1. CSNB234 ARTIFICIAL INTELLIGENCE
Chapter 3
Propositional Logic
& Predicate Logic
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2. Early Development of Symbolic Logic
English mathematician DeMorgan criticised traditional logic because it was written in natural language.
He thought that the formal meaning of a syllogistic statement was confused by the semantics of natural language.
DeMorgan and Boole both contributed to the development of Propositional Logic (or Propositional Calculus).
Using familiar algebraic symbols, they showed how certain algebraic rules were equally applicable to numbers, set and truth values of propositions.
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3. Propositional Logic (I)
Definition
Propositional Logic Sentences
Every propositional symbol and truth symbol is a sentence.
For example: true, P, Q, and R are four sentences
The negation of a sentence is a sentence
For example: P and  false are sentences
The conjunction (and) of two sentences is a sentence
For example: P  P is a sentence
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4. Propositional Logic (II)
Propositional Logic Sentences
The disjunction (or) of two sentence s is a sentence
For example: P  P is a sentence
The implication of one sentence for another is a sentence
For example: P  Q is a sentence
The equivalence of two sentences is a sentence
for example: P  Q = R is a sentence
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5. Propositional Logic Semantics
An interpretation of a set of propositions is the assignment of a truth value, either T of F, to each propositional symbol.
The interpretation or truth value for sentences is determined by:
The truth assignment of negation,  P, where P is any propositional symbol, is F if the assignment to P is T and T if the assignment to P is F.
The truth assignment of conjunction, , is T only when both conjuncts have truth value T; otherwise it is F.
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6. Propositional Calculus Semantics
The truth assignment of disjunction, , is F only when both conjuncts have truth value F; otherwise it is T.
The truth assignment of implication, , is F only when the premise or symbol before the implication is T and the truth value of the consequent or symbol after the implication is F; otherwise it is always T.
The truth assignment of equivalence, =, is T only when both expressions have the same truth assignment for all possible interpretations; otherwise it is F.
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7. Prove that ((PQR) = P  Q  R is a well-formed sentence in the propositional calculus.
Answer. Since:
P, Q and R are propositions and thus sentences
P  Q, the conjunction of two sentences, is a sentence
(P  Q) R, the implication of a sentence for another, is a sentence
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8. P, Q and R are propositions and thus sentences P and Q , the negation of two sentences, are sentences P  Q, the disjunction of two sentences, is a sentence P  Q  R, the disjunction of two sentences, is a sentence ((P  Q) R) = P  Q  R, the equivalence of two sentences, is a sentence
We get back
the
original
sentence
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9. Conclusion for the worked example
The above is our original sentence,
which has been constructed through a series of applications of legal rules and is therefore
well-formed.
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10. Constants refer to atomic propositions.
raining snowing eating hungry wet
Compound sentences capture relationships among propositions.
raining  snowing  wet
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11. The argument of a negation is called the target .
Negations: ¬ raining
The argument of a negation is called the target .
Conjunctions: (raining  snowing )
The arguments of a conjunction are called conjuncts .
Disjunctions: (raining  snowing )
The arguments of a disjunction are called disjuncts .
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12. Compound Sentences Implications: (raining  cloudy )
The left argument of an implication is the antecedent .
The right argument of an implication is called the consequent .
Reductions: cloudy  raining
The left argument of a reduction is the consequent .
The right argument of a reduction is called the antecedent .
Equivalences: raining  cloudy
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13. Rules of Algebraic Manipulation
Some Laws for Logic Use
x  y = y  x Commutativity
x  y = y  x
x  (y  z) = (x  y)  z Associativity
x  (y  z) = (x  y)  z
x  (y  z) = (x  y)  (x  z) Distributivity
x  (y  z) = (x  y)  (x  z)
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14. Semantics of Logical Operators
Negation:
Conjunction:
P P
T F
F T
P Q P  Q
T T T
T F F
F T F
F F F
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15. Semantics of Logical Operators
Disjunction:
P Q P  Q
T T T
T F T
F T T
F F F
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16. More Semantics of Logical Operators
Implication:
Reverse Implication:
Equivalence:
P Q P  Q
T T T
T F F
F T T
F F T
P Q Q  P
T T T
T F T
F T F
F F T
P Q Q  P
T T T
T F F
F T F
F F T
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17. Satisfaction
An interpretation i satisfies a sentence φ (written |=i φ ) if and only if φ i =T .
