Valid and Invalid Arguments

1
Valid and Invalid Arguments

• Section 1.3

2
Monty Python and the Holy Grail

• Quote 32 Witch Scene
• From http//www.wavsite.com/sounds.asp?ID106

• There are ways of telling whether she is a witch
• What do you do with witches? Burn them!
• What else do you burn? Wood!
• Why do witches burn? Because theyre made of
  wood!
• How do you tell if she is made of wood? Does
  wood sink in water? It floats!
• What also floats? Bread! Apples! Very small
  rocks! A duck!
• If she weighs the same as a duck, then shes made
  of wood
• And therefore a witch!

3
Monty Python and the Holy Grail

• Villagers burn Witches
• Villagers burn Wood
• If Witches are made of wood, then they float
• Bread and Apples and Very small rocks and ducks
  float
• If she weighs the same as a duck, then shes made
  of wood
• ? Shes a witch!

4
Deductive Logic

• The central concept of deductive logic is an
  argument
• An argument is a sequence of statements aimed at
  demonstrating the truth of an assertion (or
  claim)
• The statements are ordered in such a way that
  some are referred to as the premise and others
  are the conclusion
• Form of an argument
• Premise
• Premise
• ..
• ? Conclusion
• Example
• If Roland eats sweets, then he will get fat
• Roland eats sweets
• ? Roland gets fat

Sequence of statements or propositions
If p then q p ? q
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Valid Arguments

• Whenever statements are substituted that make all
  the premises True, then the conclusion is also
  True
• An argument is valid if its form is valid
• There are 2 ways to determine if an argument is
  valid
• Truth Tables
• Inference Rules

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Valid Arguments by Truth Tables

• Steps to determine validity of argument
• Identify premises and conclusions
• Construct truth table
• Find critical rows
• In each critical row, determine if conclusion is
  True
• Example
• P ? Q
• P
• ? Q

prime propositions
premises
conclusion
Valid argument because conclusion is True
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Valid Argument Example (continued)
(P ? Q) ? R Q ? (P ? R)
T T T T T F
T T T F T T F
T T F T T T T
T T F F T T T
T F T T T T
F T F T F T T
F F F F T F T
T T F F F F F
T F
All critical rows are True ? Valid
8
Invalid Arguments

• When statements are substituted that make all the
  premises True, but the conclusion is False
• Example
• P ? Q
• P
• ? Q

prime propositions
premises
conclusion
Invalid argument because conclusion is False
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Invalid Argument Example (continued)
(P ? Q) ? R P ? Q ? R
T T T T T F
T T T F T F F
T T F T T T F
T T F F T F F
F F T T T T
T T F T F T F
T T F F T F T
T T F F F F T
T F
All critical rows are not True ?
Invalid
10
Inference Rules

• A valid argument form that enables the
  elimination or the introduction of a logical
  connective
• Logical Laws and Equivalences are used to do
  Transformational Proofs (covered in previous
  section)
• Logical Equivalences and Inferences are used to
  do Deductive Proofs
• A Deductive Proof is an approach to establishing
  the validity of an argument by using a series of
  simpler arguments known to be valid

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Modus Ponens

• If both an implication and its hypothesis are
  known to be True, then the conclusion of the
  implication is also True
• Latin Method of Affirming since conclusion is
  an affirmation
• It has the form
• If P then Q
• P
• ? Q
• Example
• If the patient has a pulse, then the patients
  heart is pumping
• The patient has a pulse
• ? The patients heart is pumping

Prove to yourself with a Truth Table!!
12
Modus Tollens

• If an implication is known to be True and the
  conclusion of the implication is False, then the
  hypothesis of the implication is also False
• Latin Method of Denying since conclusion is a
  denial
• It has the form
• If P then Q
• Q
• ? P
• Example
• If I win the lottery then I am lucky
• I am not lucky
• ? I did not win the lottery

Prove to yourself with a Truth Table!!
13
Other Valid Argument Forms

• Generalization Used to make generalizations
• Form
• P Q
• ?P ? Q ?P ? Q
• Example You are asked to count the number of
  people in class that have dark (brown or black)
  hair
• Sally has brown hair
• ?Sally has brown hair or black hair

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Other Valid Argument Forms (contd)

• Specialization Used for specializing
• Form
• P Q P Q
• ?P ?Q

• Can an argument be valid if it has no critical
  rows? Yes! Consider the following
• P P
• ?P

• No critical rows, yet the conclusion is True
• True because of conjunctive simplification
  (vacuously true)

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Other Valid Argument Forms (contd)

• Elimination If you have two possibilities and
  you can rule one out, then the other must be the
  case
• Form
• P ? Q P ? Q
• Q P
• ?P ?Q

• Example
• Bill has a broken leg or a sprained ankle
• Bill does not have a sprained ankle
• ? Bill has a broken leg

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Other Valid Argument Forms (contd)

• Transitivity Chains of if then statements if
  the first implies a second the second implies a
  third, then you can conclude the first implies
  the third
• Form
• P ? Q
• Q ? R
• ? P ? R

T T T T T T T T F
T F F T F T F T
T T F F F T F F T
T T T T F T F T
F T F F T T T
T F F F T T T

• Example
• If 8 is divisible by 4, then 8 is divisible by 2
• If 8 is divisible by 2, then the sum of the
  digits of 8 is divisible by 2
• ? If 8 is divisible by 4, then the sum of the
  digits of 8 is divisible by 2

