Logic

ARTIFICIAL

INTELLIGENCE

LECTURE # 03

Artificial Intelligence 2012 Lecture 03 Delivered By Zahid Iqbal 1

Review of Last Lecture


Todays Lecture

Review of last lecture

Reasoning

Types of Reasoning

Logic


Reasoning

Reasoning is the process of deriving logical conclusions

from given facts.

Durkin defines reasoning as the process of working with knowledge, facts and problem solving strategies to draw

conclusions.

Throughout this section, you will notice how representing

knowledge in a particular way is useful for a particular

kind of reasoning.


Deductive reasoning

As the name implies, is based on deducing new

information from logically related known information.

A deductive argument offers assertions that lead

automatically to a conclusion, e.g.

If there is dry wood, oxygen and a spark, there will be a fire

Given: There is dry wood, oxygen and a spark

We can deduce: There will be a fire.

All men are mortal. Socrates is a man.

We can deduce: Socrates is mortal


Inductive Reasoning

Inductive reasoning is based on forming, or inducing a

generalization from a limited set of observations, e.g.

Observation: All the crows that I have seen in my life are black.

Conclusion: All crows are black


Comparison of deductive and inductive

reasoning

The inductive reasoning is as follows: By experience, every time I have let a ball go, it falls downwards.

Therefore, I conclude that the next time I let a ball go, it

will also come down.

The deductive reasoning is as follows: I know Newton's Laws. So I conclude that if I let a ball go, it will certainly

fall downwards.

Thus the essential difference is that inductive reasoning is

based on experience,

while deductive reasoning is based on rules, hence the

latter will always be correct.


Analogical Reasoning

Analogical reasoning works by drawing analogies

between two situations, looking for similarities and

differences, e.g.

when you say driving a truck is just like driving a car, by

analogy you know that there are some similarities in the

driving mechanism,

But you also know that there are certain other distinct

characteristics of each.


Common-sense Reasoning

Common-sense reasoning is an informal form of

reasoning that uses rules gained through experience or

what we call rules-of-thumb.

It operates on heuristic knowledge and heuristic rules.


Non-Monotonic Reasoning

Non-Monotonic reasoning is used when the facts of the

case are likely to change after some time, e.g.

Rule:

IF the wind blows

THEN the curtains sway

When the wind stops blowing, the curtains should sway

no longer.

However, if we use monotonic reasoning, this would not

happen. The fact that the curtains are swaying would be

retained even after the wind stopped blowing.


Logic

Algebra is a type of formal logic deals with number

PROPOSITIONAL LOGIC

PREDICATE CALCULUS/LOGIC


Proposition

A proposition (p, q, r, ) is simply a statement (i.e., a declarative sentence) with a definite meaning, having a truth value thats either true (T) or false (F)

Normally, a proposition is named e.g. P, Q, R etc.

Propositional Logic is the logic of compound statements built from simpler statements using Boolean connectives.


Proposition

A proposition is a statement about the world that may be either true or false.

Examples of propositions (properly formed statements): Alis car is blue. Seven plus six equals twelve. (7 + 6 = 12) Amjad is Alis uncle.

Each of the sentences is a proposition - not to be broken

down into its constituent parts. i. e., we simply assign true, say, to Amjad is Alis uncle. Artificial Intelligence 2012 Lecture 03 Delivered By Zahid Iqbal 13

Examples of non- propositions Alis uncle Seven plus four Whos there? (interrogative, question) Just do it! (imperative, command) 1 + 2 (expression with a non-true/false value)

Because we cannot assign truth value to them.


Propositional Symbols

Propositions are denoted by propositional symbols such as: P, Q, R, S,.

Truth symbols are: true (or T), false (or F).

Single propositions by themselves are not very interesting.

We need to express complex propositions/compound propositions: The book is on the table or it is on the chair.

If Socrates is a man then he is mortal.


Propositional Symbols

We can use logical connecters such as: ...and [conjunction] ...or [disjunction] ...implies [implication / conditional] ..is equivalent [biconditional] ...not [negation]

Sentences in the propositional calculus are formed from these atomic

symbols according to the syntax rules.


Operators / Connectives

An operator or connective combines one or more operand expressions into a larger expression. (E.g., + in numeric exprs.)

Unary operators take 1 operand (e.g., -3);

Binary operators take 2 operands (eg 3 4).

Propositional or Boolean operators operate on propositions or truth values instead of on numbers.


The Negation Operator

The unary negation operator (NOT) transforms a prop. into its logical negation.

E.g. If p = I have brown hair. then p = I do not have brown hair. Truth table for NOT:


p p

T F

F T

The Conjunction Operator

The binary conjunction operator (AND) combines two propositions to form their logical conjunction.

E.g. If p=I will have salad for lunch. and q=I will have steak for dinner., then pq=I will have salad for lunch and I will have steak for dinner.

Conjunction Truth Table


p q pq

F F F

F T F

T F F

T T T

The Disjunction Operator

The binary disjunction operator (OR) combines two propositions to form their logical disjunction.

