Valid and Invalid Arguments

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Valid and Invalid Arguments

• M260 2.3

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Argument

• An argument is a sequence of statements. The
  final statement is called the conclusion, the
  others are called the premises.
• ? therefore before the conclusion.

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Logical Form

• If Socrates is a human being, then Socrates is
  mortalSocrates is a human being? Socrates is
  mortal.
• If p then qp?q

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Valid Argument

• An argument form is valid means no matter what
  particular statements are substituted for the
  statement variables, if the resulting premises
  are all true, then the conclusion is also true.
• An argument is valid if its form is valid.

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Test for Validity

• Identify premises and conclusion
• Construct a truth table including all premises
  and conclusion
• Find rows with premises true (critical rows)
• If conclusion is true on all critical rows,
  argument is valid
• Otherwise argument is invalid

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Argument Validity TestExample 1

• p ? (q ? r)
• r
• ? p ? q

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premises premises conclusion
p q r q ? r p?(q?r) r p ? q
T T T
T T F
T F T
T F F
F T T
F T F
F F T
F F F
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premises premises conclusion
p q r q ? r p?(q?r) r p ? q
T T T T T F T
T T F T T T T
T F T T T F T
T F F F T T T
F T T T T F T
F T F T T T T
F F T T T F F
F F F F F T F
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premises premises conclusion
p q r q ? r p?(q?r) r p ? q
T T T T T F T
T T F T T T T
T F T T T F T
T F F F T T T
F T T T T F T
F T F T T T T
F F T T T F F
F F F F F T F
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Argument Validity TestExample 2

• p ? q ? r
• q ? p ? r
• ? p ? r

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premises premises conclusion
p q r r q?r p?r p?q?r q?p?r p ?r
T T T
T T F
T F T
T F F
F T T
F T F
F F T
F F F
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premises premises conclusion
p q r r q?r p?r p?q?r q?p?r p ?r
T T T F T T T T T
T T F T T F T F F
T F T F F T F T T
T F F T T F T T F
F T T F T F T F T
F T F T T F T F T
F F T F F F T T T
F F F T T F T T T
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premises premises conclusion
p q r r q?r p?r p?q?r q?p?r p ?r
T T T F T T T T T
T T F T T F T F F
T F T F F T F T T
T F F T T F T T F
F T T F T F T F T
F T F T T F T F T
F F T F F F T T T
F F F T T F T T T
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Rules of Inference(Valid Argument Forms)

• Modus Ponens
• Modus Tolens
• Generalization
• Specialization

• Elimination
• Transitivity
• Division into Cases
• Rule of Contradiction

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Modus Ponens

• If p then q
• p
• ? q

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Modus Ponens
premises premises conclusion
p q p?q p q
T T
T F
F T
F F
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Modus Ponens
premises premises conclusion
p q p?q p q
T T T T T
T F F T F
F T T F T
F F T F F
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Modus Ponens
premises premises conclusion
p q p?q p q
T T T T T
T F F T F
F T T F T
F F T F F
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Modus Ponens Example

• If the last digit of this number is 0, then the
  number is divisible by 10.
• The last digit of this number is a 0.
• ? This number is divisible by 10.

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Modus Tollens

• If p then q
• q
• ? p

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Modus Tollens
premises premises conclusion
p q p?q q p
T T
T F
F T
F F
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Modus Tollens
premises premises conclusion
p q p?q q p
T T T F F
T F F T F
F T T F T
F F T T T
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Modus Tollens
premises premises conclusion
p q p?q q p
T T T F F
T F F T F
F T T F T
F F T T T
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Modus Tollens Example

• If Zeus is human, then Zeus is mortal.
• Zeus is not mortal.
• ? Zeus is not human
• Modus tollens uses the contrapositive.

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Generalization

• p
• ? p?q

• q
• ? p?q

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Specialization

• p?q
• ? p

• p?q
• ? q

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Elimination

• p?q
• q
• ? p

• p ? q
• p
• ? q

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Transitivity

• p?q
• q?r
• ?p?r

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Division into Cases

• p?q
• p?r
• q?r
• ?r

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Division into Cases Example

• xgt1 or xlt-1
• If xgt1 then x2gt1
• If xlt-1 then x2gt1
• ? x2gt1

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Valid Inference ExampleStatements a, b, c.

