Lecture 2 (part 2)

1
Lecture 2 (part 2)

• Natural Deduction in Propositional Calculus
• Reading Epp Chp 1.3
• (For a math perspective of cs1104, read Epp 1.4,
  1.5)

2
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
3
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
• 5. Demo of the proof checker

This Lecture
4
2.3 Natural Deduction Ù-Intro, Ù-Elim

• Motivating Example (Ù-Intro)
• John The handbook says that we must take 5
  modules from our area of specialization.
• Peter I read also that we must take 3 cross
  faculty modules.
• John So I must take 5 modules from our area of
  specialization AND 3 cross faculty modules?
• Peter Yes.

5
2.3 Natural Deduction Ù-Intro, Ù-Elim
6
2.3 Natural Deduction Ù-Intro, Ù-Elim

• Motivating Example (Ù-Elim)
• John zzzzzzzz
• Dean For this semester, freshmen must take
  cs1101 AND one GEM course.
• John wakes up and asks Peter
• Peter, must I take a GEM course.
• Peter Yes.
• John Why?
• Peter The dean said that we must take cs1101
  AND one GEM course. So of course, you must take
  a GEM course.

7
2.3 Natural Deduction Ù-Intro, Ù-Elim
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2.3 Natural Deduction Ù-Intro, Ù-Elim
Note that p and q need not be simple propositions.
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2.3 Natural Deduction Ù-Intro, Ù-Elim
1, Ù-Elim (Left)
2, Ù-Elim (Right)
p Ù s
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2.3 Natural Deduction Ù-Intro, Ù-Elim
1, Ù-Elim (Left)
2, Ù-Elim (Left)
2, Ù-Elim (Right)
1, Ù-Elim (Right)
4,5, Ù-Intro
11
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
• 5. Demo of the proof checker

This Lecture
12
2.4 Natural Deduction Ú-Intro, Ú-Elim

• Motivating Example (Ú-Intro)
• Lecturer The pre-requisite for cs4212 is a pass
  in either cs2104 OR cs3212.
• John I have passed cs3212...
• (So it is true that I have a pass in either
  cs2104 OR cs3212). Therefore I fulfil the
  pre-requisite for cs4212.

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2.4 Natural Deduction Ú-Intro, Ú-Elim
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2.4 Natural Deduction Ú-Intro, Ú-Elim

• Motivating Example (Ú-Elim)
• The prime minister is either a criminal or
  insane.
• Suppose that he is a criminal,
• Then he ought to be locked up.
• Suppose on the other hand, that he is insane,
• Again, he ought to be locked up.
• Therefore in any case, the prime minister ought
  to be locked up.

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2.4 Natural Deduction Ú-Intro, Ú-Elim

• Motivating Example (Ú-Elim)
• 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.

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2.4 Natural Deduction Ú-Intro, Ú-Elim
Reality
Ú-Elimination (Proof by considering cases)
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2.4 Natural Deduction Ú-Intro, Ú-Elim

• Note
• i points to the or-statement
• j-k points to the nested sub-scheme (1 level
  deeper) with assumption that LHS of the or-stmt
  is true.
• l-m points to the nested sub-scheme (1 level
  deeper) with assumption that RHS of the or-stmt
  is true.
• Note that the conclusion r must be the same.

Note that p, q, r need not be simple propositions.
18
2.4 Natural Deduction Ú-Intro, Ú-Elim
2, Ú-Intro (Left)
4, Ú-Intro (Right)
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2.4 Natural Deduction Ú-Intro, Ú-Elim
2.4.2 Prove
(p Ú q) Ú r
p Ú (q Ú r)
5
5, Ú-Intro (Right)
Ú-Elim
9
9, Ú-Intro (Left)
Ú-Elim
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2.4 Natural Deduction Ú-Intro, Ú-Elim
2.4.3 Prove
2,ÚIntro(R)
6,ÙElim(R)
7,ÚIntro(L)
8,ÚIntro(L)
Ú-Elim
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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
• 5. Demo of the proof checker

This Lecture
22
2.5 Natural Deduction Reiteration

• 2.5.1 Recall (Derivation Scheme)
• Purpose Model our line of thought
• Outermost Derivation Scheme is the main line of
  thought. It corresponds to reality.
• Inner derivation schemes create an alternate
  reality - a world which exists due to the
  assumption that you have made.