Example:

p=That car has a bad engine. q=That car has a bad carburetor.

pq=Either that car has a bad engine, or

that car has a bad carburetor.


The Disjunction Operator

Note that pq means that p is true, or q is true, or both are true!

So this operation is also called inclusive or, because it includes the possibility that both p and q are true.

Disjunction Truth Table


p q pq

F F F

F T TT F TT T T

Examples

Example: BCS AI Class P = Ali is the teacher Q = Saira is the student R= AI is a course teaching in BS

P ^ Q = Ali is the teacher and Saira is the student. Q ^ R= Saira is the student and tought AI in BS

The book is on the table or it is on the chair. If Socrates is a man then he is mortal.


A Simple Exercise

Let p=It rained last night, q=The sprinklers came on last night, r=The lawn was wet this morning.

Translate each of the following into English:

p q ^ r r p r p q


The Exclusive Or Operator

The binary exclusive-or operator (XOR) combines two propositions to form their logical exclusive or.

p = I will earn an A in this course, q = I will drop this course, p q = I will either earn an A for this course, or I will drop it

(but not both!)


Exclusive-Or Truth Table

Note that pq means that p is true, or q is true, but not both!

This operation is called exclusive or, because it excludes the possibility that both p and q are true.


p q pq

F F F

F T T

T F T

T T F

The Implication Operator

The implication p q states that p implies q. It is FALSE only in the case that p is TRUE but q is FALSE. E.g., p=I am elected.

q=I will lower taxes. p q = If I am elected, then I will lower taxes

Its premise or antecedent is p and its conclusion or consequent

is q


Implication Truth Table

p q is false only when

p is true but q is not true.

Examples:

If 1+1=2, then I am richer than Bill Gates. True or False?

If the moon is made of green cheese, then I am richer than Bill Gates. True or False?


p q pq

F F T

F T T

T F F

T T T

The Biconditional Operator

The biconditional p q states that p is true if and only if (IFF) q is true.

It is TRUE when both p q and q p are TRUE. p = It is raining. q = The home team wins. p q = If and only if it is raining, the home team wins.


Biconditional Truth Table

p q means that p and q have the same truth value.

Note this truth table is the exact opposite of s! p q means (p q)

p q does not imply p and q are true, or cause each other.


p q p q

F F T

F T F

T F F

T T T

Truth Table

30

p q p q p q p p q p q p q

F F F F T T T T

F T F T T T T F

T F F T F F F F

T T T T F T T T

Precedence of Logical Operators


Precedence of Logical Operators


Operator Precedence

Some Alternative Notations


Name: not and or xor implies iffPropositional logic: Boolean algebra: p pq + C/C++/Java (wordwise): ! && || != ==C/C++/Java (bitwise): ~ & | ^Logic gates:

Propositional Calculus Sentences (Syntax)

Every propositional symbol and truth symbol is a sentence.

e. g., true, P, R.

The negation of a sentence is a sentence.

e. g., ~P, ~false

The conjunction of two sentences is a sentence.

e. g., P Q, P Q

The disjunction of two sentences is a sentence.

e. g., Q R

The implication of one sentence for another is a sentence.

e. g., P Q

The equivalence of two sentences is a sentence

e. g., P Q = R


Exercise

Fact 1: Saira likes cakes. = P

Fact 2: Saira eats cakes. = Q

P Q, PQ, Q, P Q, P Q ????????


Exercise

Fact 1: Saira likes cakes. = P

Fact 2: Saira eats cakes. = Q

P Q, PQ, Q, P Q, P Q ????????

PQ : Saira Likes cakes or eats cakes.

PQ : Saira likes cakes and eats cakes.

Q : Saira does not eat cakes.

PQ: If Saira likes cakes then he eats cakes.

PQ:Saira eats cakes if and only if he likes cakes.


Limitations of Propositional logic

We Cant describe things in terms of their properties or relationships (very limited expressive power)

Propositional logic is declarative Propositional logic is compositional: meaning of B1,1 P1,2 is derived from meaning of B1,1 and of

P1,2 We cant express rules or generalizations

If the train is late and there are no taxis, john is late for the meeting If trains are late and there are no taxis, anyone traveling by trains is late

for the meeting


Limitations

Propositions can only represent knowledge as complete sentences, e.g.

a = the balls color is blue. Cannot analyze the internal structure of the sentence.

No quantifiers are available, e.g. for-all, there-exists Propositional logic provides no framework for proving statements

such as: All humans are mortal All women are humans Therefore, all women are mortals

This is a limitation in its representational power.


References

Artificial Intelligence: Structures and Strategies for

Complex Problem Solving

Internet


End of Lecture


Puzzle Game

A farmer went to market and purchased a fox, a goose, and a bag of beans. On his way home, the farmer came to the bank of a river and hired a boat. But in crossing the river by boat, the farmer could carry only himself and a single one of his purchases - the fox, the goose, or the bag of the beans.

If left alone, the fox would eat the goose, and the goose would eat the beans.

The farmer's challenge was to carry himself and his purchases to the far bank of the river, leaving each purchase intact. How did he do it?


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