Propositional Logic

1
Propositional Logic

1. Propositional Language
2. Translations
3. Truth Tables
4. Propositional Proofs
5. Appendix Model Theory

2
Introduction

• Propositional logic studies arguments whose
  validity depends on if-then, and, or,
  not, and similar notions.
• In the following, firstly we will introduce a
  formal language that sets up the framework of
  propositional logic, and then explain the
  relationship between this language and natural
  languages.
• Secondly, we will introduce a simple model of it,
  i.e. the one made by truth-tables.
• Finally, we will introduce inference rules and
  show how to construct a proof.

3
1. Propositional Language

• vocabulary
• formation rules

PL
4
Vocabulary

• Logical constants logical connectives
• ?, ?, ?, ?, and ?.
• Logical variables propositional variables
• P, Q, R, and so on. (if necessary with
  subscripts appendedlike P1)
• Auxiliary Signs brackets
• (, and ).

5
The Function of Brackets

• Consider the case in mathematics
• 23?5?
• Either (23)?525 or 2(3?5)17.
• Similarly consider the following case
• What does ?P?R mean?
• It means either that (?P)?R or that ?(P?R).
• The function of brackets is to disambiguate the
  meaning of wffs (well-formed formula).

6
Formation Rules

1. Any capital letter is a well-formed formula.
2. The result of prefixing any wff with ? is a
   wff.
3. The result of joining any two wffs by ?, ?,
   ?, or ? and enclosing the result in
   parentheses is a wff.
4. Only that which can be generated by the rules
   (i)-(iii) in a finite number of steps is a wff in
   PL.

7
Some examples

• P?P
• P?(?R?P)
• ???R
• (?(P?R))??Q?
• ((P?R)(?Q))?P

• P??P
• P?(R?P)
• ??R
• (?(P?R))?Q
• ((P?R)?(?Q))?P

8
Construction Tree (Top-Down)

• For any wff in PL, we can construct a tree for
  it.
• Ex. P??P
• P??P (iii, ?)
• P (i) ?P (ii, ?)
• 
  
• 
  P (i)

9
Construction Tree (Bottom-Up)

• Ex. P?(R?P)
• P (i) R (i)
  P (i)
• R?P
  (iii, ?)
• P?(R?P) (iii, ?)

10
Main Connectives

• The main connective of a formula is at the top of
  the tree (Top-down)at the bottom if bottom-up.
• In another words, if the whole sentence was
  constituted at last by one connective, then we
  call this one as main connective of this
  sentence.
• Note the leaf of tree must be atomic.

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Some examples

• ?(P?R)
• ?(P?(R?P))
• (?P)?(?R)
• (?(P?R))??Q
• (?(P?R))?P

• P??P
• P?(R?P)
• ??R
• (?(P?R))?Q
• ((P?R)?(?Q))?P

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2. Translations

• As mentioned earlier, propositional logic studies
  arguments whose validity depends on if-then,
  and, or, not, and similar notions.
• It follows that PL must contain these notions in
  order to represent natural languages to certain
  degree.
• In a way, we can translate arguments in natural
  languages into wffs in PL.

13
Connectives

• ? stands for not.
• ? stands for and. (or but, though)
• ? stands for or. (inclusive-or)
• ? stands for if-then.
• ? stands for if and only if.

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Some examples

• It will rain tomorrow.
• It will not rain tomorrow.
• It will rain tomorrow and I will bring my
  umbrella.
• If it rains tomorrow, I will bring my umbrella.
• Either it rains tomorrow, or it will not rain.
• It will rain heavily if and only if the sky will
  be covered with dark clouds.

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More

• Not both Alan and Bill like to play baseball.
• If Alan took this course and Bill dropped this
  course, I would take this course.
• Only if Cindy took this course, I would take this
  course.
• Not either Alan drops this course or Bill drops
  this course.

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3. Truth Tables

• Logical connectives are represented by some truth
  functionsby which we can calculate the truth
  table of each compound wff.
• Before we introduce logical connectives, lets
  see what a function is and what kind of function
  is called truth functional.

