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Difficulty: HardSyllogism and Categorical Propositions

Given the following categorical statements:
1. All superconducting qubits are quantum processors.
2. No quantum processors are classical CMOS circuits.
3. All superconducting qubits are sub-kelvin cryo-devices.

Which of the following conclusions logically follow from the given statements under standard categorical logic rules? (Select all valid conclusions)

  1. No superconducting qubit is a classical CMOS circuit.Answer
  2. No classical CMOS circuit is a quantum processor.Answer
  3. C
    Some sub-kelvin cryo-devices are classical CMOS circuits.
  4. D
    At least one sub-kelvin cryo-device is guaranteed to be a quantum processor without assuming the physical existence of any superconducting qubit.

Answer

The valid conclusions are that no superconducting qubit is a classical CMOS circuit, and no classical CMOS circuit is a quantum processor.
The conclusion stating that no superconducting qubit is a classical CMOS circuit is logically valid because any member of the subject set is entirely included in the middle term set (quantum processors), which is completely disjoint from the predicate set (classical CMOS circuits). Additionally, the conclusion stating that no classical CMOS circuit is a quantum processor is valid by simple conversion of the universal negative premise.

Step-by-Step Solution

1
Analyze the relationship between superconducting qubits and classical CMOS circuits
Since all superconducting qubits belong to the class of quantum processors, and no quantum processor is a classical CMOS circuit, no superconducting qubit can be a classical CMOS circuit.
Valid syllogistic deduction (EAE-1 mood / Celarent).
2
Perform conversion on Statement 2 ('No quantum processors are classical CMOS circuits')
Universal negative (E) propositions convert simply: 'No A is B' logically implies 'No B is A'. Hence, no classical CMOS circuit is a quantum processor.
Immediate inference rule of simple conversion for E-propositions.
3
Evaluate particular conclusions derived solely from universal premises
Deriving particular statements ('Some...') from universal statements ('All...') without an explicit existence premise commits the existential fallacy under Boolean logic.
Universal premises hold conditionally and carry no existential import.

Key Concept

Categorical Syllogism Validity, E-Proposition Conversion, and Existential Fallacy
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Topics

Syllogism and Categorical Propositions

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