dataset_name: formal_logic
description: "The following are multiple choice questions (with answers) about formal\
  \ logic.\n\nQ: Which of the given formulas of PL is the best symbolization of the\
  \ following sentence?\nTurtles live long lives and are happy creatures, unless they\
  \ are injured.\n(A) (L \u2022 H) \u2261 I (B) (L \u2022 H) \u2228 I (C) L \u2022\
  \ (H \u2228 I) (D) L \u2022 (H \u2283 R).\nA: Let's think step by step. We refer\
  \ to Wikipedia articles on formal logic for help. Let\u2019s solve this step by\
  \ step. Let \u201CL\u201D denote \u201Cliving long\u201D, H \u201Cbeing happy\u201D\
  , and \u201CI\u201D \u201Cbeing injured\u201D. Now, consider each choice:\n(A) means\
  \ (living long AND being happy) is equivalent to (being injured). \n(B) means (living\
  \ long AND being happy) OR (being injured). \n(C) means (living long) AND (being\
  \ happy OR being injured). \n(D) means (living long) AND (being happy implies being\
  \ R), but what R denotes is not clear.\nObviously, (B) is the best symbolization\
  \ of the original sentence. The answer is (B).\n\nQ: Select the best translation\
  \ into predicate logic.George borrows Hector's lawnmower. (g: George; h: Hector;\
  \ l: Hector's lawnmower; Bxyx: x borrows y from z).\n(A) Blgh (B) Bhlg (C) Bglh\
  \ (D) Bghl\nA: Let's think step by step. We refer to Wikipedia articles on formal\
  \ logic for help. Let\u2019s solve this step by step. We are told that \u201CBxyx\u201D\
  \ means \u201Cx borrows y from z\u201D. We can rewrite \u201CGeorge borrows Hector's\
  \ lawnmower\u201D as \u201CGeorge borrows a lawnmower from Hector\u201D, which can\
  \ then be translated into predicate logic as \u201CBglh\u201D. The answer \u201C\
  Bglh\u201D appears in (C); therefore, (C) must be the correct answer. The answer\
  \ is (C).\n\nQ: \nSelect the best English interpretation of the given arguments\
  \ in predicate logic.\nDm\n(\u2200x)(Wx \u2283 ~Dx). \n(\u2200x)Wx \u2228 Ag\t/\
  \ (\u2203x)Ax\n(A) Marina is a dancer. Some weaklings are not dancers. Either everything\
  \ is a weakling or Georgia plays volleyball. So something plays volleyball. (B)\
  \ Marina is a dancer. No weakling is a dancer. Everything is either a weakling or\
  \ plays volleyball. So something plays volleyball. (C) Marina is a dancer. Some\
  \ weaklings are not dancers. Everything is either a weakling or plays volleyball.\
  \ So something plays volleyball. (D) Marina is a dancer. No weakling is a dancer.\
  \ Either everything is a weakling or Georgia plays volleyball. So something plays\
  \ volleyball.\nA: Let's think step by step. We refer to Wikipedia articles on formal\
  \ logic for help. Let\u2019s solve this step by step. Let \u201CD\u201D denote \u201C\
  being a dancer\u201D, \u201Cm\u201D denote \u201CMaria\u201D, \u201Cg\u201D denote\
  \ \u201CGeorgia\u201D, \u201CW\u201D denote \u201Cweakling\u201D, \u201CA\u201D\
  \ denote \u201Cplaying volleyball\u201D. Then, we have the following:\n1. Dm \u2192\
  \ Maria is a dance.\n2. (\u2200x)(Wx \u2283 ~Dx). \u2192 For all x, if x is a weakling,\
  \ then x is not a dancer. In other words, no weakling is a dancer.\n3. (\u2200x)Wx\
  \ \u2228 Ag\t/ (\u2203x)Ax \u2192 For all x, x is a weakling or Georgia plays volleyball.\
  \ So there exists an x that plays volleyball. \nOptions (A) and (C) do claim that\
  \ some weaklings are not dancers, but the second argument strongly states that no\
  \ weakling is a dancer. Thus, we can eliminate them. Option (B) omits the important\
  \ detail about Georgia playing volleyball. Option (D) has all the details presented\
  \ in the arguments and is the best English interpretation of the arguments. The\
  \ answer is (D).\n\nQ: Select the best translation into predicate logic: No people\
  \ drive on Mars.\n(A) ~Pd (B) (\u2200x)(Px \u2228 ~Dx) (C) (\u2200x)(Px \u2283 ~Dx)\
  \ (D) ~Dp\nA: Let's think step by step. We refer to Wikipedia articles on formal\
  \ logic for help. Let\u2019s solve this step by step. Let \u201CP\u201D denote \u201C\
  being on Mars\u201D and \u201CD\u201D denote \u201Cdriving on Mars\u201D. Then let\u2019\
  s consider each option:\nOption (A): ~Pd \u2192 d is not on Mars.\nOption (B): (\u2200\
  x)(Px \u2228 ~Dx) \u2192 For all x, x is on Mars and x do not drive on Mars.\nOption\
  \ (C): (\u2200x)(Px \u2283 ~Dx) \u2192 For all x, x is on Mars implies that x do\
  \ not drive on Mars.\nOption (D): ~Dp: \u2192 p do not drive on Mars.\nOf all these\
  \ options, Option (C) appears to be the best and most meaningful interpretation\
  \ of the argument \u201CNo people drive on Mars.\u201D The answer is (C)."
include: _mmlu_flan_cot_fewshot_template_yaml
task: mmlu_flan_cot_fewshot_formal_logic