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Set

• A set is a collection of entities (or objects).
• The principle of extensionality
• A set is defined by the members it contains.
• Ex. Suppose our domain is Alan, Bill, Cindy
• M Alan, Bill (M xx is a man)
• W Cindy (W xx is a woman)

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Membership

• Suppose we use a for Alan, b for Bill, and
  c for Cindy
• a ? M and b ? M, but c ?M.
• c ? W, but a ? W and b ? W.
• Suppose that B xx is Cindys brother. Alan is
  Cindys brother, and Bill is not.
• a ? B and b ? B.

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Subset

• If A is a subset of B, then if x ? A, then x ? B.
• A ? B
• If A is a proper subset of B, then if x ? A then
  x ? B and, for some y, such that y ? B and y ? A.
• A ? B
• Ex.
• Given that M a, b and B a, we call B is a
  proper subset of Msymbolized as B ? M.

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Intersection and Union

• A?B x?x?A and x?B
• A?B x?x?A or x?B
• Ex.
• 1, 2, 3?2, 4, 61, 2, 3, 4, 6
• 1, 2, 3?2, 4, 62

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Function Many-one Relation between Domain and
Range

• brotherhood
• Alan
• 
  Cindy
• Bill

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Suppose both Alan and Bill are Cindys brothers.

• Alan
• 
  Cindy
• Bill

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This is not a function

• 
  Alan
• Cindy
  
• 
  Bill

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Function

• 1
  1
• 0
  0

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Compound Function

• 1 1
  1
• 0 0
  0

26

• 1 1
  1
• 0 0
  0

27

• 1 1
  1
• 0 0
  0

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• 1 1
  1
• 0 0
  0

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Logical Connectives?
P ?P
1 0
0 1
30

• 1
  1
• 0
  0

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Logical Connectives?
P Q P?Q
1 1 1
1 0 0
0 1 0
0 0 0
32

• 1 1
  1
• 0 0
  0

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Logical Connectives?
P Q P?Q
1 1 1
1 0 1
0 1 1
0 0 0
34

• 1 1
  1
• 0 0
  0

35
Logical Connectives?
P Q P?Q
1 1 1
1 0 0
0 1 1
0 0 1
36

• 1 1
  1
• 0 0
  0

37
Logical Connectives?
P Q P?Q
1 1 1
1 0 0
0 1 0
0 0 1
38

• 1 1
  1
• 0 0
  0

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Truth Function

• Connectives which give rise to sentences whose
  truth value depends only on the truth values of
  the connected sentences are said to be
  truth-functional.
• Therefore, not (?), or (?), and (?),
  if-then (?), and if and only if (?) are
  truth-functional.

40
Some examples

• U ? F
• U ? ?T
• U ? ?F
• ?F ? U
• F ? U
• (?T) ? U

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U ? F
U F U?F
1 1 1
1 0 0
0 1 0
0 0 0
42
U ? ?T
U T ?T U??T
1 1 0 0
1 0 1 1
0 1 0 1
0 0 1 1
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Complex Truth Tables

• (P ? (Q ? ?R))
• ((P ? ?Q) ? R)
• ((P ? Q) ? Q)
• ((P ? ?Q) ? R)
• ?(P ? (Q ? ?R))

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(P ? (Q ? ?R))
P Q R ?R Q??R P ? (Q ? ?R)
1 1 1 0 0 1
1 1 0 1 1 1
1 0 1 0 0 1
1 0 0 1 0 1
0 1 1 0 0 0
0 1 0 1 1 1
0 0 1 0 0 0
0 0 0 1 0 0
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Functional Completeness

• A system of connectives can express all truth
  functions is said to be functionally complete.
• Ex. Define ? by ? and ?.
• Actually, we can use ? and ? to represent
  others, and we call these two functions
  functionally complete.

46

• Can you find another system of connectives which
  is also functionally complete?

47
The Truth-table Test

• Recall how valid and invalid are defined for
  arguments
• VALID no possible case has premises all true
  and conclusion false.
• INVALID some possible case has premises all
  true and conclusion false.
• Now, we can use the truth-table test on a
  propositional argument.

