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Difficulty: MediumSyllogism and Categorical Propositions

A cyber-security researcher is mapping relationships between different cryptographic systems based on the following definitive premises:

1. All theoretical omni-ciphers are unbreakable algorithms.
2. No unbreakable algorithms are publicly audited protocols.
3. Some publicly audited protocols are legacy systems.

Applying modern formal logic principles, which of the following conclusions logically follow? (Select all valid deductions)

  1. No theoretical omni-ciphers are publicly audited protocols.Answer
  2. Some legacy systems are not unbreakable algorithms.Answer
  3. C
    At least one theoretical omni-cipher exists and is an unbreakable algorithm.
  4. D
    All unbreakable algorithms are theoretical omni-ciphers.

Answer

The valid logical deductions are that no theoretical omni-ciphers are publicly audited protocols, and some legacy systems are not unbreakable algorithms.
The valid conclusions depend on chaining the categorical premises together. Because all theoretical omni-ciphers fall into the category of unbreakable algorithms, and no unbreakable algorithms intersect with publicly audited protocols, the two categories are entirely disjoint. Furthermore, since some publicly audited protocols are legacy systems, those specific legacy systems cannot belong to the unbreakable category.

Step-by-Step Solution

1
Map the categorical propositions to formal variables
Let O = Theoretical omni-ciphers, U = Unbreakable algorithms, P = Publicly audited protocols, L = Legacy systems. Premises: (1) All O are U. (2) No U are P. (3) Some P are L.
Translating statements into variables prevents real-world bias and clarifies strict logical boundaries.
2
Combine Premises 1 and 2 to evaluate the relationship between O and P
If All O are U, and No U are P, then No O are P. This confirms the deduction that no theoretical omni-ciphers are publicly audited protocols.
The transitive property of universal categorical syllogisms dictates that a subset inherits the universal exclusions of its superset.
3
Combine Premises 2 and 3 to evaluate the relationship between L and U
If Some P are L, and No U are P (which means No P are U), then Some L are not U. This confirms the deduction that some legacy systems are not unbreakable algorithms.
Applying a particular affirmative premise with a universal negative premise always yields a particular negative conclusion.
4
Evaluate remaining options for formal logical fallacies
Reject the claim that 'All U are O' as an illicit conversion. Reject the claim that 'At least one O exists' as an existential fallacy.
Logical deductions must strictly follow from the premises without assuming unstated converses or assuming the physical existence of theoretical subjects.

Key Concept

Categorical Syllogisms, Existential Fallacy, and Illicit Conversion
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Topics

Syllogism and Categorical Propositions

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