MODULE 1- RULES OF

INFERENCE

INTRODUCTION

• An argument in propositional logic is a sequence of propositions. • All but the final proposition in the argument are called

premises and the final proposition is called the conclusion. • An argument is valid if the truth of all its premises

implies that the conclusion is true.

• An argument form in propositional logic is a sequence of compound propositions involving propositional variables. • An argument form is valid no matter which particular

propositions are substituted for the propositional variables in its premises, the conclusion is true if the premises are all true.

VALID ARGUMENTS IN PROPOSITIONAL LOGIC• Example:• “If you have a current password, then you can log

onto the network.”• “You have a current password.”Therefore,• “You can log onto the network.”

Use p to represent “You have a current password” and q to represent “You can log onto the network.” Then, the argument has the form

p → qp∴ q

RULES OF INFERENCE FOR PROPOSITIONAL LOGIC

EXAMPLE

• State which rule of inference is the basis of the following argument: “It is below freezing now. Therefore, it is either below freezing or raining now.”

• State which rule of inference is the basis of the following argument: “It is below freezing and raining now. Therefore, it is below freezing now.”

• State which rule of inference is used in the argument:• If it rains today, then we will not have a barbecue

today. If we do not have a barbecue today, then we will have a barbecue tomorrow. Therefore, if it rains today, then we will have a barbecue tomorrow

USING RULES OF INFERENCE TO BUILD ARGUMENTS• When there are many premises , several rules of inference

are often needed to show that given argument is valid.• Example is as follows:• Show that the premises “It is not sunny this afternoon and it

is colder than yesterday,” “We will go swimming only if it is sunny,” “If we do not go swimming, then we will take a canoe trip,” and “If we take a canoe trip, then we will be home by sunset” lead to the conclusion “We will be home by sunset.”

• Show that the premises “If you send me an e-mail message, then I will finish writing the program,” “If you do not send me an e-mail message, then I will go to sleep early,” and “If I go to sleep early, then I will wake up feeling refreshed” lead to the conclusion “If I do not finish writing the program, then I will wake up feeling refreshed.”

• Show that the following argument is valid. If today is Tuesday, I have test on mathematics or economics. If my economics professor is sick, I will not have test in economics. Today is Tuesday and my economics professor is sick. Therefore I have a test in mathematics

RESOLUTION PRINCIPLE

• A variable or a negation of a variable is called a literal• A disjunction of literal is called a sum and conjunction

of literals is called product• For any two clauses C1 and C2, if there is a literal L1 in

C1 i.e complementary to a literal L2 in C2, delete L1 and L2 from C1 and C2 respectively and construct the disjunction of the remaining clauses. This constructed clause is a resolvent of C1 and C2

RESOLUTION PRINCIPLE (FORMAL DEFINITION)• Given S set of clauses , a resolution or deduction of C

from S is finite sequence C1, C2, … Ck of such clauses such that each C1 is either is a clause in S or a resolvent of the clause preceding C and Ck=C. A deduction of empty [] is called refutation or proof of S

EXAMPLE

• Show that the following argument is valid using resolution principle. If today is Tuesday, I have test on mathematics or economics. If my economics professor is sick, I will not have test in economics. Today is Tuesday and my economics professor is sick. Therefore I have a test in mathematics

FALLACIES

• Type of incorrect reasoning• Is the following argument valid?• If you do every problem in this book, then you will

learn discrete mathematics. You learned discrete mathematics. Therefore, you did every problem in this book.

RULES OF INTERFERENCE FOR QUANTIFIED STATEMENTS

UNIVERSAL INSTANTIATION

• P ( C ) is true, when c is the member of domain• Eg: all women are wise, that “lisa is wise” where lisa is

the member of domain women.

UNIVERSAL GENERALIZATION• We show that the quantification is true by taking a

arbitrary value C from the domain and showing that P (C ) is true.• Lisa is wise , where lisa is from domain women . Then• All women are wise

EXISTENTIAL INSTANTIATION

• Concludes that there is an element c in the domain which P © is true , if we know the quantification is true

EXISTENTIAL GENERALIZATION• Conclude that quantifiction is true when particular

element c with P (C ) is true is known

EXAMPLE

• Show that the premises “Everyone in this discrete mathematics class has taken a course in computer science” and “Marla is a student in this class” imply the conclusion “Marla has taken a course in computer science.”

• Show that the premises “A student in this class has not read the book,” and “Everyone in this class passed the first exam” imply the conclusion “Someone who passed the first exam has not read the book.”