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Quiz Game INTRODUCTION TO LOGIC FALL 2009 Concepts True/False Translations Informal Proofs Formal Proofs 100 100 100 100 100 200 200 200 200 200 300 300 300 300 300 400 400 400 400 400 500 500 500 500 500 Translations Informal Proofs Formal Proofs Concepts True/False 200 200 200 200 200 400 400 400 400 400 600 600 600 600 600 800 800 800 800 800 1000 1000 1000 1000 1000 Final Question 2000 (100) What is the domain of discourse in Tarski’s World? The blocks that appear on the checkered board. (200) Is this a sentence of FOL?: x [y Smaller(x,y) (Cube(x) Cube(y))] No. (300) When is a variable in a well-formed formula considered free? When it is not bound by (does not fall within the scope of) a quantifier. (400) What is the semantics for ? xS(x) is true iff every object satisfies S(x). (500) Is (a = b) an FO consequence of Cube(a) and Cube(a)? Explain why or why not. Yes, it is. Cube(a) and Cube(a) is a contradiction, no matter what “Cube” means, so anything follows. (200) How many quantifiers appear in this sentence?: Jay is faster than any man on his team, but some woman out there is faster. Two. (400) Is this a well-formed formula of FOL?: y(Cube(x) FrontOf(x,b)) Yes. (600) When is a sentence ambiguous? When it has more than one reading / meaning / translation into FOL. (800) How does this translate into FOL?: If a cow is chewing her cud, that means she’s happy. x [(Cow(x) ChewCud(x)) Happy(x)] (1000) Give a counterexample to this being an FO validity: xy (SameCol(x,y) SameCol(y,x)) Replace “SameCol” with nonsense predicate “Mimsy.” Let “Mimsy” mean “Loves.” Then consider a possible situation in which John loves Mary but Mary does not love John. (100) How does this translate into FOL?: It’s not the case that if Mary’s tall, Sally is also tall. (Tall(mary) Tall(sally)) (200) What is the Aristotelian form of this sentence?: Anything between two cubes is also between two tetrahedral. All A’s are B’s. (300) How does this translate into FOL?: There are some people that just can’t be pleased. x [Person(x) yPlease(y,x)] x [Person(x) CanBePleased(x)] (400) How does this sentence read in plain English?: x (Angry(x,2:00) Student(x) Fed(x,max,2:00)) No angry student fed Max at 2:00. (500) How does this translate into FOL? If something is a cube, then it is not in the same column as either a or b. x [Cube(x) (SameCol(x,a) SameCol(x,b))] (200) What is the Aristotelian form of this sentence?: Nobody who’s anybody is a quitter. No A’s are B’s. (400) How does this translate into FOL?: Everything smaller than a is a cube. x (Smaller(x,a) Cube(x)) (600) How does this sentence read in plain English?: x (LeftOf(x,a) RightOf(x,a)). Anything that’s not left of a is not right of it either. (800) How does this translate into FOL?: We’re all doomed unless Batman comes through. ComesThrough(batman) x Doomed(x) (1000) How does this translate into FOL? No object in front of a dodecahedron is small, unless there is nothing in front of it. x [y FrontOf(y,x) (z (Dodec(z) FrontOf(x,z)) Small(x))] (100) A sound argument can have false premises. False. (200) Some rabbits are black translates as: x (Rabbit(x) Black(x)) False. (300) This equivalence holds: A (B C) (A B) C True. (400) If a sentence is a logical truth, then it’s also an FO validity. False. (500) Is this sentence true or false in this world?: x y [SameCol(x,y) (Cube(x) Tet(y))] False. (200) In a world with only three small cubes, some object satisfies Large(z) Cube(z). True. (400) This equivalence holds: x (P(x) Q(x)) x P(x) x Q(x) False. (600) This equivalence holds: x (A B) x (A B) True. (800) If A is a tautological consequence of B, then A is an FO consequence of B. True. (1000) Every cat climbs some tree at some time translates as x y z [(Cat(y) Tree(z)) (Time(x) Climbs(y,z,x))] False. (100) What is the name of the inference rule used here? Existential Introduction or Generalization (200) Is the following world a counterexample to this inference? Something is large, because b is a cube. Yes. (300) What is the main method of proof used in proving this argument? Existential Elimination (400) Where does this informal proof go wrong? Let ‘c’ name some object such that Cube(c) and Small(c), from premises 1 and 2. Therefore some object is both a cube and small, by existential introduction. The first sentence: you know that something’s a cube and something’s small, but you can’t assume that the same thing is both. (500) Give an informal proof of this argument. Let ‘a’ name an arbitrary object in the domain. By universal elimination on premises 1 and 2, we know that Tet(a) Small(a) and Small(a). So it follows that Tet(a). But since the choice of a was arbitrary, we can use universal introduction to conclude x Tet(x). (200) What is the name of the inference rule used here? Anything round is either blue or white. Therefore, if the Geico Gecko is round, then he’s blue or white. Universal Elimination or Instantiation (400) What is the main method of proof used in proving this argument? Proof by Cases or Disjunction Elimination (600) What is the main method of proof used in proving this argument? General Conditional Proof (800) Describe what a counterexample world to this argument would look like. It would be a world in which no cubes exist. (1000) Give an informal proof of this argument. Suppose for contradiction that x Large(x). Then let ‘a’ name an arbitrary member of the domain. By universal elimination on the assumption and premise 1, we have Large(a) and Large(a) Small(a), so by modus ponens, Small(a). But since a was arbitrary, we can conclude x Small(x). But this contradicts premise 2. So we have a contradiction from the assumption that x Large(x); therefore, x Large(x). (100) Is this a valid proof? Yes, if a exists in the domain of discourse. (200) What is the main method of proof here? Intro (300) What’s the missing line? LeftOf(d,c) (400) What’s the missing justification? Elim: 3 (500) Give a formal proof of this argument. (200) Is this a valid proof? Yes. (400) What is the main method of proof here? Elim (600) What’s the missing line? Cube(c) (800) What’s the missing justification? Intro: 4-7 (1000) Give a formal proof of this argument. Final question Give a formal proof of this argument.