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2.5 Natural Deduction Reiteration
Wrong kind of Layout.
Right kind of Layout.
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2.5 Natural Deduction Reiteration

• 2.5.2 Reiteration
• Brings a proposition which has been derived
  previously the Outside scheme to the Inside (and
  not the other way around).
• 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. (p246
  Textbook)

25
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
• 5. Demo of the proof checker

This Lecture
26
2.6 Natural Deduction -Intro, -Elim

• Motivating Example (-Intro)
• Peter Lets assume that I get caught for drunk
  driving.
• John Well, then you would go to jail. Your
  mother would know about it and that would make
  her unhappy.
• Peter Correct. So what can we conclude?
• John IF you get caught for drunk driving, THEN
  you would make your mother unhappy.

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2.6 Natural Deduction -Intro, -Elim
Reality
Alternate reality a world of which the truth of
the statements here depends on your assumption
being true
-Introduction
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2.6 Natural Deduction -Intro, -Elim

• Motivating Example (-Elim)
• Fact1 IF it is a cat, THEN it is an animal
• Fact2 It is a cat
• Conclusion Therefore, it is an animal.
• Fact1 IF it is a cat, THEN it is an animal
• Fact2 It is NOT an animal
• Conclusion Therefore, it is NOT a cat.

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2.6 Natural Deduction -Intro, -Elim
-Elimination (LR) (Modus Ponens) Method of
Affirming
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2.6 Natural Deduction -Intro, -Elim
Note i points to the Conditional
Note that p and q need not be simple propositions.
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2.6 Natural Deduction -Intro, -Elim
1,Reiteration
4,3,-Elim(LR)
2,Reiteration
-Intro
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2.6 Natural Deduction -Intro, -Elim
2.6.2 Prove
1,Reiteration
3,2,-Elim(LR)
-Intro
1,Reiteration
8,7,-Elim(LR)
-Intro
Ù-Intro
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2.6 Natural Deduction -Intro, -Elim
2.6.3 Prove
1,Reiteration
2,Reiteration
4,5,-Elim(LR)
8,9,-Elim(LR)
Ú-Elim
-Intro
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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
• 5. Demo of the proof checker

This Lecture
35
2.7 Natural Deduction -Intro,-Elim,-Intro

• Motivating Example (-Elim)
• Boy to girl Its not that I dont love you,
  its just thatblah blah blah
• (not not love you love you)

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2.7 Natural Deduction -Intro,-Elim,-Intro
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2.7 Natural Deduction -Intro,-Elim,-Intro

• Motivating Example (-Intro)
• A I did NOT read newspapers in the kitchen.
• B Why?
• A Suppose its true that I really did read
  newspapers in the kitchen
• Now, IF I was reading newspapers in the kitchen,
  THEN my glasses will be on the kitchen table.
• So my glasses should be on the kitchen table.
• But my glasses are not there.
• So I did NOT read newspapers in the kitchen.

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2.7 Natural Deduction -Intro,-Elim,-Intro
39
2.7 Natural Deduction -Intro,-Elim,-Intro
Note that p need not be a simple proposition.
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2.7 Natural Deduction -Intro,-Elim,-Intro
2.7.1 Prove
2,Ù-Elim (L)
1,Reiteration
-Intro
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2.7 Natural Deduction -Intro,-Elim,-Intro
2.7.2 Prove
1,Reiteration
4,3,-Elim (LR)
2,Reiteration
-Elim
-Intro
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2.7 Natural Deduction -Intro,-Elim,-Intro
2.7.3 Prove
2, Ú-Intro (R)
1,Reiteration
-Intro
7, Ú-Intro (L)
1,Reiteration
-Intro
Ù-Intro
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2.7 Natural Deduction -Intro,-Elim,-Intro
2,Ù-Elim (L)
2,Ù-Elim (R)
-Intro
44
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
• 5. Demo of the proof checker

