Lecture 2 - Radford University



Lecture 6 Logic (Cont.)

Goals and Objectives:

• Understand rules of inference

• Understand how to use rules of inference to draw conclusions

Read section 1.5, p.56-73

Methods of Proof

Definition. Axioms or postulates are accepted assumptions about mathematical structures.

Definition. Rules of inference are rules that used to draw assertions based on some axioms(postulates), hypotheses and other assertions.

Definition. A theorem is a statement that can be shown to be true based

on some axioms(postulates) and hypotheses using rule of inference.

Definition. A sequence of statements that form an argument to show a

theorem is called a proof.

Definition. A lemma is a simple theorem used in the proofs of other

theorem.

Definition. A corollary is a proposition that can be established

directly from a proved theorem.

Definition. Fallacies are common forms of incorrect reasoning.

Definition. A conjecture is a statement that has an unknown truth value

I. Rule of Inference: (We draw a line to separate the hypothsis from conclusion)

Form: hypothesis

__________

Conclusion

Name Rule of Inference

____ _________________

Addition p

__

p v q

===========================================

Simplification p ^ q

__

p

===========================================

Conjunction p

q

__

p ^ q

===========================================

Modus ponents (or law of detachment)

p

p -> q

__

q

===========================================

Modus tollens [pic]q

p -> q

__

[pic]p

===========================================

Hypothetical syllogism p -> q

q -> r

__

p -> r

===========================================

Disjunctive syllogism p v q

[pic]p

__

q

Resolution p v q

[pic]p v r

__

q v r

===========================================

Questions: p. 76, Exercise #60

Use resolution to show the hypotheses « Allen is a bad boy or Hillary is a good girl” and “Allen is a good boy or David is happy” imply the conclusion “Hillary Is a good girl or David Is happy.” »

Questions: p. 73, Exercise #2

What rule of inference is used in each of these arguments?

a) Kangaroos live in Australia and are marsupials. Therefore, kangaroos are marsupials.

b) It is either hotter than 100 degrees today or the pollution is dangerous. It is less than 100 degrees outside today. Therefore, the pollution is dangerous.

c) Linda is an excellent swimmer. If Linda is an excellent swimmer, then she can work as a lifeguard. Therefore, Linda can work as a lifeguard.

d) Steve will work at a computer company this summer. Therefore, this summer Steve will work at a computer company or he will be a beach bum.

e) If I work all night on this homework, then I can answer all the exercises. If I answer all the exercises, I will understand the material. Therefore, if I work all night on this homework, then I will understand the material.

Definition. An argument is valid if whenever all hypotheses P1, …, Pn are true, then the conclusion q is also true. That is,

(P1[pic] ^ … ^ Pn) -> q

is true.

All arguments following the rules of inference are valid. That is, the proof of a theorem is valid using rules of inference.

Using Rules of Inference to Draw/Prove Conclusion

Example.

Show the following hypotheses will lead to the conclusion.

Hypotheses:

(i) It is not sunny this afternoon and it is colder than yesterday.

(ii) We will go swimming only if it is sunny.

(iii) If we do not go swimming, then we will take a canoe trip.

(iv) If we take a canoe trip, then we will be home by sunset.

Conclusion:

We will be home by sunset.

Notation:

s: It is sunny this afternoon.

c: It is colder than yesterday.

g: We will go swimming.

t: We will take a canoe trip.

h: We will be home by sunset.

Then the hypotheses are:

[pic]s ^ c, g->s, [pic]g->t, t->h

To prove the conclusion:

h

Proof:

Step Rule of Inference

____ _________________

[pic]s^c hypothesis

[pic]s simplification

g->s hypothesis

[pic]g modus tollens

[pic]g->t hypothesis

t modus ponents

t->h hypothesis

h modus ponents

Questions: p. 73, Exercise #4

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