May04 - UNIVERSITY OF TORONTO Faculty of Arts and Science...

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Unformatted text preview: UNIVERSITY OF TORONTO Faculty of Arts and Science APRIL/MAY EXAMINATIONS 2004 PHL245HIS — Glen Hoffmann Duration — 2 hours April 26, 2004 Examination Aid: Sheet with rules provided - No other aids allowed Name: Student #: 1. Define the following terms, providing an example of each (3 marks each): a) Validity in PD b) Mixed Quantifier sentence c) Truth-Functional Compound (1) Truth-Functional Falsity e) Sentence of PL Name: Student #: 2. State whether the following sentences are true or false. Briefly explain your answers (3 marks each): a) A theorem of SD is provable from no assumptions. b) An argument whose premises are all truth-functionally true is deductively valid. c) One can always demonstrate that a sentence is quantificationally indeterminate by providing appropriate interpretations. d) All formulas of predicate logic are sentences of predicate logic. e) An argument whose premises are inconsistent in PD is valid in PD. Name: Student #2 3. Use the abbreviation scheme provided to symbolize the following sentences (3 marks each): UD: Positive Integers Ex: x is even Ox: x is odd Px: x is prime ny: x is larger than y a: 1 :2 :3 :4 D-Oc' a) 1 and 2 are prime numbers, but 3 is odd if and only if 4 is even. b) No even number is prime. 0) Some even numbers are larger than any odd number. (1) All prime numbers are larger than 1. e) No number is larger than all numbers, but some number is smaller than all other numbers. Name: Student #: 4. Provide an interpretation which shows that the following argument is quantificationally invalid. Please explain your answer (10 marks): Sb {VXHPXDSXL Pb Name: Student #: 5. Construct a truth-functional expansion of the following sentence for the set of constants {‘a’, ‘b’}. Use your truth-functional expansion to show that the sentence is quantificationally indeterminate. Please explain your answer (15 marks): [~ (3y) F y v ~(3y) Dy} V (W) (Fy & Dy) Name: Student #: 6. Construct a derivation to Show that the following sentence is a theorem in PD (15 marks): (Vy) (Dy v (32) Byz) D (Vy)(32) (Py v Byz) Name: Student #: 7. Construct a derivation to show that the following argument is valid in PD+ (15 marks): (VX) [(Elx) (Gyb & nyb) 3 FX] 13x2 ngb & Hxabl (Vx) (be :3 ~Fx) 3 ~Gab Scrap Paper DERIVATION RULES OF SD Reiteration (R) I P D P ‘8c’ Rules Conjunction Introduction (&I) Conjunction Elimination (&E) P P & Q P & Q or Q > P D Q D P & Q ‘3‘ Rules Conditional Introduction (:71) Conditional Elimination (DE) P P 3 Q P Q D D P 3 Q Q ‘~’ Rules Negation Introduction (~ I) Negation Elimination (~ E) P Q ~ Q D ~ P D P ‘v’ Rules Disjunction Introduction (VI) Disjunction Elimination (vE) P P P v Q or P D P v Q D Q v P '_. R ’3 R D R ‘5’ Rules Biconditional Introduction (51) Biconditional Elimination (5E) ' P P E Q P E Q P or Q Q D D P ’i Q P D P E Q DERIVATION RULES OF SD+ All the Derivation Rules of SD and Rules of Inference Modus Tollens (MT) Hypothetical Syllogism (HS) P D Q QDR PDR D Disjunctive Syllogism (DS) PDQ ~Q D ~P PvQ ~P D Q or D PVQ ~Q P Rules of Replacement Commutation (Com) P&Q<1DQ&P PvQ<1DQvP Implication (Impl) P I) Q <1 D ~ P v Q De Morgan (DeM) ~(P&Q)<1D~PV~Q ~(PvQ)<lD~P8c~Q Transgositz'on (Trans) P 3 Q <1 D ~ Q 3 ~ P Distribution (Dist) P&(QVR)<1D(P&Q)V(P&R) PV(Q&R)<1D(PVQ)&(PVR) Association (Assoc) P&(Q&R) <1D(P&Q)&R Pv(QvR) <1l> (PVQ) VR Double Negation (DN) P <1 D ~ ~ P Idempotence (Idem) P<1DP&P P<lDPvP Exportatz'on (ExB) PD (QDR) <1D (P8cQ) DR Eo'uivalence (Eguiv) PEQ<ID(P:>Q)&(Q:>P) PEQ<}D(P&Q)V(~P&~Q) DERIVATION RULES OF PD All the Derivation Rules of SD and Universal Introduction (VI) P(a/x) D (Vx)P Provided: (i) a does not occur in an undischarged assumption. (ii) a does not occur in (Vx)P. Existential Introduction (31) P(a/x) D (3x)? Universal Elimination (VE) (Vx)P D P(a/x) Existential Elimination (3E) (3X) P ‘ P(a/x) Q D Q Provided: (i) a does not occur in an undischarged assumption. (ii) a does not occur in (3x)? (iii) a does not occur in DERIVATION RULES OF PD+ All the Derivation Rules of SD+ and of PD and Quantifier Negation ( QN) ~ (Vx)P <1 D (3x) ~ P ~ (3x)P <1 D (Vx) ~ P DERIVATION RULES OF PDE All of the Derivation Rules of PD and Universal Elimination (VE) (Vx)P i> P(t/x) where t is a closed term Identity Introduction ( =1) D l (Vx)x = x Existential Introduction (VE) P(t/x) I> (3x)P where t is a Closed term Identity Elimination (7 E) t1 = t2 t1 = t9 P or P D Paw/ta) I> P(t2//t1) where t1 and t2 are closed terms ...
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May04 - UNIVERSITY OF TORONTO Faculty of Arts and Science...

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