48
(D ? A), ?D/??A
A D D?A ?D ?A
1 1 1 0 0
1 0 1 1 0
0 1 0 0 1
0 0 1 1 1
49
(D ? A), ?D/??A
A D D?A ?D ?A
1 1 1 0 0
1 0 1 1 0
0 1 0 0 1
0 0 1 1 1
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More example

• It is in your left hand or your right hand.
• It is not in your left hand.
• ? It is in your right hand.
• (L it is in your left hand R it is in your
  right hand)
• L ? R
• ?L
• ? R

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L ? R, ?L/?R
L R L?R ?L R
1 1 1 0 1
1 0 1 0 0
0 1 1 1 1
0 0 0 1 0
52
L ? R, ?L/?R
L R L?R ?L R
1 1 1 0 1
1 0 1 0 0
0 1 1 1 1
0 0 0 1 0
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The Truth-assignment Test

• Set each premise to 1 and the conclusion 0.
• Figure out the truth value of as many letters as
  possible.
• The argument is VALID if and only if no possible
  way to assign 1 and 0 to the letters will keep
  the premises all 1 and conclusion 0.

54
L ? R, ?L/?R

• Step 1 we set each premise to 1 and the
  conclusion 0.
• L ? R 1
• ?L 1
• R 0

55
L ? R, ?L/?R

• Step 2 since premise 2 has ?L 1, we know that
  L 0.
• 0 ? R 1
• ?0 1
• R 0

56
L ? R, ?L/?R

• Step 3 since the conclusion has R 0,
• 0 ? 0 1
• ?0 1
• 0 0

57
L ? R, ?L/?R

• Step 4
• 0 ? 0 1
• ?0 1
• 0 0
• But the premise 1 cannot be true. So we cannot
  have true premises and a false conclusion.

58
Some examples

• If existence is a perfection and God by
  definition has all perfection, then God by
  definition must exist.
• God by definition has all perfections.
• ?God by definition must exist.

59

• If Newtons gravitational theory is correct and
  there is no undiscovered planet near Uranus, then
  the orbit of Uranus would be such-and-such.
• Newtons gravitational theory is correct.
• The orbit of Uranus is not such-and-such.
• ?There is an undiscovered planet near Uranus.

60
Idiomatic Arguments

• Our arguments so far have been phrased in a clear
  premise-conclusion format.
• Unfortunately, real-life arguments are seldom so
  neat and clean.
• Instead we often find convoluted wording or
  extraneous material.

61
For example

• Socrates must be mortal. After all, he is human.
  And if he is human, he is mortal.
• Socrates is human.
• If Socrates is human, he is mortal.
• ?Socrates is mortal.

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A Guide

• These often indicate premises
• Because, for, since, after all
• I assume that, as we know
• For these reasons
• These often indicate conclusions
• Hence, thus, so, therefore
• It must be, it cant be
• This proves (or shows) that

63
Some examples

• Knowledge cant be sensation. If it were, then we
  couldnt know something that we arent presently
  sensing.
• Taking the exam is a sufficient condition for
  getting an A. You didnt take the exam. This
  means you dont get an A.

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4. Propositional Proofs

• We will learn some inference rules, which state
  that certain formulas can be derived from certain
  other formulas.
• Most of these rules reflect common forms of
  reasoning.
• These rules also provide the building blocks for
  formal proofsformal proofs reduce a complex
  arguments to a series of small steps, each based
  on an inference rule.

65
S-Rules (E-Rules)

• The S-rules are used to simplify statements. In
  other words, we can infer statement without
  certain logical connective from those containing
  logical connectiveto infer a simplified formula.
• So we may also call these rules E-rulesE for
  eliminate.
• As one may notice, each connective will have one
  S-rule (E-rule) respectively.

66
ANDE-rule for ?

• Alan is hungry and so is Bill.
• Therefore, Alan is hungry.
• Therefore, Bill is hungry.

• P?Q
• ?P
• P?Q
• ?Q

67
NORE-rule for ?

• Alan is neither in school nor at home.
• Therefore, Alan is not in school.
• Therefore, Alan is not at home.

