Loading...

PPT – Announcements PowerPoint presentation | free to download - id: e2b50-ZDc1Z

The Adobe Flash plugin is needed to view this content

Announcements

- Test1 Wed 4th Sep, 1800-2000hrs, LT27
- 1800-1825 Invigilators will prepare LT for

test. Students not allowed to enter LT. (So you

just need to arrive before 1825 and wait outside

the LT) - 1825 Students will be allowed to enter LT
- 1830 (if we are on time) Test starts
- 1930 (or 1 hour from start time) Test stops,

collection of scripts counting. - Everything should finish by 2000hrs.
- Test2 Sat 28th Sep, 1400-1600hrs, LT27
- Test2 Sat 19th Oct, 1400-1600hrs, LT27
- Extra handouts/tutorial sheets are available at

pigeon hole located outside s16, level 5, General

Office.

Announcements

- Applicable for all courses Tutorial

registration - Starts next week, 5th Aug. Ends 8th Aug.
- Refer to http//www.comp.nus.edu.sg/online
- As long as there is an available slot that does

not clash with any other of your courses, you

should bid for it. (Not everyone can get the

perfect timetable) - Have a right understanding of what is a perfect

timetable Packing everything into a continuous

block may be efficient scheduling, but the brain

is usually not able to take more than 2hrs

continuous intense work without a break.

(Efficiency in scheduling does not mean

efficiency in learning.) - If the groups which you can register for are all

fully subscribed, then refer to

liqm_at_comp.nus.edu.sg (Li QiMing), who will look

into your case.

Lecture 2 (part 1)

- Natural Deduction in Propositional Calculus
- Reading Epp Chp 1.3
- (For a math perspective of cs1104, read Epp 1.4,

1.5)

Lectures 1-4 Logic and Proofs

Studying Logic, is like studying another

programming language.

SEMANTICS

Classical (Truth Tables)

Constructive (Derivations)

(System of Proof)

SYNTAX

(System of Logic)

Propositional Calculus

Lecture 1

Lecture 2

Predicate Calculus

Lecture 3

Lecture 4 Application of Constructive Proofs to

Elementary Number Theory

Overview

- 1. Motivation Arguments and Validity
- 2. Natural Deduction
- 2.1 Definition
- 2.2 Derivation Scheme
- 2.3 Ù-Intro, Ù-Elim Inference Rules
- 2.4 Ú-Intro, Ú-Elim Inference Rules
- 2.5 Reiteration
- 2.6 -Intro, -Elim Inference Rules
- 2.7 -Intro, -Elim, -Intro Inference Rules
- 2.8 -Intro, -Elim Inference Rules
- 3. Using what was proven earlier
- 4. Translation of problems

This Lecture

1. Arguments and Validity.

- 1.1 Definition
- An argument is a conditional of the form
- p1 Ù p2 ÙÙ pn q (n³0)
- (b) A valid argument is an argument which is a

tautology - p1 Ù p2 ÙÙ pn Þ q (n³0)

It is also denoted as

When n0, we write

- a q , for an argument

- a Þ q , for a valid argument, or denote it as

1. Arguments and Validity.

- 1.2 Remarks/Revision
- Resolving ambiguity Ù has higher precedence

than . - As always p1 , p2 ,, pn , q need not be simple

propositions. - Again to repeat last lecture p1 , p2 ,, pn are

known as the premises and q is known as the

conclusion - Note In this lecture, we are interested in

implication (not equivalence, which was dealt

with in the last lecture).

1. Arguments and Validity.

- An argument is a conditional of the form
- p1 Ù p2 ÙÙ pn q (n³0)
- (b) A valid argument is an argument which is a

tautology - p1 Ù p2 ÙÙ pn Þ q (n³0)

1.3 Examples Are the following arguments?

a. p Ù (q Ú r) s

Yes.

b. p Ú (q Ù r) s

Yes.

c. p Ú (q Ú r) s

Yes.

d. p Ú (q Ú r)

Yes.

e. p (q r)

Yes.

f. (p q) r

Yes.

1. Arguments and Validity

- Q Why are we interested in arguments?
- A Because our reasoning progresses in a step by

step use of IF-THEN modes. - EXAMPLE
- Where are my glasses?
- Well, if my glasses are on the kitchen table,

then I would have seen them at breakfast. But I

did not see my glasses at breakfast - Therefore my glasses are NOT on the kitchen

table. - I know that if I was reading newspapers in the

kitchen, then my glasses are on the kitchen

table. But since my glasses are NOT on the

kitchen table - Therefore, I did NOT read newspapers in the

kitchen. - But I definitely did read the papers and I only

read the papers in the kitchen or in the living

room - So therefore I must have read the papers in the

living room. - If I read the papers in the living room, then my

glasses are on the coffee table - So therefore, my glasses are on the coffee table.