This Lecture
45
Introduction and Elimination
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2.8 Natural Deduction -Intro, -Elim
Note that p and q need not be simple propositions.
47
2.8 Natural Deduction -Intro, -Elim
2.8.1 Prove
p q
q r
r p
p q
2,Reiteration
5,4,-Elim (LR)
3,Reiteration
q p
-Intro
-Intro
48
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
• 5. Demo of the proof checker

This Lecture
49
3. Using what was proven earlier.
(Left as an exercise)
-Elim
50
3. Using what was proven earlier.
1,Reiteration
4,3,-Elim (LR)
Ú-Elim
51
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
• 5. Demo of the proof checker

This Lecture
52
4. Translation of problems

• The famous detective Percule Hoirot was called in
  to solve a baffling murder mystery. He
  determined the following facts
• Lord Hazelton, the murdered man, was killed by a
  blow on the head with a brass candle stick.
• Either Lady Hazelton or a maid, Sara, was in the
  dining room at the time of the murder.
• If the cook was in the kitchen at the time of the
  murder, then the butler killed Lord Hazelton with
  a fatal dose of strychnine.
• If Lady Hazelton was in the dining room at the
  time of the murder, then the chauffeur killed
  Lord Hazelton.
• If the cook was not in the kitchen at the time of
  the murder, then Sara was not in the dining room
  when the murder was committed.
• If Sara was in the dining room at the time the
  murder was committed, then the wine steward
  killed Lord Hazelton.

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4. Translation of problems

• Lord Hazelton, the murdered man, was killed by a
  blow on the head with a brass candle stick.
• weapon_CandleStick
• Either Lady Hazelton or a maid, Sara, was in the
  dining room at the time of the murder.
• lady_dining Ú sara_dining
• If the cook was in the kitchen at the time of the
  murder, then the butler killed Lord Hazelton with
  a fatal dose of strychnine.
• cook_kitchen butlerDidIt
• cook_kitchen weapon_CandleStick
• If Lady Hazelton was in the dining room at the
  time of the murder, then the chauffeur killed
  Lord Hazelton.
• lady_dining chauffeurDidIt
• If the cook was not in the kitchen at the time of
  the murder, then Sara was not in the dining room
  when the murder was committed.
• cook_kitchen sara_dining
• If Sara was in the dining room at the time the
  murder was committed, then the wine steward
  killed Lord Hazelton.
• sara_dining stewardDidIt

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4. Translation of problems

• weapon_CandleStick
• lady_dining Ú sara_dining
• cook_kitchen butlerDidIt
• cook_kitchen weapon_CandleStick
• lady_dining chauffeurDidIt
• cook_kitchen sara_dining
• sara_dining stewardDidIt
• --------------------------------------------------
  -
• chauffeurDidIt

55
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
• 5. Demo of the proof checker

This Lecture
56

• End of Lecture.

57
Have a break

• There is an ancient story about the Sophist
  philosopher Protagoras, who agreed to instruct
  Euathlus in rhetoric so that Euathlus could
  practice law.
• Euathlus in turn agreed to pay Protagoras his fee
  only after winning his first case.
• However, upon completion of his training,
  Euathlus chose not to practise law.
• So Protagoras brought Euathlus to court and sued
  him for his fee.
• Protagoras maintained that he should be paid no
  matter what. He argued
• If you (i.e. Euathlus) win the case, I should be
  paid, by the terms of my agreement with you.
• If you lose the case, I should also be paid, by
  by order of the court.
• What would you do if you were Euathlus?

58
Have a break

• There is an ancient story about the Sophist
  philosopher Protagoras, who agreed to instruct
  Euathlus in rhetoric so that Euathlus could
  practice law.
• Euathlus in turn agreed to pay Protagoras his fee
  only after winning his first case.
• However, upon completion of his training,
  Euathlus chose not to practise law.
• So Protagoras brought Euathlus to court and sued
  him for his fee.
• Protagoras maintained that he should be paid no
  matter what. He argued
• If you (i.e. Euathlus) win the case, I should be
  paid, by the terms of my agreement with you.
• If you lose the case, I should also be paid, by
  by order of the court.

• Euathlus No, no, no
• If I win the case, then I should not pay, by
  order of the court.
• If I lose the case, then I still should not pay,
  by the terms of my agreement with Protagoras.