Math Problem Statement
answer
Solution
Let's go through each argument step by step to determine which one is not a valid argument.
Argument A:
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Premise 1:
- For every , if is true, then is true.
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Premise 2:
- There exists some such that is true.
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Conclusion:
- There exists some such that is true.
Analysis: This argument is valid. If there exists an such that is true, and for every , implies , then there must exist some such that is true.
Argument B:
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Premise 1:
- There exists some such that both and are true.
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Conclusion:
- There exists some such that is true.
Analysis: This argument is valid. If there exists some such that both and are true, it directly implies that there exists some where is true.
Argument C:
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Premise 1:
- There exists some such that is true.
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Premise 2:
- There exists some such that is true.
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Conclusion:
- There exists some such that both and are true.
Analysis: This argument is not valid. The existence of some such that is true and the existence of some (potentially different) such that is true does not guarantee the existence of a single where both and are true.
Argument D:
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Premise 1:
- There exists some such that if is true, then is true.
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Premise 2:
- is true for some specific .
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Conclusion:
- is true for the specific .
Analysis: This argument is valid. If is true and there exists some (which could be ) such that , then must be true.
Conclusion:
The invalid argument is C. The premises do not logically lead to the conclusion.
Would you like a more detailed explanation or have any questions about this solution?
Here are 5 related questions that expand on this topic:
- What are the rules of inference used to determine the validity of logical arguments?
- How do existential quantifiers differ from universal quantifiers in logical arguments?
- Can you explain the difference between validity and soundness in a logical argument?
- How can we use counterexamples to demonstrate that an argument is not valid?
- How does the structure of an argument affect its validity?
Tip: When dealing with logical arguments, always ensure the conclusion follows directly from the premises by applying the correct rules of inference.
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Math Problem Analysis
Mathematical Concepts
Predicate Logic
Quantifiers
Logical Inference
Formulas
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Theorems
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Suitable Grade Level
College
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