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Difficulty: MediumSyllogism and Categorical Propositions

Evaluate the logical consistency of the deductions based on the provided assertions. Treat the assertions as absolute truths, irrespective of factual reality.

Assertions:
(A) All algorithms are sequences.
(B) No sequence is a paradox.
(C) Some paradoxes are loops.

Deductions:
[X] No algorithm is a paradox.
[Y] Some loops are not sequences.
[Z] Some sequences are loops.

Based on the rules of syllogism, which of the following best describes the validity of these deductions?

  1. Deductions [X] and [Y] are the only valid conclusions.Answer
  2. B
    Deductions [X] and [Z] are the only valid conclusions.
  3. C
    Only deduction [X] is definitively valid.
  4. D
    All three deductions ([X], [Y], and [Z]) are logically valid.

Answer

Deductions [X] and [Y] are the only valid conclusions.
Deduction [X] is valid because algorithms are a subset of sequences, and sequences are entirely separate from paradoxes; therefore, algorithms must be separate from paradoxes. Deduction [Y] is valid because the specific paradoxes that are also loops cannot be sequences (since no paradox is a sequence), meaning at least some loops are not sequences. Deduction [Z] is invalid because no premise establishes an overlap between sequences and loops.

Step-by-Step Solution

1
Define the sets and their relationships based on the assertions using set notation.
Let AA = Algorithms, SS = Sequences, PP = Paradoxes, and LL = Loops. (A) gives A⊆SA \subseteq S. (B) gives S∩P=∅S \cap P = \emptyset. (C) gives P∩L≠∅P \cap L \neq \emptyset.
Converting abstract verbal statements into strict set-theory relationships prevents intuitive biases.
2
Evaluate Deduction [X]: 'No algorithm is a paradox.'
Since A⊆SA \subseteq S and S∩P=∅S \cap P = \emptyset, it logically follows that A∩P=∅A \cap P = \emptyset. Therefore, Deduction [X] is valid.
A subset (AA) cannot intersect with a set (PP) if its superset (SS) is entirely disjoint from that set.
3
Evaluate Deduction [Y]: 'Some loops are not sequences.'
From (C), there is at least one element xx in P∩LP \cap L. Because S∩P=∅S \cap P = \emptyset from (B), this element xx cannot be in SS. Thus, there is an element in LL that is not in SS. Deduction [Y] is valid.
The intersection of Loops and Paradoxes creates a specific subgroup of Loops. Because no Paradox is a Sequence, this subgroup of Loops cannot be Sequences.
4
Evaluate Deduction [Z]: 'Some sequences are loops.'
We know SS and PP are disjoint, and PP intersects with LL. However, no information restricts the relationship between SS and LL. They could be completely disjoint or they could intersect. Deduction [Z] is invalid.
An unstated relationship cannot be assumed true in categorical syllogisms.

Key Concept

Evaluating categorical syllogisms using set theory and mutually exclusive properties.
Estimated Time:1m 30s
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Topics

Syllogism and Categorical Propositions

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