• ?(P?Q)
• ??P
• ?(P?Q)
• ??Q

68
NIFE-rule for ?

• It is not the case that if Alan gets sick, he
  will be happier.
• Therefore, Alan gets sick.
• Therefore, Alan is not happier.

• ?(P?Q)
• ?P
• ?(P?Q)
• ??Q

69
I-Rules

• The I-rules are used to infer a conclusion from
  premises.

70
CSI-rule for ?

• Alan is not both hungry and thirsty.
• Alan is hungry.
• Therefore, Alan is not thirsty.

• ?(P?Q)
• Q
• ??P
• ?(P?Q)
• P
• ??Q

71
DSI-rule for ?

• Alan is either angry or sad about what happened
  to Bill.
• Alan is not angry.
• Therefore, Alan is sad.

• (P?Q)
• ?P
• ?Q
• (P?Q)
• ?Q
• ?P

72
MPI-rule for ?

• If Alan gets sick, he will stay at home.
• Alan gets sick.
• Therefore, Alan stays at home.
• (Alan doesnt not stay at home. Therefore, Alan
  didnt get sick.)

• (P?Q)
• P
• ?Q
• (P?Q)
• ?Q
• ??P

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Exercise

• ?(W?T)
• W

• ?Q??M
• Q

• H ? ?B
• H

• S?L
• S

74
Complex Cases

• ?((A ? B) ? C)

• (A ? B) ? C
• A ? B

75
Formal Proofs

• Formal proofs are a convenient way to test
  arguments of various systems and, in addition,
  help to develop reasoning skills.
• A formal proof breaks a complicated argument into
  a series of small steps, each based on our
  S-rules or I-rules.

76
Indirect Proof Strategy (RAA)

• First Step
• Block off the conclusion and add asm (for
  assume) followed by the conclusions simpler
  contradictory.
• Second Step
• Go through the complex wffs, to which we can
  apply S-rules or I-rules.
• Third Step
• To find a contradiction through reasoning, and
  apply RAA and derive the original conclusion.

77
S-rules and I-rules

• ?(P?Q), P??Q
• ?(P?Q), Q??P
• (P?Q), ?P?Q
• (P?Q), ?Q?P
• (P?Q), P?Q
• (P?Q), ?Q??P

• (P?Q)?P, Q
• ?(P?Q)? ?P, ?Q
• ?(P?Q)?P, ?Q
• ??P?P
• (P?Q)?(P?Q), (Q?P)
• ?(P?Q)?(P?Q), ?(P?Q)

78
For exampleFirst Step

1. T
2. T ? (B ? M)
3. M ? H
4. ?H /?B
5. asm ?B

79
For exampleSecond Step

1. T
2. T ? (B ? M)
3. M ? H
4. ?H /?B
5. asm ?B
6. ?B ? M from 1 and 2, MP
7. ?M from 5 and 6, CS
8. ?H from 3 and 7, MP

80
For exampleThird Step

1. T
2. T ? (B ? M)
3. M ? H
4. ?H /?B
5. asm ?B
6. ?B ? M from 1 and 2, MP
7. ?M from 5 and 6, CS
8. ?H from 3 and 7, MP
9. ? B from 5 4 contradicts 8, RAA

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Exercise

1. A ? B /?(?B ? ?A)

82
Exercise

1. A /?A ? B

83
Exercise with Translation

1. If we had an absolute proof of Gods existence,
   then our will would be irresistibly attracted to
   do right.
2. If our will were irresistibly attracted to do
   right, then we would have no free will.
3. Therefore, if we have free will, then we have no
   absolute proof of Gods existence.