• a. If my glasses are on the kitchen table, then I
  saw them at breakfast.
• b. I was reading the newspaper in the living room
  or I was reading the newspaper in the kitchen.
• c. If I was reading the newspaper in the living
  room, then my glasses are on the coffee table.

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Valid Inference ExampleStatements a, b, c.

• a. If my glasses are on the kitchen table, then I
  saw them at breakfast.
• b. I was reading the newspaper in the living room
  or I was reading the newspaper in the kitchen.
• c. If I was reading the newspaper in the living
  room, then my glasses are on the coffee table.

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Valid Inference ExampleSymbols p, q, r, s, t.

• p My glasses are on the kitchen table.
• q I saw my glasses at breakfast.
• r I was reading the newspaper in the living
  room
• s I was reading the newspaper in the kitchen.
• t My glasses are on the coffee table.

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Statements a, b, cin Symbols

• a. p ? q
• b. r ? s
• c. r ? t

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Valid Inference ExampleStatements d, e, f.

• d. I did not see my glasses at breakfast.
• e. If I was reading my book in bed, then my
  glasses are on the bed table.
• f. If I was reading the newspaper in the kitchen,
  then my glasses are on the kitchen table.

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Valid Inference ExampleStatements d, e, f.

• d. I did not see my glasses at breakfast.
• e. If I was reading my book in bed, then my
  glasses are on the bed table.
• f. If I was reading the newspaper in the kitchen,
  then my glasses are on the kitchen table.

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Valid Inference ExampleSymbols u, v.

• u I was reading my book in bed.
• v My glasses are on the bed table.

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Statements d, e, fin Symbols

• d. q
• e. u ? v
• f. s ? p

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Inference Example Givens

• a. p ? q
• b. r ? s
• c. r ? t

• d. q
• e. u ? v
• f. s ? p

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Deduction Sequence

• 1. p ? q from ( ) q from ( ) ? p by
  __________
• 2. s ? p from ( ) p from ( ) ?
  s by__________

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Deduction Sequence

• 1. p ? q from (a) q from (d) ? p by modus
  tollens
• 2. s ? p from (f) p from (1) ? s by modus
  tollens

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Deduction Sequence

• 3. r ? s from ( ) s from ( ) ?
  r by_____________
• 4. r ? t from ( ) r from ( ) ?
  t by_____________

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Deduction Sequence

• 3. r ? s from (b) s from (2) ? r by
  disjunctive syllogism
• 4. r ? t from (c) r from (3) ? t by modus
  ponens

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Errors in Reasoning

• Using vague or ambiguous premises.
• Circular reasoning
• Jumping to conclusions
• Converse error
• Inverse error

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Converse Error

• If Zeke is a cheater, then Zeke sits in the back
  row. Zeke sits in the back row. ? Zeke is a
  cheater.
• p?qq ? p

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Inverse Error

• If interest rates are going up,then stock market
  prices will go down.Interest rates are not going
  up? Stock market prices will not go down.
• p?qp ? q

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Inverse Error

• If I intend to sell my house, then I will need a
  permit for this wall.I do not intend to sell my
  house.? I do not need a permit for this wall.
• p?qp ? q

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Validity vs. Truth

• Valid arguments can have false conclusions if one
  of the premises is false.
• Invalid arguments can have true conclusions.

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Valid but False

• If John Lennon was a rock starthen John Lennon
  had red hair.
• John Lennon was a rock star.
• ? John Lennon had red hair.

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Invalid but True

• If New York is a big city,then New York has tall
  buildings.
• New York has tall buildings.
• ? New York is a big city.

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Contradiction Rule

• If the supposition that p is false leads to a
  contradiction then p is true.
• p ? c, where c is a contradiction.? p

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Contradiction Rule

• If the supposition that p is false leads to a
  contradiction then p is true.
• p ? c, where c is a contradiction.? p

premise conclusion
p p c p?c p
T F F T T
F T F F F
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Rule of Contradiction Example

• Knights tell the truth, Knaves lie.
• A says B is a knight.
• B says A and I are opposite types.
• What are A and B?
• (Hint Suppose A is a Knight.)

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Rules of Inference(Valid Argument Forms)

• Modus Ponens
• Modus Tolens
• Disjunctive Addition
• Conjunctive Simplification

• Disjunctive Syllogism
• Hypothetical Syllogism
• Division into Cases
• Rule of Contradiction