Examkin

Question

Difficulty: MediumSyllogism and Categorical Propositions

Examine the following statements and candidate conclusions. Assuming all given statements are true, even if they appear to contradict commonly known facts, identify which conclusion(s) logically follow(s).

Statements:
1. All quantum sensors are diagnostic tools.
2. Some diagnostic tools are portable devices.
3. No portable devices are calibration instruments.

Conclusions:
I. Some quantum sensors are portable devices.
II. Some diagnostic tools are not calibration instruments.
III. No quantum sensors are calibration instruments.
IV. Some portable devices are diagnostic tools.

Which of the following represents the correct evaluation?

  1. A
    Only I, II, and IV follow
  2. Only II and IV followAnswer
  3. C
    Only II, III, and IV follow
  4. D
    Only I and III follow

Answer

Only conclusions II and IV logically follow from the given statements.
Conclusion II is valid because Premise 2 establishes that a subset of diagnostic tools are portable devices, and Premise 3 states no portable devices are calibration instruments. Therefore, that specific subset of diagnostic tools can never be calibration instruments. Conclusion IV is valid as it is a direct logical conversion of the particular affirmative Premise 2 ('Some A are B' directly implies 'Some B are A').

Step-by-Step Solution

1
Analyze the relationship between quantum sensors, diagnostic tools, and portable devices based on Premises 1 and 2.
All quantum sensors (Q) are entirely within the set of diagnostic tools (D). Some diagnostic tools (D) intersect with portable devices (P). However, Q and P are not guaranteed to intersect.
This sets up the first part of the mental Venn diagram and proves that Conclusion I (Some Q are P) is invalid because the overlap is only a possibility, not a certainty. Conclusion IV (Some P are D) is immediately proven valid as a direct conversion of Premise 2.
2
Integrate Premise 3 (No portable devices are calibration instruments) into the analysis.
The set of portable devices (P) and calibration instruments (C) are completely disjoint. Therefore, any item belonging to P can never belong to C.
This constraint is necessary to evaluate the remaining conclusions.
3
Evaluate Conclusion II based on the established relationships.
Conclusion II is valid. Because some diagnostic tools are portable devices, and no portable devices can be calibration instruments, that specific overlapping subset of diagnostic tools cannot be calibration instruments.
It applies the negative constraint from Premise 3 to the intersection established in Premise 2.
4
Evaluate Conclusion III based on the established relationships.
Conclusion III is invalid. While P and C are completely separate, there is no rule preventing C from overlapping with Q (quantum sensors).
A definitive negative conclusion cannot be drawn without a chain of negative premises linking the two specific categories.

Key Concept

Syllogistic Reasoning and Venn Diagram Analysis
Rate this question

Topics

Syllogism and Categorical Propositions

More questions from this topic