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Other Valid Argument Forms (contd)

• Proof by Division into Cases If you know one
  proposition or another is True, and in either
  case a certain conclusion follows, then the
  conclusion must be True
• Form
• P ? Q
• P ? R
• Q ? R
• ? R
• Example
• Wayne walks or runs 3 miles a day
• If Wayne walks 3 miles, then Wayne gets exercise
• If Wayne runs 3 miles, then Wayne gets exercise
• ? Wayne gets exercise

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A More Complex Deduction

• A set of premises and conclusions are given. Use
  Inference Rules to deduce the conclusion from the
  premises giving a reason for each step.
• Given
• B ? D ? E
• E
• C ? D
• ? C

B ? D ? E Premise E Premise ? (B ? D) Modus
Tollens (1)
(B ? D) (1) B D DeMorgan ?
D Specialization (2)
C ? D Premise D (2) ? C Modus Tollens
(Conclusion)
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Fallacies

• A Fallacy is an error in reasoning that results
  in an invalid argument
• Three common fallacies are
• Using vague or ambiguous premises
• Begging the question (assuming what is to be
  proved without deriving it from the premises)
• Jumping to a conclusion (without adequate
  grounds)
• Two other fallacies that resemble Modus Ponens
  and Modus Tollens but are not in fact valid
• Inverse Error
• Converse Error

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Converse Error

• Example
• If the butler did it, then he has blood on his
  hands
• The butler had blood on his hands
• ? The butler did it
• Argument form
• P ? Q
• Q
• ? P
• Fallacy is called converse error because the
  conclusion of the argument would follow from the
  premises if the premise P ? Q were replaced by
  its converse
• Remember, an implication is not logically
  equivalent to its converse i.e. P ? Q ? Q ? P

P Q P ? Q Q P T T T T T T
F F F T F T T T
F F F T F F
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Inverse Error

• Example
• If the butler is nervous, then he did it
• The butler is very mellow
• ? The butler did not do it
• Argument form
• P ? Q
• P
• ? Q
• Fallacy is called inverse error because the
  conclusion of the argument would follow from the
  premises if the premise P ? Q were replaced by
  its inverse
• Remember, an implication is not logically
  equivalent to its inverse i.e. P ? Q ? P ? Q

P Q P ? Q P Q T T T F F T
F F F T F T T T
F F F T T T
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Contradiction

• Suppose that P is a statement whose truth you
  wish to deduce If you can show that the
  supposition that statement P is false leads
  logically to a contradiction, then you can
  conclude that p is True
• Form
• P ? c
• P
• Example
• Percule Hoirot murder mystery

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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 candlestick.
• 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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Translation of Problems

• Lord Hazelton, the murdered man, was killed by a
  blow on the head with a brass candlestick.
• Weapon_candlestick
• Either Lady Hazelton or a maid, Sara, was in the
  dining room at the time of the murder.
• Lady_dining V 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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Translation of Problems

• Weapon_candlestick
• Lady_dining ? Sara_dining
• Cook_kitchen ? butlerDidIt
• Cook_kitchen ? Weapon_candlestick
• Lady_dining ? chauffeurDidIt
• Cook_kitchen ? Sara_dining
• Sara_dining ? stewardDidIt

WC LD V SD CK ? BD CK ? WC LD ? CD CK ? SD SD
? STD
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Translation of Problems

• Case 1 Sara_dining (SD)
• CK ? WC Premise
• WC Premise
• CK Modus Tollens (1)
• CK ? SD Premise
• SD Premise
• CK Modus Tollens (2)
• CK ? BD Premise
• CK (2)
• BD Modus Ponens (3)
• SD ? STD Premise
• SD Premise
• STD Modus Ponens (4)

• Case 2 Lady_dining (LD)
• LD ? CD Premise
• LD Premise
• CD Modus Tollens

chauffeurDidIt!
(3) and (4) cant both be right butlerDidIt and
stewardDidIt - Contradiction! So, LD ? c ? LD
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Other Examples

• Nerds are good at math
• Buffy is good at math.
• ? Buffy is a nerd.

P x is a nerd Q x is good at math P ? Q Q ?P
- Invalid by converse error!
P x is tired Q x finds it difficult to
study P ? Q Q ? P - Valid by Modus Tollens!
It is difficult to study whenever I am tired I
found it easy to study today. ? I am not tired
today.
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Other Examples

• Animals at the bottom of the food chain are very
  nervous
• People are not at the bottom of the food chain.
• ? People are not nervous

P x is at the bottom of the food chain Q x
is nervous P ? Q P ?Q - Invalid by inverse
error!
Im always in a good mood when I feel good. I
feel great today. ? Im in a good mood.
P x feels good Q x is in a good mood P ?
Q P ?Q - Valid by Modus Ponens!
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Other Examples

• P ? Q ? R
• S ? Q
• T
• P ? T
• P ? R ? S
• ? Q

P ? Q ? R Premise P ? Q (2) ? R Modus Ponens
(3) P (1) R (3) ? P ? R Conjunction
(4) P ? R ? S Premise P ? R (4) ? S Modus
Ponens (5) S ? Q Premise S (5) ?
Q Elimination

• P ? T Premise
• T Premise
• ? P Modus Tollens (1)
• P (1)
• ? P ? Q Generalization (2)