84
Translation

1. we had an absolute proof of Gods existence ?
   our will would be irresistibly attracted to do
   right
2. our will were irresistibly attracted to do
   right ? we would have no free will
3. ? we have free will ? we have no absolute
   proof of Gods existence

85
Translation

1. P ? I
2. I ? ?F /? F ? ?P

86
Proof

1. P ? I
2. I ? ?F /? F ? ?P
3. asm ?(F ? ?P) First Step!

87
Proof

1. P ? I
2. I ? ?F /? F ? ?P
3. asm ?(F ? ?P)
4. ?F from 3, NIF
5. ???P from 3, NIF
6. ?P from 5, DN
7. ?I from 1 and 6, MP
8. ??F from 2 and 7, MP

Second Step!
88
Proof

1. P ? I
2. I ? ?F /? F ? ?P
3. asm ?(F ? ?P)
4. ?F from 3, NIF
5. ???P from 3, NIF Third
6. ?P from 5, DN
   Step!
7. ?I from 1 and 6, MP
8. ??F from 2 and 7, MP
9. ?F ? ?P from 3 4 contradicts 8, RAA

89

1. If racism is clearly wrong, then either its
   factually clear that all races have equal
   abilities or its morally clear that similar
   interests of all beings ought to be given equal
   consideration.
2. It is not factually clear that all races have
   equal abilities.
3. If its morally clear that similar interests of
   all beings ought to be given equal consideration,
   then similar interests of animals and humans
   ought to be given equal consideration.
4. Therefore, if racism is clearly wrong, then
   similar interests of animals and humans ought to
   be given equal consideration.

90

1. racism is clearly wrong ? either its
   factually clear that all races have equal
   abilities or its morally clear that similar
   interests of all beings ought to be given equal
   consideration
2. ?it is factually clear that all races have equal
   abilities
3. its morally clear that similar interests of all
   beings ought to be given equal consideration ?
   similar interests of animals and humans ought to
   be given equal consideration
4. ?racism is clearly wrong ? similar interests
   of animals and humans ought to be given equal
   consideration

91

1. W ? its factually clear that all races have
   equal abilities ? its morally clear that
   similar interests of all beings ought to be given
   equal consideration
2. ?it is factually clear that all races have equal
   abilities
3. its morally clear that similar interests of all
   beings ought to be given equal consideration ?
   similar interests of animals and humans ought to
   be given equal consideration
4. ?W ? similar interests of animals and humans
   ought to be given equal consideration

92

1. W ? (F? M)
2. ?F
3. M ? A /?W ? A

93

1. W ? (F? M)
2. ?F
3. M ? A /?W ? A
4. asm ?(W ? A)

94

1. W ? (F? M)
2. ?F
3. M ? A /?W ? A
4. asm ?(W ? A)
5. ?W from 4, NIF
6. ??A from 4, NIF
7. ? F ? M from 1 and 5, MP
8. ?M from 2 and 7, DS
9. ?A from 3 and 8, MP

95

1. W ? (F? M)
2. ?F
3. M ? A /?W ? A
4. asm ?(W ? A)
5. ?W from 4, NIF
6. ??A from 4, NIF
7. ? F ? M from 1 and 5, MP
8. ?M from 2 and 7, DS
9. ?A from 3 and 8, MP
10. W ? A from 4 6 contradicts 9, RAA

96
Harder Proofs

1. If the butler was at the party, then he fixed the
   drinks and poisoned the deceased.
2. If the butler was not at the party, then the
   deceased would have seen him leave the mansion
   and would have reported this.
3. The deceased did not reported this.
4. Therefore, the butler poisoned the deceased.

97

1. the butler was at the party ? he fixed the
   drinks and poisoned the deceased
2. the butler was not at the party ? the deceased
   would have seen him leave the mansion and would
   have reported this
3. ?The deceased did reported this
4. ? the butler poisoned the deceased

98

1. A ? he fixed the drinks ? poisoned the
   deceased
2. ?the butler was at the party ? the deceased
   would have seen him leave the mansion ? would
   have reported this
3. ?The deceased did reported this
4. ? the butler poisoned the deceased

99

1. A ? (F ? P)
2. ?A ? (S ? R)
3. ?R /?P
4. asm ?P
5. ?

100
Strategy Expanded

• Make another assumption when youre stuck.
• Make an assumption that breaks a complex wff.

101

1. A ? (F ? P) Pick a complex wff
2. ?A ? (S ? R) (1 or 2), and pick
3. ?R /?P left or right side, and
4. asm ?P assume it or its
5. asm ?A break 1 negation!

