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Difficulty: HardSyllogism and Categorical Propositions

Examine the following categorical premises and determine the validity of the derived conclusions:

Premises:
1. All synthetic room-temperature superconductors are diamagnetic materials.
2. No diamagnetic materials are ferromagnetic alloys.

Conclusions:
I. No synthetic room-temperature superconductor is a ferromagnetic alloy.
II. Some diamagnetic materials are synthetic room-temperature superconductors.
III. No ferromagnetic alloy is a synthetic room-temperature superconductor.

Which of the candidate conclusions logically follow(s) from the given premises under modern categorical logic?

  1. A
    All conclusions I, II, and III follow
  2. Only conclusions I and III followAnswer
  3. C
    Only conclusion I follows
  4. D
    Only conclusion II follows

Answer

Only conclusions I and III follow
Conclusion I is logically valid because if all elements of SS belong to set DD, and set DD shares no elements with set FF, then set SS can share no elements with set FF. Conclusion III is also valid because a universal negative proposition ('No SS is FF') undergoes valid simple conversion to 'No FF is SS'. Conclusion II is invalid under standard Boolean categorical logic because deriving a particular statement ('Some DD are SS') from universal premises without asserting existence constitutes an existential fallacy. Therefore, the option stating that only conclusions I and III follow is correct.

Step-by-Step Solution

1
Represent premises symbolically in standard categorical form
Let SS = synthetic room-temperature superconductors, DD = diamagnetic materials, FF = ferromagnetic alloys. Premise 1: All SS are DD (S⊆DS \subseteq D). Premise 2: No DD are FF (D∩F=∅D \cap F = \emptyset).
Symbolic representation allows clear visual and set-theoretic verification of categorical relationships.
2
Evaluate Conclusion I: No SS is FF
Since S⊆DS \subseteq D and D∩F=∅D \cap F = \emptyset, it follows directly that S∩F=∅S \cap F = \emptyset. Thus, 'No synthetic room-temperature superconductor is a ferromagnetic alloy' is logically valid (Figure 1 Celarent syllogism).
A subset of a set disjoint from FF must also be completely disjoint from FF.
3
Evaluate Conclusion II: Some DD are SS
Deriving a particular proposition ('Some DD are SS') from universal statements ('All SS are DD') requires assuming that the set SS is non-empty (S≠∅S \neq \emptyset). Under modern Boolean logic, universal premises do not carry existential import. Hence, Conclusion II is invalid.
Assuming non-emptiness of universal categories without explicit existential premises is an existential fallacy.
4
Evaluate Conclusion III: No FF is SS
From Conclusion I ('No SS is FF'), applying simple conversion for EE-type categorical propositions yields 'No FF is SS'. Thus, Conclusion III logically follows.
Universal negative (EE) propositions convert validly without changing truth value.

Key Concept

Existential Fallacy and Categorical Syllogism Conversions
Estimated Time:1m 30s
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Topics

Syllogism and Categorical Propositions

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