Two statements are given below followed by two conclusions numbered as I and…

2023

Two statements are given below followed by two conclusions numbered as I and II respectively. Consider the given statements as true even if they seem to be not. After reading all the conclusions, determine which of the given conclusions logically follows, disregarding commonly known facts.

Statements:

I. All arrows are bows.

II. Some bows are javelins.

Conclusions:

I. Some arrows are javelins.

II. No arrow is javelin.

  1. A.

    If only conclusion I follows.

  2. B.

    If only conclusion II follows.

  3. C.

    If either conclusion I or II follows.

  4. D.

    If neither conclusion I nor II follows.

Show answer & explanation

Correct answer: C

For two premises where one is a universal affirmative (All A are B) and the other is a particular affirmative (Some B are C) sharing the middle term B, the standard rule of mediate inference is that no single conclusion about A and C follows with certainty — this is the classic 'no valid conclusion' case for an A + I combination, because the members of B described by 'some' need not overlap with the members of B described by 'all.' Separately, when two candidate conclusions about the very same subject and predicate are a complementary I-E pair — one worded as 'Some X are Y' and the other as 'No X is Y' — the pair is logically exhaustive: across every arrangement of the underlying sets, at least one of the two must be true, even when the premises themselves settle nothing about X and Y directly.

Here, Statement I ('All arrows are bows') is the universal affirmative and Statement II ('Some bows are javelins') is the particular affirmative, sharing the middle term 'bows.' By the rule above, this All + Some combination does not fix any certain relationship between 'arrows' and 'javelins' on its own. Conclusion I ('Some arrows are javelins') is a particular affirmative and Conclusion II ('No arrow is javelin') is a universal negative — both share the same subject ('arrows') and predicate ('javelins'), so together they form exactly the complementary I-E pair described above.

  1. Arrange the three groups so that the arrows and the javelin-bows are entirely separate parts of 'bows': then no arrow is a javelin, and Conclusion II holds.

  2. Arrange the three groups so that the arrows and the javelin-bows overlap even partially: then some arrows are javelins, and Conclusion I holds.

Every arrangement consistent with both premises falls into one of these two cases — never a case where both conclusions are simultaneously false, and never a case where both are simultaneously true. This confirms the complementary-pair rule for this pair of conclusions.

Hence, either Conclusion I or Conclusion II necessarily follows.

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