Unanswerable Questions: Identifying and Classifying Paradoxes

When dealing with questions in the digital realm, it's often intriguing to ponder how we classify those that resist simple answers. On platforms like Quora, you might find a question marked 'No answers yet' even if it eventually gains responses. This article aims to delve into the classification of unanswerable questions, particularly focusing on paradoxes that can lead to such a situation. We'll explore three categories of paradoxes outlined by philosopher Willard Van Orman Quine and discuss how these can help us determine whether a question is truly unanswerable.

Understanding Paradoxes

Paradoxes are statements or problems that seem to be self-contradictory or lead to a logical dilemma. They often arise in ambiguity, logical errors, or challenges in definitions, prompting questions about their solvability.

Quine's Classification of Paradoxes

Willard Van Orman Quine, a prominent philosopher, divides paradoxes into three categories. Let's explore each one:

Veridical Paradoxes

Definition: Veridical paradoxes have correct answers or sensible explanations.

Example: Consider the classic problem from Gilbert and Sullivan's The Pirates of Penzance. A character claims to be 21 years old but has only experienced 5 birthdays. This seems paradoxical until one realizes the character was born on a leap year. Once the leap year is explained, the problem is resolved, and the paradox is no longer paradoxical.

Falsidical Paradoxes

Definition: Falsidical paradoxes result from hidden mistakes or logical fallacies.

Example: The famous mathematical paradox involving division by zero:

1. x 1
2. x x^2
3. Multiplying both sides by x: x^2 x^3
4. Subtract 1 from both sides: x^2 - 1 x^3 - 1
5. Factoring the right-hand side: (x - 1)(x 1) (x - 1)x^2
6. Dividing both sides by (x - 1): x 1 x^2
7. Subtract 1 from both sides: x 0
8. Therefore, 1 2

The error lies in the division by zero in step 6, indicating a falsidical paradox. Upon recognizing the mistake, the paradox is resolved.

Genuine Antimony

Definition: Genuine antimony arises from fundamental issues in definitions or logical structures.

Example: Consider the word "heterological," which is defined as "not describing itself." However, if "heterological" were heterological, it would describe itself, which is a contradiction. Similarly, if it were not heterological, it would be describing itself, also a contradiction. This forms a genuine antimony due to the inherent logical structure, rather than a simple mistake or unintuitive result.

Applying Quine's Categories to Unanswerable Questions

When encountering a seemingly unanswerable question, it's crucial to carefully examine the underlying paradox. Here are steps to determine if a question is truly unanswerable:

Check for Logical Errors: Look for hidden mistakes or logical fallacies, especially in mathematical or logical reasoning. Consider the Definition and Context: Analyze if the problem arises from fundamental issues in definitions or logical structures. Seek Feedback: Discuss the question with others to ensure you haven't overlooked any subtle points or logical flaws.

By carefully examining the paradox or logical structure, you can determine whether a question is veridical, falsidical, or genuinely antinomic.

Load-Specific Questions and Proof by Contradiction

Some questions, especially those containing loaded elements, may be unanswerable due to inherent contradictions. For example, if a question involves a logical contradiction, such as:

Suppose there is an answer to the question 'What is the last digit of pi?' If there were an answer, it would be one of 0 to 9. However, the last digit of pi is undefined because pi is an irrational number. Hence, the assumption that the question has an answer leads to a contradiction.

In such cases, the question is inherently unanswerable because any attempt to answer would result in a logical contradiction.

Understanding and applying these principles can help in identifying and resolving paradoxes, making questions more answerable and reducing ambiguity in digital platforms like Quora.