Title: CSNB234 ARTIFICIAL INTELLIGENCE
1CSNB234ARTIFICIAL INTELLIGENCE
Chapter 3 Propositional Logic Predicate Logic
(Chapter 2, pp. 45-76, Textbook) (Chapter 8, pp.
240-253, Ref. 3) Read online supplementary
slides
Instructor Alicia Tang Y. C.
2Early 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.
3Propositional 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
4Propositional 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
5Propositional Logic (III)
- 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.
6Propositional Logic (IV)
- 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.
7A Worked Example
- 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
8A Worked Example ..cont
- 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
9Conclusion 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.
10Constant Compound Sentences in Propositional
Logic
- Constants refer to atomic propositions.
- raining snowing eating hungry wet
- Compound sentences capture relationships among
propositions. - raining ? snowing ? wet
11Compound Sentences
- 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 .
12Compound 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
13Rules 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)
14Semantics of Logical Operators
P ?P T F F T
P Q P ? Q T T T T F F F T
F F F F
15Semantics of Logical Operators
P Q P ? Q T T T T F T F T
T F F F
16More Semantics of Logical Operators
Implication Reverse Implication Equ
ivalence
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
17Satisfaction
An interpretation i satisfies a sentence f
(written i f ) if and only if f 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.
18Truth 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.
19Logical Equivalence
Two sentences are logically equivalent if and
only if they logically entail each
other. Examples (p) p (p ? q ) p ? q
de Morgans law (p ? q ) p ? q de
Morgans law (p ? q ) p ? q
20Problems
- 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)
21Truth Tables
- The interpretation of any expression in
propostional logic can be specified in a truth
table. An example of a truth table is shown here
22Example of validity Problem to solve
Problem (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
23 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.
24Conversion 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
25Predicate Calculus(Predicate Logic)
26Predicate 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.
27Predicate 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
28Predicate 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
29Predicate 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
30English 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).
31Rule All purple mushrooms are poisonous. ?X
(purple(X) ? mushroom(X) ? poisonous(X)) Fact
Tom loves Jerry. loves(tom, Jerry).
32Quiz Translate the following English Statements
into Predicate Expressions
All people that are not poor and are intelligent
are happy. Students who like to read books are
not stupid. Batman is knowledgeable and he is
wealthy. Tweety can fly if it is not fried and
has wings.
33Exercise 1
Everybody likes something. There is something
whom everybody likes.
34Answers to Exercise 1
- Everybody likes something.
- "x.y. likes(x,y)
- There is something whom everybody likes.
- y."x. likes(x,y)
35Exercise 2
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)
36Answers to Exercise 2
37Quiz Convert each of the following predicate
logic to English sentences
- ?X loves(X, superman) ? loves(superman, X)
- food(laksa)
-
- ?X food(X) ? like(arul, X)
-
- ?X ?Y eat(X, Y) ? alive(X) ? food(Y)
- ?X eat(haswan, X) ? eat(hasman, X)
38Stages involved in Proof Theory
- Stage 1
- 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
39An Example
Consider the argument 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
40- 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)
41Exercise 3
- 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
42Predicate logic for the 8 facts in Exercise 3
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)
43Answers to Exercise 3
?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