#4 formal methods – predicate logic

Preparedby:SharifOmarSalem–ssalemg@gmail.com
Prepared by: Sharif Omar Salem – ssalemg@gmail.com
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In general Propositional Logic is not enough to
describe properties and its related specs.

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Variables
 A variable is a symbol that stands for an individual in a collection or set.
For example, the variable x may stand for one of the days. We may let x =
Monday or x = Tuesday, etc.
 A collection of objects is called the domain of a variable.
 For the above example, the days in the week is the domain of variable x.
 Months have 30 days.
 Domain or Set is Months of the year ≔ x
 Individuals or objects are Jan, Feb, …… Dec.
 Property or Predicate is “ has 30 days” ≔ P
 Predicate Formula P(x)
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Quantifiers:
 Quantifiers are phrases that refer to given quantities.
 Two kinds of quantifiers: Universal and Existential
Universal Quantifier: represented by 
 The symbol is translated as and means “for all”, “given any”, “for each,” or “for
every,” and is known as the universal quantifier.
 All Days have 24 hours.
 Predicate Formula (x)P(x)
Existential Quantifier: represented by 
 The symbol is translated as and means variously “for some,” “there exists,”
“there is a,” or “for at least one”.
 Some months has 30 days.
 Predicate Formula (x)P(x)
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Predicate
 It is the verbal statement that describes the property of a variable.
 Usually represented by the letter P, the notation P(x) is used to represent
some unspecified property or predicate that x may have
 e.g. P(x) = x has 30 days.
 P(April) = April has 30 days.
 The collection of objects that satisfy the property P(x) is called the domain
of interpretation.
 Truth value of expressions formed using quantifiers and predicates
 What is the truth value of (x)P(x)
 x is all the months
 P(x) = x has less than 32 days.
 The above formula is true since no month has 32 days.
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 Truth of expression (x)P(x)
1. P(x) is the property that x is yellow, and the domain of interpretation is the
collection of all flowers:
2. P(x) is the property that x is a plant, and the domain of interpretation is the
collection of all flowers:
3. P(x) is the property that x is positive, and the domain of interpretation
consists of integers:
 Can you find one interpretation in which both (x)P(x) is true and (x)P(x) is
false?
 Can you find one interpretation in which both (x)P(x) is true and (x)P(x) is
false?
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not true
not true
true
Case 1 as mentioned above
Not possible

 A predicate Formula for a single variable is known as unary
predicate
All days has 24 hours ≔ (x)P(x)
 A predicate Formula for two variables is known as Binary
predicate
For every University there exists a talent students .
≔ (x) (y) Q(x,y)
 A predicate Formula for N variables is known as N-ary predicate.
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 Formal definition: An interpretation for an expression involving
predicates consists of the following:
1. A collection of objects, called the domain of interpretation,
which must include at least one object.
2. An assignment of a property of the objects in the domain to
each predicate in the expression.
3. An assignment of a particular object in the domain to each
constant symbol in the expression.
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 Predicate wffs can be built similar to propositional wffs using logical
connectives with predicates and quantifiers.
 Examples of predicate wffs
 (x)[P(x)  Q(x)]
 (x) ((y)[P(x,y) V Q(x,y)]  R(x))
 S(x,y) Λ R(x,y)
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“Every person is nice”
can be rephrased as
“For any thing, if it is a person, then it is nice.”
P(x) ≔ “x is a person”
Q(x) ≔ “x is nice” the statement can be symbolized as
For any thing, if it is a person, then it is nice
(x) [ P(x)  Q(x) ]
 “All persons are nice” or “Each person is nice” will also have the
same symbolic formula.
  always related with  (implication)
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 “There is a nice person”
can be rewritten as
“There exists something that is both a person and nice.”
In symbolic form,
(x)[P(x) Λ Q(x)].
 Variations: “Some persons are nice” or “There are nice persons.”
  always related with Λ (conjunction)
 What would the following form mean for the example above?
(x)[P(x)  Q(x)] ???????
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 Example for forming symbolic forms from predicate symbols
 All dogs chase all rabbits ≔
For anything, if it is a dog, Then for any other thing, if it is a rabbit,
then the dog chases it ≔
(x)[ D(x)  (y)( R(y)  C(x,y) ) ]
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D(x) ≔ “x is a dog”
R(y) ≔ “y is a rabbit”
C(x,y) ≔ “x chases y”

