehensions, so we can apply terms in µ
expressions. Expressions of the form,
•return the result of applying the term t to the
unique element of the set X which satisfies the
predicate p
•gives the value Washington
)|:( tpXx 
)capital(c) richest is c |Country :c( 
Page 45

46. Uniqueness
•If there is no unique element which possesses
the relevant property, then the result is
undefined
•and
•are both undefined.
)n even isn | N:n (
2

)capital(c)10,000,000 (c) population |Country :c ( 
Page 46

47. Exercise
•Write the following in terms of µ
expressions
–The tallest mountain in the world
–The height of the tallest mountain
–The oldest person in the world
–The nationality of the oldest person in the
world
–The smallest natural number
Page 47

48. Solution
m))nN:m(|N:n(
y(p))nationalitin worldoldest is p|People:p(
in worldoldest is p|People:p(
height(m))in world tallest is m, Mountain, :m(
in world) tallest is m|Mountain:m(








Page 48

49. Equational Reasoning
•Two methods of reasoning
–Equational reasoning
–Natural deduction
•Also,
tpX:x
tp |X:x
tpX:x
tp | X:x




Page 49

50. Equational Reasoning
•Law 8
If a predicate holds for all elements of a set,
then it holds for some of them
•Law 9
If a predicate holds for exactly one element
of a set, then it holds for at least one of them
pXxpXx  ::
pXxpXx  ::
1
Page 50

51. Equational Reasoning
•Law 10
If p holds for all elements of X and t is a term of type
X, then p holds for t
Example,
The predicate
holds for all natural numbers. The term 3+4 is of type N
and as such,
holds.
p(t)X)tp(x)X:x(  
odd isx 2xprime(x) 
odd is 432434)prime(3 
Page 51

52. Equational Reasoning
•Law 11
This law states that if t is a term of type X
and p holds of t, then we can conclude that
p holds for some elements of X
•Example
The proposition 7εN is true and furthermore
7 is prime. As such,
))(:())(( xpXxtpXt 
prime isn N:n
Page 52

53. Equational Reasoning
•The quantification of any variables with regards to propositions
which are always true or always has absolutely no effect. This
notion is captured by following laws,
•Law 12
•Law 13
•Law 14
•Law 15
truetrue)X:x( 
falsefalse)X:x( 
truetrue)X:x( 
falsefalse)X:x( 
Page 53

54. Equational Reasoning
•The predicate is concerned
with three variables: x, y and z.
•x is bound and y and z are free. As x does not
appear in the predicate y=z, we are free to
assume that the existential quantification of x
has no effect on the truth or falsity of that
predicate.
•We conclude that following equivalence holds.
zyxpXx
zyxpXx


))(:(
))((:
zyp(x)X:x 
Page 54

55. Equational Reasoning
•Law 16
It states that as x does not appear free in q,
the following holds
•Example,
)):(():( qpXxqpXx 
)10)3:(()103:(  lnNnlnNn
Page 55

56. Equational Reasoning
•Law 17
It states that as x does not appear free in q,
the following holds,
•Example
)):(():( qpXxqpXx 
)10)3:(()103:(  lnNnlnNn
Page 56

57. Equational Reasoning
•Law 18
It states as x does not appear free in q, the
following holds
•Example
)):(():( qpXxqpXx 
10)lspam) likes pPerson:p((
10)lspam likes pPerson:p(


Page 57

58. Equational Reasoning
•Law 19
It states as x does not appear free in q, the
following holds
•Example
)):(():( qpXxqpXx 
10)lspam) likes pPerson:p((
10)l spam likes pPerson:p(


Page 58

59. Equational Reasoning
•Example: A theorem in predicate logic
•Proof:
):():( qpXxqpXx 
law) onal(propositi qpX:x
law) onal(propositi )(:
5) (law )(:
:




qpXx
qpXx
qpXx
Page 59

60. Equational Reasoning
•Example Theorem
18) (law qpX:x
3) (law qpX:x
8) (law q)X:x(p)X:x(
5) (law q)X:x(p)X:x(
law) onal(propositi q)X:x(p)X:x(
q))X:x(p)X:x((
q)pX:x(q))X:x(p)X:x((







Page 60

61. Exercise
)):():(():( qXxpXxqpXx 
Page 61

62. Solution
18) (law q)X:x(p)X:x(
5) (law ):():(
3) (law ):()X:x(
law) onal(propositi :
:





qXxpXx
qXxp
qpXx
qpXx
Page 62

63. Natural deduction
•Natural deduction can also be extended to
predicate logic
•Natural deduction rules for universal
quantification
•Rule1 states that if p holds for all values of X,
then provided that t is of type X- p holds for t
elim) (
)/(
x t:


xtp
pXx 
Page 63

64. Natural deduction
•Example:
All natural numbers which are prime and greater than
2 are odd. 7 is prime and greater than 2 and we may
conclude that it is odd.
elim) (
)7(
N7 27prime(7) odd(n)2nprime(n)|N:n


odd

Page 64

65. Natural deduction
•Rules for existential quantification
If xεX such that p holds for x, then if we
are able to define further predicate, q in
which x does not appear free, then we
may conclude q
Page 65
q
qpXx  |:

66. Natural deduction
•Intro rule
•If p is true for any term t of set X, then we
conclude that, holdspXx:
intro) (
pX:x
p[t/x] Xt



Page 66
intro) (
prime(n)N:
prime(7) N7

n


67. Natural deduction
•Example
intro) (
qp|X:x
q[t/x] p[t/x] Xt



Page 67

68. The one-point rule
•It states that p is a person that likes Emily, but also
that the name of person is Rick
•We can conclude that rick ε Person and rick likes
emily are both true
•Law 20
X t p[t/x]
txpX:x
p(t/x)) x(t t)xpX:x(


 
rickpemily likes pPerson:p 
Page 68

69. The one-point rule
•Example

txpX:x
Xtp[t/x]
car red a ownsbecky Personbecky
beckyp car red a owns pPerson:p
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beckyp car red a owns pPerson:p
Personbeckycar red a ownsbecky
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70. References
•Discrete Mathematics with Examples
by Simpson
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71. Thank you!
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