1. Review for Final Exam 80-210: Logic & Proofs August 6, 2009 Karin Howe

2. Definitions you should know… Validity and invalidity (remember kangaroos!) Types of statements: tautology, contradictory, contingent

3. Definitions An argument is valid iff: it is impossible for all of its premises to be true and its conclusion false. An argument is invalid iff: it is possible that all of its premises to be true, and yet the conclusion is false.

4. Definitions A statement is a _______ iff _________: –Tautology (logically true)… …iff it is true on every truth-value assignment (or interpretation) Examples: (P   P), (  x)(P(x)   P(x)) –Contradictory formula (logically false)… …iff it is false on every truth-value assignment (or interpretation) Examples: (P &  P), (  x)(P(x) &  P(x)) –Contingent … … iff it is false on some truth-value assignments (or interpretations), and true on others Example: (P & Q), (  x)(P(x) & Q(x))

5. Skills you should be comfortable with… Translating statements (conjunctions, disjunctions, conditionals, negations, universals, and existentials) Well-formed formulas (WFFs), and parse trees (propositional and predicate) Constructing and interpreting truth tables or interpretations Truth trees (for propositional and predicate statements or arguments) –For determining validity/invalidity –For determining whether a statement is a tautology/contradictory statement/contingent statement Constructing counter-examples (for propositional and predicate arguments) Derivations (for propositional and predicate arguments) –Fill-in-the-blanks –Complete derivations (by hand)

6. Practice: Propositional Translations 1.Provided that he does well on the LSAT, John will apply to law school. 2.If I had some bread I could make a ham sandwich if I had some ham. 3.Show me a guy who doesn’t know which side his bread is buttered on and I’ll show you a guy with a slippery sandwich. 4.Myra will quit unless she is promoted. 5.Beth and Carmen are not both eligible. 6.Beth and Carmen are both not eligible.

7. Practice: Predicate Translations 1.Provided that he does well on the LSAT, John will apply to law school. oW(x,y) = x will do well on y, A(x,y) = x will do apply to y, j = John, l = LSAT, s = law school 2.Any waitress has met cheap people. oM(x,y) = x has met y, W(x) = x is a waitress, C(x) = x is a cheap person 3.One is not a prime. oP(x) = x is a prime, 1 = one

8. Practice: Predicate Translations 1.Unaccompanied children will be given a free kitten. oU(x) = x is an unaccompanied child, K(x) = x is a kitten, F(x) = x is free, G(x,y) = x is given y 2.A person who is nice to you, but rude to the waiter, is not a nice person. oN(x,y) = x is nice to y, R(x,y) = x is rude to y, N(x) = x is nice, P(x) = x is a person 3.Show me a doctor who makes house calls and I’ll show you a PhD that’s selling vacuum cleaners oD(x) = x is a doctor, H(x) = x makes house calls, P(x) = x is a PhD, V(x) = x sells vacuum cleaners

9. Practice: WFFs (Propositional) 1.(  {(I & J)  [K  (C  R)]}) 2.{  [  (  J  B   D]} 3.{  [(F [  X)  L]  [(U  G)  (K  B)]} 4.[I  {S  [(X & I)  {V]})  (  I  N)] 5.{  [N  (Y & { (U  U)  [M & (Z  V)]})]} 6.({  (V  J)  [(E  Q)  A]}  W)

10. Practice: WFFs (Predicate) 1.(  z)(P(a) & Q(z)) 2.(  z)P(a) 3.(  z)(  a)(  R(z,a) &  R(a,z)) 4.(  z)(  R(z,a) &  R(a,z)) 5.(  y)  L(y,y) & M(a) 6.(   x)(Q(x)   P(x)) 7.  (  x)(Q(x)   P(x)) 8.(  z)(  x)(R(x,x) & R(x,x)) 9.(  z)(Ex)(  R(z,x) &  R(x,z))

11. Truth Tables: Propositional Logic pqp & qpq p  qp  q p pp TTTTTTTF TFFTFTFT FTFFTT FFFFFF pq p  qp  q pq p  qp  q TTTTTT TFFTFF FTTFTF FFTFFT

12. Memorizing the characteristic truth tables Conjunctions: A conjunction is true only when both conjuncts are true. Disjunctions: A disjunction is false only when both disjuncts are false. Conditionals: Conditionals are false only when it is the case that the antecedent is true and the consequent false. Negations: The truth value of a negation is always the opposite of the truth value of the un-negated statement. Biconditionals: Biconditionals are true when both sides match in truth value.