 Some dogs chase all rabbits ≔
There is something that is a dog and for any other thing, if that thing is a rabbit,
then the dog chases it ≔
(x)[D(x) Λ (y)(R(y)  C(x,y) ) ]
 Only dogs chase rabbits ≔
For any things, If it chase rabbits, then it is a dog.
≔ For any things and for any other things if the other things is rabbits and
chases by the first thing, then that first thing is a dog
≔ For any two things, if one is a rabbit and the other chases it, then the other is
a dog
≔ (y) (x)[R(y) Λ C(x,y)  D(x)]
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D(x) ≔ “x is a dog”
R(y) ≔ “y is a rabbit”
C(x,y) ≔ “x chases y”

 Everything is fun ≔ (x)A(x)
 Negation will be “it is false that everything is fun,” ≔ [(x)A(x)]
 i.e. “something is non-fun.” ≔ (x)[A(x)]
 In symbolic form, [(x)A(x)] ↔ (x)[A(x)]
 Similarly negation of “Something is fun” ≔ (x)A(x)
 Negation will be “it is false that Something is fun,”
“Nothing is fun” ≔ [(x)A(x)]
 i.e. “Everything is boring.” ≔ (x)[A(x)]
 Hence, [(x)A(x)] ↔ (x)[A(x)]
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What is the negation of the following statements?
Some pictures are old and faded.
All people are tall and thin.
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Every picture is neither old nor faded.
Someone is short or fat.

 Write wffs that express the following statements:
 All players are good.
 Some good players, score goals.
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For anything, if it is a player, then it is good ≔ (x)[S(x)  I (x)]
There is something that is good and it is a player and it score goals
≔ (x)[I(x) Λ S(x) Λ M(x)]
 S(x)≔ x is a player
 I(x)≔ x is good
 M(x)≔ x scores goals

 Everyone who scores goals is a bad player.
 Only good player, scores goals.
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For anything, if that thing scores goals, then it is a player and it is not good
≔ (x)[ M(x)  S(x) Λ (I (x)) ]
For any thing, if it scores goals , then it is a player and it is good
≔ (x)(M(x)  S(x) Λ I(x))
 S(x)≔ x is a player
 I(x)≔ x is good
 M(x)≔ x scores goals

 Similar to a tautology of propositional logic.
 Truth of a predicate wff depends on the interpretation.
 A predicate wff is valid if it is true in all possible interpretations just like a
propositional wff is true if it is true for all rows of the truth table.
 A valid predicate wff is intrinsically true
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Truth values
True or false – depends on the truth
value of statement letters
True, false or neither (if the wff
has a free variable)
Intrinsic truth
Tautology – true for all truth values
of its statements
Valid wff – true for all
interpretations
Methodology
(Validity Porve)
Truth Table/Proof sequence using
rules
Proof sequence using
rules/others
Free Variable (x)[P(x,y)  (y) Q(x,y)]
variable y is not defined for P(x,y) hence y is called a free variable. Such
expressions might not have a truth value at all.