102

1. A ? (F ? P)
2. ?A ? (S ? R)
3. ?R /?P
4. asm ?P
5. asm ?A break 1
6. ? S ? R from 2 and 5, MP
7. ? S from 6, AND
8. ? R from 6, AND

103

1. A ? (F ? P)
2. ?A ? (S ? R)
3. ?R /?P
4. asm ?P
5. asm ?A break 1
6. ? S ? R from 2 and 5, MP
7. ? S from 6, AND
8. ? R from 6, AND
9. ?A from 5 3 contradicts 8, RAA

104

1. A ? (F ? P)
2. ?A ? (S ? R)
3. ?R /?P
4. asm ?P
5. asm ?A break 1
6. ? S ? R from 2 and 5, MP
7. ? S from 6, AND
8. ? R from 6, AND
9. ?A from 5 3 contradicts 8, RAA
10. ?F ? P from 1 and 9, MP
11. ?P from 10, AND

105

1. A ? (F ? P)
2. ?A ? (S ? R)
3. ?R /?P
4. asm ?P
5. asm ?A break 1
6. ? S ? R from 2 and 5, MP
7. ? S from 6, AND
8. ? R from 6, AND
9. ?A from 5 3 contradicts 8, RAA
10. ?F ? P from 1 and 9, MP
11. ?P from 10, AND
12. ?P from 4 4 contradicts 11, RAA

106
Final Proof Strategy

• First Step (START)
• Add asm followed by the conclusions simpler
  contradictory.
• Second Step (SI)
• Apply S-rules and I-rules to the complex wffs
• If you get a contradiction, then go to the Third
  Step.
• If you cant derive anything further but there is
  a complex wff, then go to the Fourth Step.
• If you cant derive anything further and every
  complex wff are used, then go to the Fifth Step.
• Third Step (RAA)
• Apply the RAA rule.
• Fourth Step (ASSUME)
• Fifth Step (REFUTEInvalid Argument)

107
Final Proof Strategy

• Fourth Step (ASSUME)
• Pick a complex wff (having one of these forms
  ?(A?B), A?B, or A?B.).
• Assume one side or its negation, and then go to
  Second Step.
• Fifth Step (REFUTEInvalid Argument)
• Construct a refutation box containing any simple
  wffs.
• Assign each wff 1 or 0, by which we can show this
  argument is invalid (all premises true and
  conclusion false).

108
Final Proof Strategy

• First Step (START)
• Add asm followed by the conclusions simpler
  contradictory.
• Second Step (SI)
• Apply S-rules and I-rules to the complex wffs
• If you get a contradiction, then go to the Third
  Step.
• If you cant derive anything further but there is
  a complex wff, then go to the Fourth Step.
• If you cant derive anything further and every
  complex wff are used, then go to the Fifth Step.
• Third Step (RAA)
• Apply the RAA rule.
• Fourth Step (ASSUME)
• Pick a complex wff (having one of these forms
  ?(A?B), A?B, or A?B.).
• Assume one side or its negation, and then go to
  Second Step.
• Fifth Step (REFUTEInvalid Argument)
• Construct a refutation box containing any simple
  wffs.
• Assign each wff 1 or 0, by which we can show this
  argument is invalid (all premises true and
  conclusion false).

109
Exercise

1. B ? A
2. B ? A /??(A ? ?A)

110
Exercise

1. B ? A
2. B ? A /??(A ? ?A)
3. asm A ? ?A

111
Exercise

1. B ? A
2. B ? A /??(A ? ?A)
3. asm A ? ?A
4. asm B break 1
5. ?A from 2 and 4, MP
6. ??A from 3 and 5, MP
7. ??B from 4 5 contradicts 6, RAA

112
Exercise

1. B ? A
2. B ? A /??(A ? ?A)
3. asm A ? ?A
4. asm B break 1
5. ?A from 2 and 4, MP
6. ??A from 3 and 5, MP
7. ??B from 4 5 contradicts 6, RAA
8. ?A from 1 and 7, DS
9. ??A from 3 and 8, MP
10. ??(A ? ?A)from 3 8 contradicts 9, RAA

113
Exercise with Translation

1. If determinism is true and Dr. Freudlov correctly
   predicts what I will do, then if she tells me her
   prediction Ill do something else.
2. If Dr. Freudlov tells me her prediction and yet
   Ill do something else, then Dr. Freudlov doesnt
   correctly predict what Ill do.
3. Therefore, if determinism is true, then Dr.
   Freudlov doesnt correctly predict what Ill do
   or else she wont tell me her prediction.

