Mixed particular and universal with attribute layering: Using 'Some swords are sharp' and 'All swords are rusty', test whether the conclusions (I) 'Some rusty things are sharp' and (II) 'Some rusty things are not sharp' must be true.
Verbal Reasoning
Logical Deduction
Difficulty: Easy
Choose an option
Answer
Correct Answer: Only conclusion I follows
Explanation
Given data
- Premise 1: Some Swords are Sharp (∃ Sword ∩ Sharp).
- Premise 2: All Swords are Rusty (Sword ⊆ Rusty).
- Conclusions: (I) Some Rusty things are Sharp. (II) Some Rusty things are not Sharp.
Concept/Approach
Map attributes: If a subset (Swords) lies entirely within Rusty, then any particular Sword (sharp or not) is a Rusty thing. From 'Some Swords are Sharp' it follows that some Rusty things (those sharp swords) are Sharp. But we cannot assert existence of non-sharp swords.
Step-by-step evaluation
1) From Premise 2, every Sword is Rusty.2) Premise 1 gives at least one element that is both Sword and Sharp.3) Therefore that element is also Rusty ∧ Sharp ⇒ (I) holds.4) (II) requires an element that is Rusty but not Sharp; premises do not guarantee such an element (all swords could be sharp for all we know).Verification/Alternative
Countermodel for (II): Let all Swords be Sharp; both premises hold, but there is no Rusty ∧ ¬Sharp element. Hence (II) is not necessary.
Common pitfalls
- Assuming from 'Some swords are sharp' that some swords are not sharp; such a negation is unwarranted.
Final AnswerOnly conclusion I follows.