Valid and Invalid Arguments

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 Test Example 1 p  (q  r) ~r  p  q

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

premises conclusion p q r q  r p(qr) ~r p  q T T T T F 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(qr) ~r p  q T T T T F T T F T F T T F F F T T F T F F F T F F F

Argument Validity Test Example 2 p  q  ~r q  p  r  p  r

premises p q r ~ r q~r pr p r T T T T T F T F T T F F F T T F T F 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

premises p q r ~ r q~r pr p r T T T F T T T F T F T T F F F T T conclusion p q r ~ r q~r pr pq~r qpr p r T T T F T T T F T F T T F F F T T F T F F F T F F F

premises p q r ~ r q~r pr p r T T T F T T T F T F T T F F F T T conclusion p q r ~ r q~r pr pq~r qpr p r T T T F T T T F T F T T F F F T T F T F F F T F F F

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 pq p q T T T F F T F F

Modus Ponens premises conclusion p q pq p q T T T T F F F T F F

Modus Ponens premises conclusion p q pq p q T T T T F 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 pq ~q ~p T T T F F T F F

Modus Tollens premises conclusion p q pq ~q ~p T T T F T F F T F F

Modus Tollens premises conclusion p q pq ~q ~p T T T F T F F T F F

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  pq q  pq

Specialization pq  p pq  q

Elimination pq ~q  p p  q ~p  q

Transitivity pq qr pr

Division into Cases pq pr qr r

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 Example Statements 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 Example Statements 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 Example Symbols 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, c in Symbols a. p  q b. r  s c. r  t

Valid Inference Example Statements 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 Example Statements 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 Example Symbols u, v. u =I was reading my book in bed. v = My glasses are on the bed table.

Statements d, e, f in 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  t from ( ) r from ( )  t by_____________

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

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. pq q  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. pq ~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. pq ~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 star then 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 ~pc T 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