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 (x)P(x)  (x)P(x)
 This is valid because if every object of the domain has a certain property, then
there exists an object of the domain that has the same property.
 (x)P(x)  P(a)
 Valid – quite obvious since “a” is a member (object) of the domain of x.
 (x)P(x)  (x)P(x)
 Not valid since the property cannot be valid for all objects in the domain if it is
valid for some objects of than domain. Can use a mathematical context to
check as well.
 Say P(x) = “x is even,” then there exists an integer that is even but not every
integer is even.
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 (x)[P(x) V Q(x)]  (x)P(x) V (x)Q(x)
 Invalid, can prove by mathematical context by taking P(x) = x is even
and Q(x) = x is odd.
 In that case, the hypothesis is true but not the conclusion is false
because it is not the case that every integer is even or that every integer
is odd.
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P(x) ≔ x is a painter
L(x,y) ≔ x loves y
a ≔ Alice
b ≔ Bob
Alice is a painter. ≔ P(a)
Bob loves Alice ≔ L(b,a)
Alice loves Bob ≔ L(a,b)
Bob is not a painter ≔ ¬P(b) or [P(b)]˜
If Bobisapainterthen AlicelovesBob ≔ P(b)L(a,b)
BobisapainteronlyifAliceisnotapainter ≔ P(b) ¬P(a)

a ≔ Alice
b ≔ Bob
Everyone is a painter ≔ (x)P(x)
Someone is a painter ≔ (x)P(x)
Not everyone is a painter ≔ ¬(x)P(x)
No one is a painter ≔ (x) ¬P(x)
Everyone loves Bob ≔ (x)L(x,b)
Alice loves someone ≔ (x)L(a,x)
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W(x) ≔ x is a worker
a ≔ Alice
b ≔ Bob
Every painter loves Bob ≔ (x)[P(x)  L(xb)]
Some worker is a painter ≔ (x)[W(x) ∧ P(x)]
Alice loves every worker ≔ (x)[W(x)  L(ax)]
Some painters are not workers ≔ (x)[P(x) ∧ ¬W(x)]
No painters are workers ≔ (x)[P(x)  ¬W(x)]
Not every painter loves Alice ≔ ¬(x)[P(x)  L(x,a)]
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W(x) ≔ x is a worker
a ≔ Alice
b ≔ Bob
Every worker loves some painter ≔ (x){W(x)  (y)[P(y) ∧ L(x,y)]}
Some painter loves some worker ≔ (x){P(x) ∧ (y)[W(y) ∧ L(x,y)]}
No worker loves every worker
≔ (x){W(x) 
¬(y)[W(y)L(x,y)]}
EveryworkerwhoisalsoapainterlovesBob ≔ (x){[W(x) ∧ P(x)]  L(x,b)}
Some worker loves both Bob and Alice ≔ (x){W(x) ∧ [L(x,b) ∧ L(x,a)]}
EveryworkerwholovesBobalsolovesAlice ≔ (x){[T(x) ∧ L(x,b)]L(x,a)}
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 What is the truth of the following wffs where the domain consists of
integers:
(x)[L(x)  O(x)] where O(x) is “x is odd” and L(x) is “x < 10”?
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 What is the truth of the following wffs where the domain consists of
integers:
 (x)[L(x)  O(x)] where O(x) is “x is odd” and L(x) is “x < 10”?
 Using predicate symbols and appropriate quantifiers, write the
symbolic form of the following English statement:
 D(x) is “x is a day” M is “Monday” T is “Tuesday”
 S(x) is “x is sunny” R(x) is “x is rainy”
 Some days are sunny and rainy.
 It is always a sunny day only if it is a rainy day.
 It rained both Monday and Tuesday.
 Every day that is rainy is not sunny.
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 A valid argument for predicate logic need to be a tautology.
 The meaning and the structure of the quantifiers and predicates
determines the interpretation and the validity of the arguments
 Basic approach to prove arguments:
The same proof sequence methodology we use in propositional
logic. We use the same rules beside another four inference rules.
Four new inference rules
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Two rules to strip the qualifiers Two rules to reinsert the qualifiers

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From To Name/Appr. Conditions
(x)P(x) ⊢ P(t) ui ≔ Universal
Instantiation
- If t is a variable, it must not fall within the
scope of a quantifier for t
(x)P(x) ⊢ P(t) ei ≔ Existential
Instantiation
- Where t is not previously used in a proof
sequence.
- Must be the first rule used that introduces t.
P(x) ⊢ (x)P(x) ug ≔ Universal
Generalization
P(x) has not been
- deduced from any hypotheses in which x is a
free variable
- or deduced by ei from any wff in which x is a
free variable
P(x) ⊢ (x)P(x)
P(a) ⊢ (x)P(x)
eg ≔ Existential
Generalization
- To go from P(a) to (x)P(x), x must not appear
in P(a)
- Where t is a sub-variable or constant symbol (object).