A sentence is satisfiable if and only if there is some interpretation that satisfies it.
A sentence is valid if and only if every interpretation satisfies it.
A sentence is unsatisfiable if and only if there is no interpretation that satisfies it.
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18. Truth Tables
A truth table is a table of all possible values for a set of propositional constants.
p q r
T T T
T T F
T F T
T F F
F T T
F T F
F F T
F F F
Each interpretation of a language is a row in the truth table for that language.
For a propositional language with n logical constants,there are 2 n interpretations.
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19. Logical Equivalence
Two sentences are logically equivalent if and only if they logically entail each other.
Examples:
¬(¬p) p
¬(p  q ) ¬p  ¬q de Morgan’s law
¬(p  q ) ¬p  ¬q de Morgan’s law
(p  q ) ¬p  q
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20. Problems
There can be many, many interpretations for a propositional language.
Remember that, for a language with n constants, there are 2n possible interpretations.
Sometimes there are many constants among premises that are irrelevant to the conclusion Much work wasted.
Solution: use other kind of proof theory,
such as refutation proof (later part)
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21. Example of validity: Problem to solve
Problem: what sentence is this (p  q) (q  r)?
Solution:
p q r (p  q) (q  r) (p q )  (q  r )
T T T T T T
T T F T F T
T F T F T T
T F F F T T
F T T T T T
F T F T F T
F F T T T T
F F F T T T
It is a valid sentence!
All
values
are
“true”
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22. Clausal Form
Propositional resolution works only on expressions in clausal form.
Fortunately, it is possible to convert any set of propositional calculus sentences into an equivalent set of sentences in clausal form.
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23. Conversion to Clausal Form
Implications Out:
P  Q Ø P Ú Q
P  Q P Ú Ø Q
P  Q (Ø P Ú Q) Ù (P Ú ØQ )
Negations In:
Ø Ø P P
Ø (P Ù Q) Ø P Ú Ø Q
Ø (P ÚQ ) Ø P Ù Ø Q
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24. Predicate Calculus (=Predicate Logic)
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25. Predicate Calculus (I)
In Proposition Logic, each atomic symbol (P, Q, etc) denotes a proposition of some complexity. There is no way to access the components of an individual assertion. Through inference rules we can manipulate predicate calculus expressions, accessing their individual components and inferring new sentences.
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26. Predicate Calculus (II)
In Predicate Calculus, there are two ways variables may be used or quantified. In the first, the sentence is true for all constants that can be substituted for the variable under the intended interpretation. The variable is said to be universal quantified.
Variables may also be quantified existentially. In this case the expression containing the variable is said to be true for at least one substitution from the domain of definition. Several relationships between negation and the universal and existential quantifiers are given below:
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27. Predicate Calculus (III)
Predicate calculus sentences
Every atomic sentence is a sentence
if s is a sentence, then so is its negation, s
if s1 and s2 are sentences, then so is their conjunction, s1  s2
if s1 and s2 are sentences, then so is their disjunction, s1  s2
if s1 and s2 are sentences, then so is their implication, s1  s2
if s1 and s2 are sentences, then so is their equivalence, s1 = s2
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28. Predicate Calculus (IV)
If X is a variable and s is a sentence, then X s is a sentence
If X is a variable and s is a sentence, then X s is a sentence
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29. English Sentences Represented In Predicate Calculus:
Some people like fried chicken.
X (people(X)  likes(X, fried_chicken)).
Nobody likes income taxes.
 X likes(X, income_taxes).
X  likes(X, income_taxes).
