>[!abstract] >A paradox is a statement or line of reasoning that appears self-contradictory or logically impossible. Its conclusion may either be true, false, or have no stable truth value within a given system. Paradoxes arise when intuition, language, or assumptions clash with formal logic or reality, exposing limits in understanding or inconsistencies in conceptual frameworks. Paradoxes play a central role in philosophy, mathematics, and science by challenging assumptions and prompting conceptual refinement. >[!note] Notes >I have identified and annotated two main taxonomies of paradoxes based on their domain of pathology: >1. **Ontological taxonomy** (*where* the reasoning fails or surprises us) after [[Ramsey, 1926|Ramsey (1926)]]: >- [[Logical paradox|Logical paradoxes]] expose contradictions in formal reasoning (e.g., set theory, self-reference). >- [[Semantic paradox|Semantic paradoxes]] expose contradictions in meaning and language (e.g., truth statements, vagueness). >2. **Epistemic validity** (*how* the reasoning fails or surprises us) after [[Quine, 1976|Quine (1976)]]: >- [[Veridical paradox|Veridical paradoxes]] form a true conclusion and their reasoning are valid. They are paradoxes only for as long as they resist human intuition; once properly understood, the tension disappears. >- [[Falsidical paradox|Falsidical paradoxes]] form a false conclusion due to their invalid reasoning. They are paradoxes only for as long as their fallacy remains hidden; once the logical flaw has been spotted, the tension disappears. >- [[Antinomic paradox|Antinomic paradoxes]] form a self-contradicting conclusion based on valid reasoning. They reveal fundamental inconsistencies in their underlying logical or conceptual system. Different systems might dissolve that tension. Because these two taxonomies are [[Orthogonality|orthogonal]] to each other, it is possible to combine them into a matrix, and populate it with examples: | Paradox type | [[Logical paradox]] | [[Semantic paradox]] | | ----------------------------------------------------- | -------------------------------------------------------------------------------------------------------------------------------- | ------------------------------------------------------------------------------------------- | | [[Veridical paradox]]<br>(counterintuitive truth) | - [[Banach-Tarski paradox]]<br>- [[Bertrand's box paradox]]<br>- [[Monty Hall problem]] | - [[Sorites paradox]] <br>- [[Three prisoners problem]]<br>- [[Unexpected hanging paradox]] | | [[Falsidical paradox]]<br>(contains a hidden fallacy) | - [[Galileo's paradox]]<br>- [[Isosceles triangle paradox]]<br>- [[Proof that 1 = 2]] | - [[Heap paradox]]<br>- [[Zeno's paradoxes]] | | [[Antinomic paradox]]<br>(contradicts itself) | - [[Barber paradox]]<br>- [[Burali-Forti paradox]]<br>- [[Cantor's paradox]]<br>- [[Curry's paradox]]<br>- [[Russell's paradox]] | - [[Berry paradox]]<br>- [[Grelling-Nelson paradox]]<br>- [[Liar's paradox]] | >[!related] >- **North** (upstream): — >- **West** (similar): — >- **East** (different): — >- **South** (downstream): —