41. If no rain (P) => soccer (Q). Not soccer (Not Q). Therefore, not "no rain" (Not P)= raining. Valid: negating the consequent.

42. If raining (P) => wet (Q). Wet (Q) . . . invalid. Affirming the consequent.

43. If green (P) => no pinch (Q). Pinched = Not "no pinch" (Not Q). Therefore, not green (Not P). Valid: negating the consequent.

44. If all assignments (P) => pass (Q). Missed two = not "all assignments" (Not P) . . . invalid. Negating the antecedent.

45. If menudo (P) => full (Q1) and no sleep (Q2). If no sleep (Q2) => not on time (R). Menudo (P). Therefore, full (Q1), no sleep (Q2), and not on time (Q3). Valid: affirming the antecedent of a chain argument. The argument only concludes that the speaker won't be able to make it to class tomorrow, but being too full and not being able to sleep are valid conclusions, as well.

46. If leave (P) => not stay (Q). Not leave (Not P) . . . invalid. Negating the antecedent.

47. If early (P) => not tired (Q). Not tired (Q) . . . invalid. Affirming the consequent.

48. If earlier (P) => free (Q). Not earlier (Not P) . . . invalid. Negating the antecedent.

50. If sweater (P) => warm (Q). No sweater (Not P) . . . invalid. Negating the antecedent.

51. All chocolate eaters (X) are pimply (Y). All teenagers (Z) are chocolate eaters (X). So, all teenagers (Z) are pimply (Y). Valid. No negative conclusion; the distributed term in the conclusion (Z) is distributed in the second premise; and the middle term (X) is distributed in the first premise.

52. If romance (P) => enjoy (Q). Not romance (Not P) . . . invalid. Negating the antecendent.

53. All lawful (X) are maximum 2 and 2 (Y). Sam (Z) is 6 = Sam is not 2, and to negate the and we only need to negate one, so this is: No Z is Y. Therefore, No Z is X = Sam is not lawful.

54. Notice the effects of or: If 5:00 or no traffic (P) => home by dark (Q). No traffic = P, because we only have to affirm one element of the "or." Therefore, Q, home by dark.

55. Notice the effects of "only" and "and": All aloof and unfriendly (X) are mistreated (Y). No mine (Z) are mistreated (Y). Therefore, no mine (Z) are aloof and unfriendly (X). Valid, as far as it goes. "My cats" might be aloof, or they might be unfriendly. All we can conclude here is that they are not both, since negating an "and" negates only one.

56. If not study (P) => not pass (Q) Studied = Not "not study" (Not P) . . . invalid. Negating the antecedent.

57. Some classes (X) are "suck" or "no-go" (Y1 or Y2). Critical thinking (Z) is "no-go" (Y1). Therefore, critical thinking (Z) is "suck" (Y2). The X term appears in only one of the claims, instead of two. In addition, the middle term (Y) is not distributed. Either of those reasons would be sufficient to conclude that the argument is invalid.

58. All the best teams (X) are playoff teams (Y). Some (two) of the playoff teams (Y) are Super Bowl teams (Z). Therefore, Super Bowl teams (Z) are the best teams (X). Since this means "all Super Bowl teams are the best teams," this argument is invalid, because "Super Bowl" (Z) is distributed in the conclusion, but not in the premise. In addition, the middle term (Y), "playoff teams," is not distributed in either premise, also invalidating the argument.

59. Both premises need restating. The first premise begins with the phrase "only some." Let's take an example: "Only some of the students are passing" means "Not all the students are passing," or "Some of the students are not passing." Notice, therefore, that in non-universal claims, "only" has a different effect than in universal claims. The second premise must be restated using a state-of-being verb, in this case a form of "should" since this is an advocatory claim. The choice is between "No one should be a not-worthwhile course taker," and "No not-worthwhile course should be required." Choose the latter, because it repeats one of the terms from the previous premise. Some courses are not-worthwhile. No not-worthwhile courses should be required. Therefore, some courses should not be required. The distributed term in the conclusion, "required," is distributed in the second premise; there is a negative in the conclusion and in the second premise; and the middle term, "not-worthwhile" is distributed in the second premise. Therefore, this is a valid argument. (Note that the negative in "not-worthwhile" is part of the term; you can avoid the confusion by substituting "worthless" for "not-worthwhile."

60. Some trained (X) are vicious (Y1) or friendly (Y2). Some friendly (Y2) are not good pets (Z). Therefore, some trained (X) are not good pets (Z). If the first premise said simply, "Some trained are friendly," this still would not be valid, because the middle term, "friendly," is not distributed. But it is also invalidated by the disjunction, "or," in the first premise, because that means that no well-trained dog is necessarily friendly, and therefore the second premise, concerning friendly dogs, may not apply to the first.

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