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Announcements. Test#1 : Wed 4th Sep, 1800 ... ( So you just need to arrive before 1825 and wait ... Announcements [Applicable for all courses]: Tutorial ... – PowerPoint PPT presentation

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Title: Announcements


1
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.

2
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.

3
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)

4
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
5
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
6
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

7
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).

8
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.
9
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.

10
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.

11
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

12
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
13
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)
14
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.

15
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.

16
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

17
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.

18
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.

19
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!!!

20
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.

21
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.

22
  • End of part 1 of lecture 2
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