Rules of Inference.pptx

Rules of Inference

CS 111

Nature & Importance of Proofs

• In mathematics, a proof is:

– A sequence of statements that form an argument.

– Must be correct (well-reasoned, logically valid) and complete (clear, detailed) that rigorously & undeniably establishes the truth of a mathematical statement.

• _{Why must the argument be correct & complete?} – Correctness prevents us from fooling ourselves.

Rules of Inference

•

_{Rules of inference are patterns of logically}

valid deductions from hypotheses to

conclusions

.

• “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”

Inference Rules - General Form

•

_{Inference Rule}

_–

– Pattern establishing that if we know that a set of

hypotheses are all true, then a certain related

conclusion statement is true.

Hypothesis 1

Hypothesis 2 …

Inference Rules & Implications

•

_{Each logical inference rule corresponds to an}

implication that is a tautology.

•

Hypothesis 1

Inference rule

Hypothesis 2 …



conclusion

•

_{Corresponding tautology:}

((Hypoth. 1)  (Hypoth. 2)  …)  conclusion

Modus Ponens

p

p



q (p



(p



q))



q



q

You have a current password

If you have a current password, then you can log onto the network

_________________________________

Modus Tollens



q

p



q (



q



(p



q))



q



p

• “You cannot log onto the network”

• “If you have a current password, then you can log onto the network”

therefore

• “You have not a current password”

Some Inference Rules

p

Rule of Addition



p



q

p



q

Rule of Simplification

Some Inference Rules

p q

 pq

Rule of Conjunction

01/06/2021 L.Niepel 2016 9

• _{“It is cloudy.}

• _{It is raining now.}

Syllogism Inference Rules

p



q

Rule of hypothetical

q



r

syllogism



p



r

p



q

Rule of disjunctive



p

syllogism

Resolution Inference Rule

p

∨

q

￢

p

∨

r

∴

q

∨

r

The rule is based on the tautology:

((p

∨

q)

∧

(

￢

p

∨

r))

→

(q

∨

r)

This rule is frequently used in automatic proving

of theorems

Formal Proofs

• A formal proof of a conclusion C, given premises

p₁, p₂,…,p_n consists of a sequence of steps, each of which applies some inference rule to premises or to previously-proven statements (as hypotheses) to yield a new true statement (the conclusion).

• A proof demonstrates that if the premises are true,

Formal Proof – Example 1

• Suppose we have the following premises:

“It is not sunny and it is cold.”

“if it is not sunny, we will not swim”

“If we do not swim, then we will canoe.” “If we canoe, then we will be home early.”

• Given these premises, prove the theorem

“We will be home early” using inference rules.

Proof Example 1

cont.

•

_{Let us adopt the following abbreviations:}

sunny = “It is sunny”; cold = “It is cold”;

swim = “We will swim”; canoe = “We will canoe”; early = “We will be home early”.

•

Then, the premises can be written as:

(1)



sunny



cold

(2)



sunny





swim

(3)



swim



canoe

(4) canoe



early

Proof Example 1

cont.

•

_{We construct an argument as follows:}

step reason

(1) sunny  cold Premise

(2) sunny Simplification using (1)

(3) sunny  swim Premise

(4) swim Modus ponens for (2), (3)

(5) swim  canoe Premise

(5) canoe Modus ponens for (4), (5) (6) canoe  early Premise

(7) Early Modus ponens for (5), (6)

Proof Example

2.

•

_{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.”

Proof Example 2

cont.

•

_{Let us adopt the following abbreviations:}

mail= “you send me an e-mail message”; finish = “I will finish writing the program”;

sleep = “I will go to sleep early”;

refreshed = “I will wake up feeling refreshed

Then, the premises can be written as:

(1) mail finish

(2)



mail



sleep

(3) sleep



refreshed

Proof Example 2

cont

.

Step Proved by

1. mail finish Premise #1.

2 finish  mail contrapositive of 1. 3. mail  sleep Premise #2.

4. finish  sleep Hypothetical syllogism (2)(3). 5. sleep refreshed Premise #3.

Proof by resolution – Example 3.

Show that the premises (pq) r and rs imply the conclusion p s.

Solution:

(pq) r  (p r)  (q r) premise 1 rs  (r s) premise 2

(p r) simplification from premise1

From (p r)  (r s) using resolution follows p s.

Inference Rules for Quantifiers

• _{x P(x)}

P(o) (substitute any object o)

• _{P(g) (for g a general element of}

discourse)

x P(x)

• _{x P(x)}

P(c) (substitute a new constant c)

• _{P(o) (substitute any extant object o)} x P(x)

Universal instantiation

Universal generalization

Existential instantiation

Existential generalization

Example

“Everyone in this discrete math class has taken a course in computer science” and “Fatmah is a student in this class” imply “Fatmah has taken a course in computer science”

D(x): “x is in discrete math class”

C(x): “x has taken a course in computer science” x (D(x)  C(x)) Premise 1

D(Fatmah) Premise 2

 C(Fatmah)

Example – cont.

Step Proved by

1. x (D(x)  C(x)) Premise #1.

2. D(Fatmah)  C(Fatmah) Univ. instantiation. 3. D(Fatmah) Premise #2.

Universal modus ponens.

We can use together universal instantiation and rule Modus ponens:

x (P(x)  Q(x)) Premise 1

P(a), where a is a particular element in domain

__________________________________________

 Q(a)

Example.

Assume that for all positive numbers n, if n is greater than 4, then n2 is less the 2n. Use universal modus

ponens to show that 1002 < 2 100.

n>4 (n2 <2n)

n= 100

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