1. Arguments and Validity

- Q How do we show whether an argument is valid?
- p1 Ù p2 ÙÙ pn Þ q
- A1 Draw the truth table. Check whether it is a

tautology (last lecture). - Disadvantage Very tedious. WHY?
- Consider the case when you have to prove
- (p r Ú s) Ù (u Ù s t) Ù (r t Ù u) Þ (p

Ú q) Ú (s q) - (1) n simple propositions require 2n truth table

row entries. - (here we have 6 simple propositions 64 rows!)
- (2) n logical connectives require n

sub-expression computations. (here we have 15

connectives, so we have 15 columns!) - Total number of entries 64 x 15 960.

1. Arguments and Validity

- Q How do we show whether an argument is valid?
- p1 Ù p2 ÙÙ pn Þ q
- A2 CONSTRUCT/DERIVE the conclusion from the

hypothesis USING valid inference rules (this

lecture). - p1 Ù p2 ÙÙ pn Þ Þ Þ Þ q

1. Arguments and Validity

- An example The following 2 are valid arguments

which will be used as inference rules - (1) a Ù b Þ a (2) a Ù b Þ b

Q Are you sure they are valid? A Yes. Draw the

truth table.

(a Ù b) a

(a Ù b) b

T

T

1. Arguments and Validity

- An example The following 2 are valid arguments

which will be used as inference rules - (1) a Ù b Þ a (2) a Ù b Þ b

Q Now, show that the following is also a valid

argument. a Ù ((b Ù c) Ù d) Þ c

a Ù ((b Ù c) Ù d)

Þ (b Ù c) Ù d (2)

Þ b Ù c (1)

Þ c (2)

1. Arguments and Validity

- Q How do we show whether an argument is valid?
- p1 Ù p2 ÙÙ pn Þ q
- A2 CONSTRUCT/DERIVE the conclusion from the

hypothesis USING valid inference rules (this

lecture). - p1 Ù p2 ÙÙ pn Þ Þ Þ Þ q
- In this way we REASON our way from the hypothesis

to the conclusion. - Advantage Closely models our reasoning

processes, i.e. the way we think. It tells us

(and teaches us) how to think/reason correctly.

1. Arguments and Validity

- The formalization of logical deduction,

especially as it has been developed by Frege,

Russell, and Hilbert, is rather far removed from

the forms of deduction used in practice in

mathematical proofs. Considerable formal

advantages are achieved in return. In contrast,

I intend to set up a formal system which comes as

close as possible to actual reasoning. - - Gerhard Gentzen (1909 -1945), Investigations

into Logical Deduction (1935). - This formal system is now known as Gentzens

Natural Deduction.

2. Natural Deduction - Definition.

- 2.1 Natural Deduction
- An argument is valid, i.e.
- p1 Ù p2 ÙÙ pn Þ q
- if there is a DERIVATION SCHEME from the

premises p1 , p2 ,, pn to the conclusion q

through the use of the following INFERENCE RULES - (a) Ù-Intro (e) -Intro (i) -Intro
- (b) Ù-Elim (f) -Elim (j) -Elim
- (c) Ú-Intro (g) -Intro (k) -Intro
- (d) Ú-Elim (h) -Elim (l) Reiteration

2. Natural Deduction - Derivation Scheme.

- 2.2 Definition A derivation scheme is of the

form

- where p1 pn are propositions and
- each si is either
- A proposition derived using inference rule Ri
- Another (sub) derivation scheme.

2. Natural Deduction - Derivation Scheme.

- Example
- At this point in time, either John is at the

movies, or he is out playing tennis. - Lets assume hes at the movies.
- John told me that he never watches movies wearing

spectacles. Because the specs restrict hes

field of view. - Therefore John is wearing contact lens.
- Lets assume hes playing tennis.
- Tennis is a sport.
- John always wears contact lens when he is playing

any sport. - Therefore John is wearing contact lens.
- Therefore, in any case, John is wearing contact

lens now.

2. Natural Deduction - Derivation Scheme.

- Example
- At this point in time, either John is at the

movies, or he is out playing tennis. - Lets assume hes at the movies.
- John told me that he never watches movies wearing

spectacles. Because the specs restrict hes

field of view. - Therefore John is wearing contact lens.
- Therefore, in any case, John is wearing contact

lens now. - Wrong Deduction!!!

2. Natural Deduction - Derivation Scheme.

- Remarks
- The derivation scheme models our depth of thought

in our mind, in as much as - the frame/stack pointer keeps track of the depth

of execution in a computer. - Our thoughts can nest arbitrarily deep, in as

much as the execution call of procedures can nest

arbitrarily deep.

2. Natural Deduction - Derivation Scheme.

- In his book Godel, Escher, Bach, Douglas

Hofstadter introduces the fantasy rule for

mathematical proof. - Hofstadter points out that when you start a

mathematical argument with if, let, suppose, you

are stepping into a fantasy world where not only

are all the facts of the real world true but

whatever you are supposing is also true. - Once you are in that world, you can suppose

something else. That sends you into a subfantasy

world where not only is everything in the fantasy

world true but also the new thing you are

supposing. - Of course you can continue stepping into new

subfantasy worlds in this way indefinitely. - You return one level closer to the real world

each time you derive a conclusion that makes a

whole if-then or universal statement true. - Your aim in a proof is to continue deriving such

conclusions until you return to the world from

which you made your first supposition. - - Recommended Text, p246.

- End of part 1 of lecture 2