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30. All students passed examination. X (student(X)  pass_exam(X))
Rules:
All students passed examination.
X (student(X)  pass_exam(X))
All purple mushrooms are poisonous.
X (purple(X)  mushroom(X)  poisonous(X))
Facts:
Tom loves Jerry.
loves(tom, Jerry).
Ali is a student.
student(ali).
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31. Batman is knowledgeable and he is wealthy.
Exercise #1: Translate The Following English Statements Into Predicate Expressions
Batman is knowledgeable and he is wealthy.
All people that are not poor are happy.
Students who like to read books are intelligent.
Tweety can fly if it is not fried and has wings.
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32. Exercise #2 Everybody likes something.
There is something whom everybody likes.
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33. Answers to Exercise #1 Everybody likes something. "x.$y. likes(x,y)
There is something whom everybody likes.
$y."x. likes(x,y)
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34. Exercise #3   X p(X) =  X  p(X)  Y q(Y) =  Y  q(Y)
For predicates p & q, and variables X and Y:
Write the following in English
  X p(X) =  X  p(X)
 Y q(Y) =  Y  q(Y)
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35. Exercise #4: Convert each of the following predicate logic to English sentences
food(laksa)
X loves(X, superman)  loves(superman, X)
X food(X)  like(arul, X)
X eat(haswan, X)  eat(hasman, X)
X Y eat(X, Y)  alive(X)  food(Y)
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36. Exercise #5
Convert each of the following into Predicate Calculus equivalence:
Marcus was a man
Marcus was a Pompeian
All Pompeians were Romans
Caesar was a ruler
All Romans were either loyal to Caesar or hated him
Everyone is loyal to someone
people only try to assassinate rulers they are not loyal to
Marcus tried to assassinate Caesar
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37. Predicate logic for the 8 facts in Exercise #5
1. man(Marcus)
2. pompeian(Marcus)
3.  X. pompeian(X)  roman(X)
4. ruler(Caesar)
5.  X. roman(X)  loyalto(X, Caesar)  hate(X, Caesar)
6.  X. Y. loyalto(X,Y)
7.  X.  Y. person(X)  ruler(Y)  tryassassinate(X,Y)
  loyalto(X,Y)
8. tryassasinate(Marcus, Caesar)
9.  X. man(X)  person(X)
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38. Answers to Exercise #5 loyato(Marcus, Caesar)
(using 7, substitution, & apply M.P)
person(Marcus)  tryassassinate(Marcus, Caesar)
 ruler(Caesar)
using (4)
using (8)
person(Marcus)
(using 9, substitution & apply M.P)
man(Marcus)
using (1)
nil
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39. person(Marcus)  tryassassinate(Marcus, Caesar)  ruler(Caesar)
loyato(Marcus, Caesar)
(using 7, substitution, & apply M.P)
person(Marcus)  tryassassinate(Marcus, Caesar)  ruler(Caesar)
using (4)
person(Marcus)  tryassassinate(Marcus, Caesar)
using (8)
person(Marcus)
(using 9, substitution & apply M.P)
man(Marcus)
using (1)
nil

40. Towards the Resolution and Refutation Proof
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41. Stages involved in Proof Theory
convert all axioms into prenex form
i.e. all quantifiers are at the front
Stage 2
purge existential quantifiers
this process is known as skolemization
Stage 3
drop universal quantifiers
as they convey no information
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42. An Example Consider the arguments: All men are mortal (given premise)
Superman is a man (given premise)
Superman is mortal (goal to test)
The argument gets formalised as:
X man(X)  mortal(X)
man(Superman)
mortal(Superman) (goal)
And has, as its conflict set in Clausal form:
 man(X)  mortal(X) (1)
man(Superman) (2)
 mortal(Superman) (3)
Negation of goal
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43. Apply resolution to derive at a contradiction: We get:
man(Superman) from (1) & (3)
and,
direct contradiction from (2) & (4)
The conclusion is that “the goal is true”
(i.e. superman is mortal)
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