114

1. determinism is true and Dr. Freudlov correctly
   predicts what I will do ? if she tells me her
   prediction Ill do something else
2. Dr. Freudlov tells me her prediction and yet
   Ill do something else ? Dr. Freudlov doesnt
   correctly predict what Ill do
3. ?determinism is true ? Dr. Freudlov doesnt
   correctly predict what Ill do or else she wont
   tell me her prediction

115

1. determinism is true ? Dr. Freudlov correctly
   predicts what I will do ? she tells me her
   prediction ? Ill do something else
2. Dr. Freudlov tells me her prediction ? Ill
   do something else ? Dr. Freudlov doesnt
   correctly predict what Ill do
3. ?determinism is true ? Dr. Freudlov doesnt
   correctly predict what Ill do ? she wont tell
   me her prediction

116

1. determinism is true ? Dr. Freudlov correctly
   predicts what I will do ? she tells me her
   prediction ? Ill do something else
2. Dr. Freudlov tells me her prediction ? Ill
   do something else ? ?Dr. Freudlov correctly
   predicts what Ill do
3. ?determinism is true ? ?Dr. Freudlov
   correctly predicts what Ill do ? ?she will
   tell me her prediction

117

1. (D ? P) ? (T ? E)
2. (T ? E) ? ?P /?D ? (?P ? ?T)

118

1. (D ? P) ? (T ? E)
2. (T ? E) ? ?P /?D ? (?P ? ?T)
3. asm ?(D ? (?P ? ?T))

119

1. (D ? P) ? (T ? E)
2. (T ? E) ? ?P /?D ? (?P ? ?T)
3. asm ?(D ? (?P ? ?T))
4. ?D from 3, NIF
5. ??(?P ? ?T) from 3, NIF

120

• (D ? P) ? (T ? E)
• (T ? E) ? ?P /?D ? (?P ? ?T)
• asm ?(D ? (?P ? ?T))
• ?D from 3, NIF
• ??(?P ? ?T) from 3, NIF
• ?P from 5, NOR
• ?T from 5, NOR
• ??(T ? E) from 2 and 6, MP
• ??E from 7 and 8, CS

121

• (D ? P) ? (T ? E)
• (T ? E) ? ?P /?D ? (?P ? ?T)
• asm ?(D ? (?P ? ?T))
• ?D from 3, NIF
• ??(?P ? ?T) from 3, NIF
• ?P from 5, NOR
• ?T from 5, NOR
• ??(T ? E) from 2 and 6, MP
• ??E from 7 and 8, CS
• asm ?(D ? P) break 1

122

• (D ? P) ? (T ? E)
• (T ? E) ? ?P /?D ? (?P ? ?T)
• asm ?(D ? (?P ? ?T))
• ?D from 3, NIF
• ??(?P ? ?T) from 3, NIF
• ?P from 5, NOR
• ?T from 5, NOR
• ??(T ? E) from 2 and 6, MP
• ??E from 7 and 8, CS
• asm ?(D ? P) break 1
• ??P from 4 and 10, CS
• ? D ? P from 10 6 contradicts 11, RAA

123

• (D ? P) ? (T ? E)
• (T ? E) ? ?P /?D ? (?P ? ?T)
• asm ?(D ? (?P ? ?T))
• ?D from 3, NIF
• ??(?P ? ?T) from 3, NIF
• ?P from 5, NOR
• ?T from 5, NOR
• ??(T ? E) from 2 and 6, MP
• ??E from 7 and 8, CS
• asm ?(D ? P) break 1
• ??P from 4 and 10, CS
• ? D ? P from 10 6 contradicts 11, RAA
• ? T ? E from 1 and 12, MP
• ? E from 7 and 13, MP
• ? D ? (?P ? ?T) from 3 9 contradicts 14, RAA

124
Refutation

• A Refutation
• A set of truth conditions making all premises
  true and conclusion false.
• Showing that the argument is invalid.
• When assuming the negation of conclusion, it
  wont lead to a contradiction.
• By the truth-assignment of every simple wff, we
  can construct a refutation box.

