The proposition ‘No historians are non-mathematician’ is equivalent to which of the following proposition?
All historians are mathematicians
To solve this logical reasoning problem, we need to analyze the proposition "No historians are non-mathematicians" and determine its equivalent statement from the given options.
The original statement "No historians are non-mathematicians" can be rewritten using logical negation. Let's break it down:
- The phrase "non-mathematicians" is the logical negation of "mathematicians".
- The original proposition states that there are no historians who are non-mathematicians, meaning all historians must be mathematicians to hold this statement true.
Thus, logically, "No historians are non-mathematicians" is equivalent to saying "All historians are mathematicians". This means every historian is a mathematician, therefore confirming that they do not belong to the non-mathematician group.
Let's look at the options and see which statement matches this conclusion:
- All historians are mathematicians - This directly states that historians belong fully to the group of mathematicians, which aligns with our derived logic.
- No historians are mathematicians - This contradicts our derived statement, so it cannot be correct.
- Some historians are mathematicians - This implies not all historians are mathematicians, contradicting the derived statement.
- Some historians are not mathematicians - This indicates a portion of historians do not belong to the mathematicians group, which is contrary to "All historians are mathematicians".
After evaluating all options, the equivalent proposition is clearly "All historians are mathematicians". Therefore, the correct answer is:
All historians are mathematicians
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