According to Russell's "On Denoting", the proposition "The prime number between 7 and 11 is NOT larger than 12" would be true, if _______.
According to Russell's "On Denoting", the proposition "The prime number between 7 and 11 is NOT larger than 12" would be true, if _______.
Show Hint
- The proposition "The prime number between 7 and 11 is larger than 12" is false
The expression "the prime number between 7 and 11" has a primary occurrence in the proposition
The expression "the prime number between 7 and 11" has a secondary occurrence in the proposition
- There were only one prime number between 7 and 11
The Correct Option is C, D
Solution and Explanation
Under Russell's analysis, "The F is not G" has two readings:
• Secondary occurrence (wide–scope negation): $\neg\exists!x\,[F(x)\land G(x)]$. If the definite description denotes nothing, the whole statement is true. For "the prime between 7 and 11", in fact no such prime exists; hence with secondary occurrence the proposition is true. (C)
• Primary occurrence (narrow–scope negation): $\exists!x\,[F(x)\land \neg G(x)]$. This requires that there be exactly one such prime and that it is not larger than 12. If (counterfactually) there were exactly one prime between 7 and 11, it would certainly be $\le 10$, hence "not larger than 12" would hold; the proposition would then be true. (D)
Why (A) is not sufficient: The falsity of the converse does not by itself determine the truth of the given sentence under Russell's analysis.
Why (B) is wrong: With primary occurrence but no such prime (the actual case), the proposition is false, not true.
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