125
Construct a Refutation Box

• First Step and Second Step are the same as
  constructing a proof.
• However, we cant apply RAA, for we cant derive
  a contradiction by applying S-rules and I-rules
  to wffs.
• So we assign each simple wff 1 or 0 in a way
  that makes all premises true but conclusion false.

126
For example

1. T
2. T ? (B ? M)
3. M ? H /?B

127

• T
• T ? (B ? M)
• M ? H /?B
• asm ?B
• ? B ? M from 1 and 2, MP
• ?M from 4 and 5, DS
• ?H from 3 and 6, MP
• No
  contradiction!

128

• T
• T ? (B ? M)
• M ? H /?B
• asm ?B
• ? B ? M from 1 and 2, MP
• ?M from 4 and 5, DS
• ?H from 3 and 6, MP
• To make premises true and conclusion false
• T 1, B 0

129

• T
• T ? (B ? M)
• M ? H /?B
• asm ?B
• ? B ? M from 1 and 2, MP
• ?M from 4 and 5, DS
• ?H from 3 and 6, MP
• To make premises true and conclusion false
• T 1, B 0
• Since T ? (B ? M) 1 and, given that T 1 and B
  0, M 1.

130

• T
• T ? (B ? M)
• M ? H /?B
• asm ?B
• ? B ? M from 1 and 2, MP
• ?M from 4 and 5, DS
• ?H from 3 and 6, MP
• To make premises true and conclusion false
• T 1, B 0
• Since T ? (B ? M) 1 and, given that T 1 and B
  0, M 1.
• To make M ? H 1, given M 1, H 1.

131

• T
• T ? (B ? M)
• M ? H /?B
• Under the truth-assignment of T 1, B 0, M
  1, and H 1, this argument is invalid.

132
Exercise

• 1. A ? B /?A

133
Exercise

1. A ? B /?A
2. asm ?A

134
Exercise

1. A ? B /?A
2. asm ?A
3. ?B from 1 and 2, DS

135
Exercise

• A ? B /?A
• asm ?A
• ?B from 1 and 2, DS
• Assign A 0 and A ? B 1

136
Exercise

• A ? B /?A
• asm ?A
• ?B from 1 and 2, DS
• Assign A 0 and A ? B 1
• So B 1

137
Exercise

• A ? B /?A
• asm ?A
• ?B from 1 and 2, DS
• Under the truth-assignment of A 0 B 1, this
  argument is invalid.

138
5. Appendix Model Theory

• Two approaches to Validity
• Proof Theory an argument is valid if and only if
  there is a formal proof of it. (an argument is a
  series of small steps, each based on S-rules and
  I-rules)
• Model Theory an argument is valid if and only if
  there is a truth-table showing that no possible
  case has premises all true and conclusion false.

139
Model Theory

• A model is an interpretation of formal language,
  such as propositional language.
• As introduced at the beginning, PL contains two
  components
• Vocabulary
• Formation Rules
• Therefore, a model must interpret the vocabulary
  of PL.

140
Interpretation

• Vocabulary contains two parts
• Logical constants (logical connectives)
• Logical variables (propositional letters)
• A model interprets logical connectives by truth
  functions and propositional letters by truth
  values (1 or 0).
• Recall the truth-assignment test
• An argument is valid if and only if no possible
  truth-assignment can make all premises true and
  conclusion false.

141
Formal Explanation

• A model for PL is an interpretation function I
• I(?) f?(x) 1 ? v(x)
• I(?) f?(ltx, ygt) v(x) ? v(y)
• I(?) f?(ltx, ygt) v(x) v(y) ? v(x) ? v(y)
• I(?) f?(ltx, ygt) v(y) ? v(x) ? v(y)
• I(PL) v(PL)
• (v PL?0, 1)