Some white are blue. All cakes are white. No blue is black. Which conclusions follow: I. Some cakes are not black. II. Some white cannot be black. III. Some cakes are blue. IV. All... Some white are blue. All cakes are white. No blue is black. Which conclusions follow: I. Some cakes are not black. II. Some white cannot be black. III. Some cakes are blue. IV. All white being cake is a possibility?
Understand the Problem
The question involves logical reasoning based on given statements and conclusions about colors and cakes. It requires evaluating which conclusions logically follow from the provided statements.
Answer
Only conclusions I, II, and IV follow from the statements.
Answer for screen readers
Only conclusions I, II, and IV follow from the statements.
Steps to Solve
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Interpret the Statements
The statements provided are:
- Some white are blue.
- All cakes are white.
- No blue is black.
From these statements, we can infer relationships between white, blue, cakes, and black.
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Analyze the Conclusions
We need to evaluate each conclusion based on the statements:
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Conclusion I: Some cakes are not black.
- Since all cakes are white and no blue is black, cakes cannot be black. Thus, this conclusion is valid.
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Conclusion II: Some white cannot be black.
- From the statement "No blue is black" and "Some white are blue," we can conclude that some of the white (those that are blue) cannot be black. Hence, this conclusion is also valid.
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Conclusion III: Some cakes are blue.
- All cakes are white, and only some white are blue. Therefore, we cannot definitively say that some cakes are blue. This conclusion is not valid.
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Conclusion IV: All white being cake is a possibility.
- There’s no information in the statements that rules this possibility out. Thus, this conclusion is valid.
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Summarize Valid Conclusions
Based on the analysis, conclusions I, II, and IV follow from the given statements.
Only conclusions I, II, and IV follow from the statements.
More Information
This problem illustrates the power of logical reasoning and deduction, demonstrating how to determine relationships among groups based on given premises. It is essential in areas like mathematics, philosophy, and programming.
Tips
- People often assume relationships without fully analyzing the implications of each statement. Always ensure to follow logical connections.
- Misinterpreting what "some" means; "some" does not imply "all," which can lead to incorrect conclusions.
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