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Difficulty: MediumSyllogism and Categorical Propositions

An engineering team is reviewing material classifications based on a newly established standardized database. The database enforces the following absolute rules:

Statements:
1. All crystalline structures are brittle materials.
2. No brittle materials are malleable alloys.
3. Some malleable alloys are highly conductive elements.

Based on the information above, which of the given conclusions logically follow(s)?

Conclusions:
I. No crystalline structures are malleable alloys.
II. Some highly conductive elements are not brittle materials.
III. Some brittle materials are not crystalline structures.

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

Answer

Only conclusions I and II follow
Conclusion I is valid because the set of crystalline structures is entirely within brittle materials, which are completely separate from malleable alloys. Conclusion II is valid because some highly conductive elements are malleable alloys, and since no malleable alloys are brittle materials, this overlapping portion cannot be brittle. Conclusion III is invalid because 'All A are B' does not necessitate that there are B's which are not A; the sets could be identical.

Step-by-Step Solution

1
Evaluate Conclusion I (No crystalline structures are malleable alloys).
Conclusion I is logically valid.
Premise 1 states all crystalline structures belong to the brittle materials group. Premise 2 states that no brittle materials can be malleable alloys. Therefore, crystalline structures can never be malleable alloys.
2
Evaluate Conclusion II (Some highly conductive elements are not brittle materials).
Conclusion II is logically valid.
Premise 3 establishes an overlap between malleable alloys and highly conductive elements. Since Premise 2 states no malleable alloys are brittle materials, the specific highly conductive elements located in that overlap can never be brittle materials.
3
Evaluate Conclusion III (Some brittle materials are not crystalline structures).
Conclusion III is invalid.
Premise 1 states 'All crystalline structures are brittle materials'. In formal logic, this allows for the possibility that the two sets are perfectly identical. We cannot assume the existence of brittle materials outside the crystalline structure category without explicit proof.

Key Concept

Evaluating multi-statement syllogisms using subset definitions, disjoint sets, and avoiding invalid conversions.
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Topics

Syllogism and Categorical Propositions

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