13. Interpretations: Predicate Logic All of the usual truth table rules for the connectives will still apply Definition: Truth and falsity with respect to an interpretation: 1.If  is a 0-place predicate letter, then  is true iff I(  ) = T. 2.If  is of the form  (x 1, …, x n ) where  is a n-place predicate letter (with n > 0), and x 1, …, x n are n terms, then  is true on I iff is in I(  ) 3.If  is of the form (  u) , then  is true on I iff for each member a of the domain of discourse  is true on I[a/u], and false otherwise. 4.If  is of the form (  u) , then  is true on I iff there is at least one member a of the domain of discourse such that  is true on I[a/u,] and false otherwise.

14. Practice: Determining Truth/Falsity Domain: Natural numbers Dictionary: P(x) = x is prime, E(x) = x is even, L(x,y) = x is less than y, K(x) = x is a kangaroo, B = brown is my favorite color Interpretation: –I(P) = {2,3,5,7,…} –I(E) = {2,4,6,…} –I(G) = {,,,,, …} –I(K) =  –I(B) = F True or false? 1.(  x)(P(x) & E(x)) 2.(  x)[(P(x) & E(x)) > L(1,x)] 3.(  x){B > (  x)[(E(x) & P(x)) & L(2,x)]} 4.(  x)(E(x) v K(x)) 5.(  x)(  y)L(x,y)

15. Truth Trees: Propositional Rules (p & q) p q (p  q) p q (p  q)  p q p  q p  p q  p  (p & q)  p  q (p  q)pq(p  q)pq  (p  q) p  q  p p  ( p  q) p  p  q q

16. Truth Trees: Predicate Rules (  x)(P(x) & Q(x)) (P(a) & Q(a))* * Constant introduced must be new to the branch (  x)(P(x)  Q(x)) (P(a)  Q(a)) a = b P(a) P(b)  (  x)(P(x) & Q(x)) (  x)  (P(x) & Q(x))  (  x)(P(x)  Q(x)) (  x)  (P(x)  Q(x)) ¬a = a *

17. Practice: Truth Trees (Propositional) Valid or invalid? 1.  A &  B   (A & B) 2.H  I   I   H 3.D  (F  E)  D  E 4.C  D, D  E  E  D 5.  (H  I)   H   I 6.  (H  I)   (H  I)

18. Practice: Truth Trees (Propositional) Tautology, contradictory, or contingent? 1.(N  O) & (N   O) 2.P  [(  P & Q)  (  P &  Q)] 3.(S   S) & ( S  S) 4.{[(N  G) & (N  R)] & (G  R)}  R 5.  (K &  L) & (  K  L) 6.  (Z   A)  (  Z & A)

19. Practice: Truth Trees (Predicate) Valid or invalid? 1.(  x)(A(x)  B(x))  (  x)(A(x) & B(x)) 2.(  x)F(x)  (  x)(G(x)  (  x)(F(x)  G(x)) 3.  (  x)(  y)G(x,y)  (  x)  G(x,x) 4.(  x)(x=a  J(x))  (J(a) 5.(P(a) & Q(a)), (Q(b)   P(a))  a=b 6.(  x)B(x), (  x)A(x)  (  x)(A(x) & B(x)) 7.(  x)[S(x)  (  y)(A(y) &  T(y,x))]  (  x)[A(x) & (  y)(S(y)   T(x,y))]

20. Practice: Truth Trees (Predicate) Tautology, contradictory, or contingent? 1.(a=b &  a=b) 2.(  B(f)   (  x)B(x)) 3.((  x)C(x) & (  x)  C(x)) 4.((  x)A(x)  (  x)A(x)) 5.((  x)A(x) & (  x)  B(x)) 6.((  x)C(x) & (  x)  C(x)) 7.(  x)(x=a   x=a)

21. Basic Rules (Propositional): (In)troduction Rules ConjunctionDisjunctionConditional &I  IL  IR II p q__________ p & q &I p________ q  p  IL p________ p  q  IR p  q  I p A. q

22. Basic Rules (Propositional): (In)troduction Rules NegationBiconditional II II II p  p___________   I  p  I p  q  I p A.  p A. q q A. p

23. Basic Rules (Predicate): (In)troduction Rules Universal Introduction Existential Introduction Identity Introduction II II =I p1. (P(v)  Q(v))*_______ (  x)(P(x)  Q(x))  I * Restrictions: 1. v is a variable 2. v does not occur in (  x)(P(x)  Q(x)) 3. v does not occur free in any assumption on which line p1 depends P(a) & Q(a)_____ (  x)(P(x) & Q(x))  I. a = a =I

24. Basic Rules (Propositional): Elimination (Out) Rules ConjunctionDisjunctionConditional &EL&ER EE EE p & q_____ p &EL p & q_____ q &ER p  q r  E p  q p________ q  E p A. r q A. r

25. Basic Rules (Propositional): Elimination (Out) Rules NegationBiconditional EE  EL  ER p  E p  q q______________ p  EL p  q p_____________ q  ER  p A. 

26. Basic Rules (Predicate): Elimination (Out) Rules Universal Elimination Existential Elimination Identity Elimination EE EE =E (  x)(P(x)  Q(x))__ (P(a)  Q(a))  E (  x)(P(x) & Q(x)) p2. p3.   E * Restrictions: 1.v is a variable 2.v does not appear in (  x)(P(x) & Q(x)). 3.v does not occur in , 4.v does not occur free in any lines that p2 depends on. P(a) a=b______ P(b) =E (P(v) & Q(v))* A. 

27. Propositional Derived Rules: Hammers Disjunctive SyllogismModus Tollens DSLDSRMT p  q  q_______ p p  q  p_______ q p  q  q_______  p

28. Predicate Derived Rules: Hammers Universal Modus TollensUniversal Disjunctive Syllogism  MT  DSL  DSR (  x)(P(x)  Q(x)) (  x)  Q(x)_______ (  x)  P(x)  MT (  x)(P(x)  Q(x)) (  x)  Q(x)_______ (  x)P(x)  DSL (  x)(P(x)  Q(x)) (  x)  P(x)_______ (  x)Q(x)  DSR Existential Disjunctive Syllogism  DSL  DSR (  x)(P(x)  Q(x)) (  x)  Q(x)_______ (  x)P(x)  DSL (  x)(P(x)  Q(x)) (  x)  P(x)_______ (  x)Q(x)  DSR

29. Propositional Derived Rules: Abracadabra! CutExportation CutExp& Exp  p  q  p_  r___ q  r (p & q)  r p  (q  r) (p & q)  r

30. Propositional Derived Rules: Abracadabra! Definition of the Conditional Hypothetical Syllogism Transitivity Def  EDef  I HSTrans p  q____  p  q  p_  q___ p  q q  r_____ p  r p  q ::  q   p

31. Propositional Derived Rules: Abracadabra! Commutativity Comm& Comm  Comm  p & q :: q & p p  q :: q  pp  q :: q  p DeMorgan's Laws DeM  (p & q) ::  p   q  (p  q) ::  p &  q

32. Predicate Derived Rules: Abracadabra! Definition of the Universal Quantifier Def  E Def  I (  x)(P(x)  Q(x))_________  (  x)  (P(x)  Q(x)) Def  E  (  x)  (P(x)  Q(x))____ (  x)(P(x)  Q(x)) Def  I Definition of the Existential Quantifier Def  E (  x)(P(x) & Q(x))_________  (  x)  (P(x) & Q(x)) Def  E  (  x)  (P(x) & Q(x))____ (  x)(P(x) & Q(x)) Def  I

33. Predicate Derived Rules: Abracadabra! Negated UniversalExistentially Quantified Negation   (  x)(P(x)  Q(x))_________ (  x)  (P(x)  Q(x))  (  x)  (P(x)  Q(x))____  (  x)(P(x)  Q(x))  Negated ExistentialUniversally Quantified Negation   (  x)(P(x) & Q(x))_________ (  x)  (P(x) & Q(x))  (  x)  (P(x) & Q(x))____  (  x)(P(x) & Q(x)) 

34. Practice: Derivations (Propositional) 1.[A  (B  C)]  [(A  B)  C] 2.  (A &  B),  (B &  C)   (A &  C) 3.(X  Y), [(X  Z)   A], (Y  Z), (A  B)  B 4.  Q, R   (Q   R) 5.[(B  C)  D],  D  (B &  C) 6.(O  P)  [  (O &  P) &  (  O & P)]

35. Practice: Derivations (Predicate) 1.(  x)(I(x)  J(x)), (  x)(J(x)  K(x)), (  x)(K(x)  L(x))  (  x)(I(x)  L(x)) 2.(  x)(M(x)  N(x)), (  x)(M(x) & P(x)), (  x)(N(x)  O(x))  (  x)(O(x) & P(x)) 3.  (  x)(B(x)   A(x))  (  x)(B(x) & A(x)) 4.(  x)(F(x)  (  y)M(y,x))  (  x)(  y)(F(x)  M(y,x)) 5.(  x)(  y)L(y,x)  (  x)(  y)L(x,y)