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Valid and Invalid Arguments
M260 2.3
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.
Logical Form
• If Socrates is a human being, then Socrates is mortal;Socrates is a human being; Socrates is mortal.
• If p then q;p;q
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.
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
Argument Validity TestExample 1
• p (q r)• ~r p q
premises conclusion
p q r q r p(qr) ~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
premises conclusion
p q r q r p(qr) ~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
premises conclusion
p q r q r p(qr) ~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
Argument Validity TestExample 2
• p q ~r
• q p r p r
premises conclusion
p q r ~ r q~r pr pq~r qpr 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
premises conclusion
p q r ~ r q~r pr pq~r qpr 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
premises conclusion
p q r ~ r q~r pr pq~r qpr 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
Rules of Inference(Valid Argument Forms)
• Modus Ponens• Modus Tolens• Generalization• Specialization
• Elimination• Transitivity• Division into Cases• Rule of Contradiction
Modus Ponens
• If p then q;
• p; q
Modus Ponens
premises conclusion
p q pq p q
T T
T F
F T
F F
Modus Ponens
premises conclusion
p q pq p q
T T T T T
T F F T F
F T T F T
F F T F F
Modus Ponens
premises conclusion
p q pq p q
T T T T T
T F F T F
F T T F T
F F T F F
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.
Modus Tollens
• If p then q;
• ~q; ~p
Modus Tollens
premises conclusion
p q pq ~q ~p
T T
T F
F T
F F
Modus Tollens
premises conclusion
p q pq ~q ~p
T T T F F
T F F T F
F T T F T
F F T T T
Modus Tollens
premises conclusion
p q pq ~q ~p
T T T F F
T F F T F
F T T F T
F F T T T
Modus Tollens Example
• If Zeus is human, then Zeus is mortal.
• Zeus is not mortal. Zeus is not human
• Modus tollens uses the contrapositive.
Generalization
• p pq
• q pq
Specialization
• pq p
• pq q
Elimination
• pq• ~q p
• p q• ~p q
Transitivity
• pq
• qrpr
Division into Cases
• pq• pr
• qrr
Division into Cases Example
• x>1 or x<-1
• If x>1 then x2>1
• If x<-1 then x2>1 x2>1
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.
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.
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.
Statements a, b, cin Symbols
• a. p q
• b. r s• c. r t
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.
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.
Valid Inference ExampleSymbols u, v.
• u =I was reading my book in bed.
• v = My glasses are on the bed table.
Statements d, e, fin Symbols
• d. ~q
• e. u v
• f. s p
Inference Example Givens
• a. p q• b. r s• c. r t
• d. ~q• e. u v• f. s p
Deduction Sequence
• 1. p q from ( )~q from ( ) ~p by __________
• 2. s p from ( )~p from ( ) ~s by__________
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
Deduction Sequence
• 3. r s from ( )~s from ( ) r by_____________
• 4. r tfrom ( )r from ( ) t by_____________
Deduction Sequence
• 3. r s from (b)~s from (2) r by disjunctive syllogism
• 4. r tfrom (c)r from (3) t by modus ponens
Errors in Reasoning
• Using vague or ambiguous premises.
• Circular reasoning
• Jumping to conclusions
• Converse error
• Inverse error
Converse Error
• If Zeke is a cheater, then Zeke sits in the back row. Zeke sits in the back row. Zeke is a cheater.
• pqq p
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.
• pq~p ~q
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.
• pq~p ~q
Validity vs. Truth
• Valid arguments can have false conclusions if one of the premises is false.
• Invalid arguments can have true conclusions.
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.
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.
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
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 ~pc p
T F F T T
F T F F F
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.)
Rules of Inference(Valid Argument Forms)
• Modus Ponens• Modus Tolens• Disjunctive Addition• Conjunctive
Simplification
• Disjunctive Syllogism• Hypothetical
Syllogism• Division into Cases• Rule of Contradiction
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