Strategies for Dealing with Paradoxes
How can we deal with paradoxes? Here are some strategies:
1. Identifying Hidden Assumptions
Many paradoxes are based on hidden or unconscious assumptions. By identifying and questioning these assumptions, we can often resolve the contradiction.
Example: In Zeno's paradoxes it is implicitly assumed that an infinite number of steps cannot be completed in a finite time. Modern mathematics, however, shows that an infinite series can have a finite sum.
2. Distinguishing Levels of Language
Semantic paradoxes such as the liar paradox can often be solved by distinguishing between object language and metalanguage.
Example: According to Tarski's theory of truth, the truth of a statement can only be discussed in a metalanguage, not in the language of the statement itself.
3. Making Vague Concepts Precise
Sorites paradoxes can be addressed by making vague concepts precise or by introducing thresholds.
Example: Instead of asking "Is this a heap?", we could ask more precise questions such as "How many grains of sand does this heap contain?" or define boundary ranges.
4. Contextual Analysis
Some paradoxes dissolve when we take into account the context in which they arise.
Example: The Ship of Theseus can be viewed differently depending on the context: from a legal, physical or historical perspective, different answers might be appropriate.
5. Accepting the Limits
Sometimes we have to accept that certain paradoxes point to fundamental limits of our concepts, language or logic.
Example: Gödel's incompleteness theorems show that we have to accept the limits of formal systems: not every true statement can be proved within a system.
6. Developing New Theoretical Frameworks
Some paradoxes have led to the development of entirely new theoretical frameworks.
Examples:
- Russell's paradox led to type theory and axiomatic set theory
- Zeno's paradoxes inspired the development of infinitesimal calculus
- The prisoner's dilemma led to new approaches in game theory and evolutionary biology