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Quick Questions?� What is the negation of “Some students in your class

use e-mail”?

� Some students in your class do not use e-mail.

� All students in your class use e-mail.

� All students in your class do not use e-mail.

Quick Questions?� What is the negation of “Some friendly people are not

angry”?

� Some friendly people are angry.

� Some unfriendly people are angry.

� All friendly people are angry.

� All unfriendly people are angry.

� All friendly people are not angry.

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Chapter 1, Part III: Proofs

Summary� Valid Arguments and Rules of Inference

� Proof Methods

� Proof Strategies

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Section 1.6

Example of Valid Arguments� We have the two premises: (Let P(x), Q(x), and R(x) be the

propositional functions “x is a lion,” “x is fierce,” and “x drinks coffee,” respectively.)

� “All lions are fierce.” ∀x (P(x)→ Q(x))

� “Some lions do not drink coffee.” ∃x (P(x) ∧ R(x))

� And the conclusion:

� “Some fierce creatures do not drink coffee.”

∃x (Q(x) ∧ R(x))

� How do we get the conclusion from the premises?

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Example of Valid Arguments� We have the two premises:

� “If you have a current password, then you can log onto the network.” ( p → q )

� “You have a current password.” (proposition p)

� And the conclusion:

� “You can log onto the network.” (proposition q)

� How do we get the conclusion from the premises?

→

The Argument� We can express the premises (above the line) and the

conclusion (below the line) in propositional logic as an argument:

� We will see shortly that this is a valid argument.

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Valid Arguments

� The rules of inference are the essential building block in the construction of valid arguments.

� We will show how to construct valid arguments in two stages; first for propositional logic and then for predicate logic.

1. Propositional Logic

Inference Rules

2. Predicate Logic

Inference rules for propositional logic plus additional inference rules to handle variables and quantifiers.

Arguments in Propositional Logic

� An argument in propositional logic is a sequence of propositions. All but the final proposition are called premises. The last statement is the conclusion.

� An argument is valid if the truth of all its premises implies that the conclusion is true.

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Arguments in Propositional Logic� 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.

Arguments in Propositional Logic� The argument form with premises p1 ,p2, …,pn and

conclusion q is valid, when (p1 ∧ p2 ∧ … ∧ pn ) → q is a tautology.

� The key to showing that an argument in propositional

logic is valid is to show that its argument form is valid.

� Inference rules are relatively simple argument forms that will be used to construct more complex argument forms.

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Rules of Inference for Propositional

Logic: Modus Ponens

Example:Let p be “It is snowing.”Let q be “I will study discrete math.”

“If it is snowing, then I will study discrete math.”“It is snowing.”

“Therefore , I will study discrete math.”

Corresponding Tautology:(p ∧ (p →q)) → q

Modus Tollens

Example:Let p be “it is snowing.”Let q be “I will study discrete math.”

“If it is snowing, then I will study discrete math.”“I will not study discrete math.”

“Therefore , it is not snowing.”

Corresponding Tautology:(p∧(p →q))→q

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Hypothetical Syllogism

Example:Let p be “it snows.”Let q be “I will study discrete math.”Let r be “I will get an A.”

“If it snows, then I will study discrete math.”“If I study discrete math, I will get an A.”

“Therefore , If it snows, I will get an A.”

Corresponding Tautology:((p →q) ∧ (q→r))→(p→ r)

Disjunctive Syllogism

Example:Let p be “I will study discrete math.”Let q be “I will study English literature.”

“I will study discrete math or I will study English literature.”“I will not study discrete math.”

“Therefore , I will study English literature.”

Corresponding Tautology:(p∧(p ∨q))→q

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Addition

Example:Let p be “I will study discrete math.”Let q be “I will visit Las Vegas.”

“I will study discrete math.”

“Therefore, I will study discrete math or I will visit Las Vegas.”

Corresponding Tautology:p →(p ∨q)

Simplification

Example:Let p be “I will study discrete math.”Let q be “I will study English literature.”

“I will study discrete math and English literature”

“Therefore, I will study discrete math.”

Corresponding Tautology: (p∧q) →p

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Conjunction

Example:Let p be “I will study discrete math.”Let q be “I will study English literature.”

“I will study discrete math.”“I will study English literature.”

“Therefore, I will study discrete math and I will study English literature.”

Corresponding Tautology:((p) ∧ (q)) →(p ∧ q)

Resolution

Example:Let p be “I am taking CSCI 1301.”Let q be “I am taking CSCI 2610.”Let r be “I am taking CSCI 1302.”

“I am not taking CSCI 1301 or I am taking CSCI 1302.”“I am taking CSCI 1301 or I am taking CSCI 2610.”

“Therefore, I am taking CSCI 1302 or I am taking CSCI 2610.”

Corresponding Tautology:((p ∨ r ) ∧ (p ∨ q)) →(q ∨ r)

Resolution plays an important role in AI and is used in Prolog.

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Using Rules of Inference to Build

Valid Arguments� A valid argument is a sequence of statements. Each statement is

either a premise or follows from previous statements by rules of inference. The last statement is called conclusion.

� A valid argument takes the following form:S1

S2

.

.

.

Sn

C

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Fallacies

Mistakes in proof by applying contingencies instead of tautologies are fallacies.

� Affirming the conclusion (Common Mistake)

If it is Sunday afternoon, then I go to the movies

I go to the movies

∴ It is Sunday afternoon

It’s Saturday Night!!

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Fallacies

� Denying the hypothesis

If an animal flies then the animal is a bird

The animal does not fly

∴ The animal is not a bird

It’s a penguin!!

Valid ArgumentsExample 1111: From the single proposition

Show that q is a conclusion.

Solution: