Review for Final Exam : Logic & Proofs August 6, 2009 Karin Howe
Definitions you should know… Validity and invalidity (remember kangaroos!) Types of statements: tautology, contradictory, contingent
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.
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))
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)
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.
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
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
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)
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))
Truth Tables: Propositional Logic pqp & qpq p qp q p pp TTTTTTTF TFFTFTFT FTFFTT FFFFFF pq p qp q pq p qp q TTTTTT TFFTFF FTTFTF FFTFFT
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.
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.
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)
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)pq(p q)pq (p q) p q p p ( p q) p p q q
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 *
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)
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)
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))]
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)
Basic Rules (Propositional): (In)troduction Rules ConjunctionDisjunctionConditional &I IL IR II p q__________ p & q &I p________ q p IL p________ p q IR p q I p A. q
Basic Rules (Propositional): (In)troduction Rules NegationBiconditional II II II p p___________ I p I p q I p A. p A. q q A. p
Basic Rules (Predicate): (In)troduction Rules Universal Introduction Existential Introduction Identity Introduction II II =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
Basic Rules (Propositional): Elimination (Out) Rules ConjunctionDisjunctionConditional &EL&ER EE EE p & q_____ p &EL p & q_____ q &ER p q r E p q p________ q E p A. r q A. r
Basic Rules (Propositional): Elimination (Out) Rules NegationBiconditional EE EL ER p E p q q______________ p EL p q p_____________ q ER p A.
Basic Rules (Predicate): Elimination (Out) Rules Universal Elimination Existential Elimination Identity Elimination EE EE =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.
Propositional Derived Rules: Hammers Disjunctive SyllogismModus Tollens DSLDSRMT p q q_______ p p q p_______ q p q q_______ p
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
Propositional Derived Rules: Abracadabra! CutExportation CutExp& Exp p q p_ r___ q r (p & q) r p (q r) (p & q) r
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
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
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
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))
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)]
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)