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Difficulty: Very hardSyllogism and Categorical Propositions

Consider the following three categorical statements:

1. All photonic switches are optical waveguides.
2. All optical waveguides are dielectric structures.
3. No dielectric structures are superconducting conduits.

Evaluating these premises under modern formal logic (where universal propositions do not carry existential import), which of the following conclusions logically follow(s)?

Conclusions:
I. No photonic switches are superconducting conduits.
II. Some dielectric structures are photonic switches.
III. Some superconducting conduits are optical waveguides.

Which of the following option choices is correct?

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

Answer

Only conclusion I follows.
The correct response identifies that only the first conclusion is logically sound. By combining the universal affirmative premises, we establish that all photonic switches are dielectric structures. Since no dielectric structures are superconducting conduits, it unconditionally follows that no photonic switches are superconducting conduits. Conclusion II relies on the existential fallacy by assuming subject classes are non-empty, and Conclusion III directly contradicts the derived negative relationship.

Step-by-Step Solution

1
Analyze the relation between Photonic Switches (P), Optical Waveguides (W), and Dielectric Structures (D).
Statement 1 (All P are WAll\ P\ are\ W) and Statement 2 (All W are DAll\ W\ are\ D) chain syllogistically to give All P are DAll\ P\ are\ D (Universal Affirmative, A-type).
Hypothetical syllogism rule: Barbara AAA-1 figure.
2
Combine the intermediate result (All P are DAll\ P\ are\ D) with Statement 3 (No D are SNo\ D\ are\ S).
Since all PP are within DD, and no DD overlaps with SS, it logically necessitates that no PP overlaps with SS (No P are SNo\ P\ are\ S).
Celarent EAE-1 syllogistic figure yields a valid Universal Negative conclusion.
3
Evaluate Conclusion I (No P are SNo\ P\ are\ S).
Conclusion I matches the validly derived proposition No P are SNo\ P\ are\ S. Thus, Conclusion I is valid.
Direct syllogistic deduction.
4
Evaluate Conclusion II (Some D are PSome\ D\ are\ P) under modern formal logic.
Deriving Some D are PSome\ D\ are\ P from All P are DAll\ P\ are\ D requires assuming that the subject set PP is non-empty. In modern Boolean standard logic, universal propositions (All A are BAll\ A\ are\ B) lack existential import.
Inferring a particular proposition (I-type) from universal premises without explicit existence claims commits the existential fallacy.
5
Evaluate Conclusion III (Some S are WSome\ S\ are\ W).
From Statement 2 (All W are DAll\ W\ are\ D) and Statement 3 (No D are SNo\ D\ are\ S), we deduce No W are SNo\ W\ are\ S (which converts simply to No S are WNo\ S\ are\ W). Therefore, Some S are WSome\ S\ are\ W is false.
Direct contradiction of valid E-proposition.

Key Concept

Existential Fallacy in Universal Syllogisms
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Topics

Syllogism and Categorical Propositions

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