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From Name/Appr.
(x)P(x) ⊢ P(t) ui ≔ Universal Instantiation
(x)P(x) ⊢ P(t) ei ≔ Existential Instantiation
P(x) ⊢ (x)P(x) ug ≔ Universal Generalization
P(x) ⊢ (x)P(x)
eg ≔ Existential Generalization
P(a) ⊢ (x)P(x)
R, R  S ⊢ S mp ≔ Modus Ponens
R  S, S ⊢ R’ mt ≔ Modus Tollens
R, S ⊢ R Λ S con ≔ Conjunction
R Λ S ⊢ R, S sim ≔ Simplification
R ⊢ R V S add ≔ Addition
P  Q, Q  R ⊢ P  R hs ≔ Hypothetical syllogism
P V Q, P ⊢ Q ds ≔ Disjunctive syllogism
(P Λ Q)  R ⊢ P  (Q R) exp ≔ Exportation
P, P ⊢ Q inc ≔ Inconsistency
P Λ (Q V R) ⊢ (P Λ Q) V (P Λ R) dist ≔ Distributive
P V (Q Λ R) ⊢ (P V Q) Λ (P V R) dist ≔ Distributive

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From Name/Appr.
R V S ↔ S V R comm ≔ Commutative
R Λ S ↔ S Λ R comm ≔ Commutative
(R V S) V Q ↔ R Λ (S Λ Q) ass ≔ Associative
(R Λ S) Λ Q ↔ R V (S V Q) ass ≔ Associative
(R V S) ↔ R Λ S (De-Morgan) ≔ De-Morgan’s Laws
(R Λ S) ↔ R V S (De-Morgan) ≔ De-Morgan’s Laws
R  S ↔ R V S imp ≔ Implication
R ↔ (R) dn ≔ Double Negation
PQ ↔ (P  Q) Λ (Q  P) equ ≔ Equivalence
Q  P ↔ P  Q cont≔ Contraposition
P ↔ P Λ P self ≔ Self-reference
P V P ↔ P self≔ Self-reference
Deduction Method
{P1 Λ P2 Λ ... Λ Pn  R  Q} ⊢ {P1 Λ P2 Λ ... Λ Pn Λ R  Q}

