PRACTICE MIDTERM EXAM – ANSWER KEY NAME _________________________________PHIL 110, SPRING 2014HAMILTON2/28/2014PART 1: CIRCLE TRUE OR FALSE FOR EACH OF THE FOLLOWING STATEMENTS. (1 point each)1. F If the premises of an argument are all true in some particular world, the argument must be valid and sound for that world.2. T The conclusion of an invalid argument may be true in some particular world.3. T If an argument is truly sound in some particular world, it must be a valid argument.4. T ¬(P ᴧ Q) ¬P ᴠ ¬Q5. F ¬¬¬P P6. F P ᴠ Q is a logical truth.7. T If two sentences are tautologically equivalent to each other, then they are logically equivalent to each other.8. T The FOL predicate Larger is transitive.9. F The only way for the complex sentence P ᴧ Q to be false in some world is for both P and Q individually to be false in that world. 10. T The sentence (¬Cube(a) ᴧ Tet(b)) ᴠ ¬(Cube(a) ᴧ ¬Tet(b)) considered as a whole is a disjunction.11. T A truth claim (sentence) is logically possible if it is true in one or more possible worlds.12. F The proof rule ‘Identity Introduction’ is used to assert a new identity statement of the sort a = b.13. T SameSize(a,b) is a logical consequence of SameSize(b,a) in virtue of the meaning of the predicate involved.14. T The symbol ¬ is a Boolean connective.15. T A truth table is able to demonstrate instances of tautological consequence.16. T The premise of a subproof requires no justification by a proof rule.17. F The proof rule ‘Disjunction Introduction’ is also called ‘Proof by Cases.’18. T A sentence of FOL is demonstrably a tautology if in a truth table the truth-values directly under the sentence’s main connective are true in all rows.19. T Each row of a truth table represents one possible combination of truth-values of whatever distinct atomic sentences are contained within the complex sentence(s) being evaluated in the table.20. F In the Tarski’s World blocks language, ‘RightOf’ is a unary predicate.21. T The following statement is an atomic sentence of FOL: Between(a,b,c)22. F One way to demonstrate that an argument is invalid is to provide a counterexample world in which the premises of the argument are false but the conclusion is true.23. F The following is a complex sentence of FOL: SameSize(a ᴧ b)24. T The following sentence is a contradiction as considered in the blocks language for Tarski’s World: FrontOf(d,e) ᴧ SameRow(e,d)25. T The following set of four sentences constitutes a contradiction as considered in the blocks language for Tarksi’s World: (1) Larger(a,b) (2) Larger(b,c) (3) Larger(c,d) (4)Larger(d,e)26. F Sentences may contradict each other only if each of the sentences individually is a contradiction.27. F The proof rule ¬Intro is based on the fact that anything and everything follows from a contradiction.28. T The proof rule ⊥ Intro provides a means of flagging a contradiction that has arisen in the course of the proof.29. F Once a subproof has been discharged, any individual line of that subproof may serveas input to the proof rules that justify subsequent steps of the proof.30. T One element of a good strategy for tackling proofs is to first think through whether the argument you intend to prove is really valid.PART 2: Write a good translation of each of the following sentences into FOL. (2 points each)31. “a is a small cube to the right of b”Translation into FOL: ______Small(a) ᴧ Cube(a) ᴧ RightOf(a,b)_________________________32. “a is neither small nor a tetrahedron, and it’s in the same row as b”Translation into FOL: _______ ¬ (Small(a) ᴠ Tet(a)) ᴧ SameRow(a,b)____ ___________________33. “a and b are both in front of c”Translation into FOL: __________FrontOf (a,c) ᴧ FrontOf(b,c)_________________________34. “a is small, but b is not”Translation into FOL: __________Small(a) ᴧ ¬Small (b)_________________________________Part 3: In the blank next to each of the following four sentences, put the one capital letter (from the accompanying diagram) that labels the innermost logical region to which the sentence belongs. You may use the same letter more than once. (2 points each)A: logical possibility35. SameSize(b,a) ᴠ ¬SameSize(b,a) ___C__36. SameSize(b,a) ᴠ ¬SameSize(a,b) ___B__37. SameSize(b,a) ᴧ SameSize(a,b) ___A__38. ¬SameSize(a,b) ᴧ ¬SameSize(b,a) ___A__Part 4: In the box, draw a world that provides a counterexample to the following argument. Clearly indicate the sizes of your objects. (4 points)39. P1: Large(a) ᴠ Large(b)P2: Large(a) ᴠ Large(c)Concl: Large(a) ᴧ (Large(b) ᴠ Large(c))Part 5: Fill in all of the missing truth-values in the truth-table below (using standard format) and answer the questions about the table that follow. (0.5 pts for each truth-value; 2 pts per question )40.Is thelast complex sentence (i.e, on the far right) a tautological consequence of the other two complexsentences? YES _ X ___ NO ____41. If you answered NO for #40, put an arrow () next to a row of the table that provides a counterexample showing that the tautological consequence does not hold.If you answered YES for #40, explain how you determined that the answer is ‘yes’: __________ THE LAST SENTENCE IS TRUE UNDER ITS MAIN CONNECTIVE IN THE ONE ROW (4 TH ) ___ WHERE BOTH OF THE FIRST TWO SENTENCES ARE TRUE UNDER THEIR MAIN CONNECTIVES _42. Is the last complex sentence in the table tautologically equivalent to the first complex sentence?YES ____ NO _ X __Part 6: Circle T or F to indicate whether each of the following sentences of the expanded blocks language is TRUE or FALSE in the world provided. (1 point each)43. T FrontOf(rm(bm(a)), b)44. T fm(fm(b)) = lm(rm(g))45. F ¬(LeftOf(f, lm(h)) ᴠ LeftOf(h, rm(a))) (tet) B: logical necessityC: tautology-- OR – b caa(large)a da bcA B B ᴠ ¬ A ¬ B ¬ (A ᴧ B)T T T F F F TT F F F T T FF T T T F T FF F T T T T FbcadgfehPart 7: Circle T or F to indicate whether each of the following sentences of the first-order language of set theory is TRUE or FALSE given the following definitions. (1 point each)In a world in which . . .a names the set {5, 10, 15}b names the set {5, 10}c names the set {5, {5, 10, 15}}d names 1546. T d ∈
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