 Prove the following argument:
 All flowers are plants. Sunflower is a flower. Therefore, sunflower is
a plant.
 P(x) is “ x is a flower”
 a is a constant symbol (Sunflower)
 Q(x) is “x is a plant”
 The argument is (x)[P(x)  Q(x)] Λ P(a)  Q(a)
 The proof sequence is as follows:
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 Prove the argument (x)[P(x)  Q(x)] Λ [Q(y)]  [P(y)]
 Proof sequence:
 Prove the argument (x)P(x)  (x)P(x)
 Proof sequence:
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 Prove the argument (x)[P(x)  Q(x)] Λ (x)P(x)  (x)Q(x)
 Proof sequence:
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 Prove the argument
(x)[P(x) Λ Q(x)]  (x)P(x) Λ (x)Q(x)
 Proof sequence:
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(y)[P(x)  Q(x,y)]  [P(x)  (y)Q(x,y)]
 Proof sequence:
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 A temporary hypothesis can be inserted into a proof sequence.
If T is inserted as a temporary hypothesis
and eventually W is deduced from T and other hypotheses,
then the wff T  W has been deduced from other hypotheses
and must be reinserted into the proof sequence
 P1,P2,…..  Original hypothesis.
 T  Temporary hypothesis.
 W  the result of Implementing rules with T,P1,P2,…..
 Then the formula T  W must be reinserted into the proof
sequence
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[P(x)  (y)Q(x,y)]  (y)[P(x)  Q(x,y)]
 Proof sequence:
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 Prove the Formula
[(x)A(x)] ↔ (x)[A(x)]
 To prove equivalence, implication in each direction should be
proved
{[(x)A(x)] → (x)[A(x)]} Λ {[(x)A(x)] → (x)[A(x)]}
 Proof sequence for [(x)A(x)]  (x)[A(x)]
1. [(x)A(x)] hyp
2. A(x) temp. hyp
3. (x)A(x) 2, eg
4. A(x)  (x)A(x) temp. hyp discharged
5. [A(x)] 1, 4, mt
6. (x)[A(x)] 5, ug
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 Proof sequence for (x)[A(x)]  [(x)A(x)]
1. (x)[A(x)] hyp
2. (x)A(x) temp. hyp
3. A(a) 2, ei
4. [A(a)] 1, ui
5. [(x)[A(x)]] 3, 4, inc
6. (x)A(x)  [(x)[A(x)]] temp. discharged
7. [((x)[A(x)])] 1, dn
8. [(x)A(x)] 6, 7, mt
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 Every ambassador speaks only to diplomats, and some
ambassadors speak to someone. Therefore, there is a diplomat.
 Q: Prove the upper argument using Predicate Logic?
 Step 1: Determine Key statements and present by predicate logic
 A(x) ≔ x is an ambassador;
 D(y) ≔ y is a diplomat
 S(x,y) ≔ x speaks to y;
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 A(x) ≔ x is an ambassador
 S(x,y) ≔ x speaks to y
 Step 2: Convert verbal argument to predicate logic (hypothesis,
conclusion and form)
Every ambassador speaks only to diplomats
≔ For everyone and every others if this one is an ambassador
and he speaks to the other one, then that other one is a diplomat.
≔ (x) (y)[(A(x) Λ S(x,y))  D(y)]
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 A(x) ≔ x is an ambassador
 S(x,y) ≔ x speaks to y
Some ambassadors speak to someone
≔ There exist someone and some others that this one is an
ambassador and he speaks to others.
≔ (x)(y)(A(x) Λ S(x,y))
there is a diplomat
≔ (x)D(x)
Argument formula
(x) (y)[(A(x) Λ S(x,y))  D(y)] Λ (x)(y)(A(x) Λ S(x,y)) 
(x)D(x)
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Step 3: Prove the Predicate form using the prove sequence rules
(equivalence, inference and deduction)

 Every cat is bigger than every mouse. Tom is a cat. Jerry is either a cat or a
mouse. Tom is bigger than jerry . Therefore, Jerry is a mouse.
 “A student who is registered for this course is not eligible. Every student
registered for this course has pass the prerequisite. Therefore a student who
has pass the prerequisite is not eligible.”
 “All good players play to some famous team. Messi is a good player and he
plays to FC Barcelona. Therefore, FC Barcelona is a famous team.”
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 Propositional Logic is not enough to describe properties and its related specs.
 Predicate Logic Syntax are variables, quantifiers, predicate and connectives.
 A collection of objects is called the domain of a variable.
 Two kinds of quantifiers: Universal and Existential.
 The collection of objects that satisfy the property P(x) is called the domain of
interpretation.
•  always related with  (implication)
•  always related with Λ (conjunction)
• Negation of statements represented by the formula
 [(x)A(x)] ↔ (x)[A(x)]
 [(x)A(x)] ↔ (x)[A(x)]
 Bound variable is the variable within the scope of the quantifier. Free variable is
the one out of the scope.
 A predicate wff is valid if it is true in all possible interpretations
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- Prove a predicate formula means that it is tautology.
- Proof sequence Methodology is one of the methods used for proving predicate
formula.
- Proof rules from prepositional Logic used in predicate logic prove sequence in
addition to another four rules.
- A temporary Hyp. can be inserted as T then must be discharged as T implies W.
- Proving a verbal argument pass a three systematic steps
- Find the atomic predicates , translate argument to a predicate formula , then
use proof rules to prove the translated predicate formula.
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#4 formal methods – predicate logic

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