Key Logical Paradoxes

Logical paradoxes reveal the tricky side of reasoning, challenging our understanding of truth, sets, and definitions. They highlight contradictions in self-reference and vague concepts, pushing the boundaries of what we learn in Formal Logic I and II.

1. Liar Paradox

   • A statement that declares itself to be false, such as "This statement is false."
   • It creates a contradiction: if the statement is true, then it must be false, and vice versa.
   • Challenges the principles of truth and reference in formal logic.

2. Russell's Paradox

   • Arises in set theory, questioning whether a set can contain itself.
   • The set of all sets that do not contain themselves leads to a contradiction.
   • Highlights issues in naive set theory and the need for more rigorous foundations.

3. Sorites Paradox

   • Involves vague predicates, such as "heap," and the problem of defining when a collection of grains becomes a heap.
   • Demonstrates the difficulty in making precise distinctions in cases of gradual change.
   • Challenges the law of excluded middle in formal logic.

4. Barber Paradox

   • A barber who shaves all those who do not shave themselves creates a self-referential contradiction.
   • If the barber shaves himself, he must not shave himself, and vice versa.
   • Illustrates the complexities of self-reference and set membership.

5. Berry Paradox

   • Concerns the definition of the smallest natural number not definable in fewer than eleven words.
   • The phrase itself defines such a number, leading to a contradiction.
   • Highlights issues with self-reference and definability in formal logic.

6. Curry's Paradox

   • Involves a self-referential statement that leads to a contradiction through implication.
   • A statement like "If this statement is true, then 2 + 2 = 5" creates a logical inconsistency.
   • Challenges the principles of implication and truth in formal logic.

7. Epimenides Paradox

   • A Cretan who states, "All Cretans are liars," creates a self-referential contradiction.
   • If he is telling the truth, then he is a liar, and if he is lying, then he is truthful.
   • Explores the complexities of truth-telling and self-reference.

8. Grelling–Nelson Paradox

   • Concerns the classification of adjectives as "autological" (self-descriptive) or "heterological" (not self-descriptive).
   • The adjective "heterological" leads to a contradiction when applied to itself.
   • Highlights issues of self-reference and classification in language.

9. Unexpected Hanging Paradox

   • A judge tells a condemned prisoner he will be hanged on a weekday but not on the last day of the week.
   • The prisoner deduces he cannot be hanged unexpectedly, leading to a contradiction when he is hanged.
   • Explores concepts of knowledge, expectation, and surprise in logical reasoning.

10. Zeno's Paradoxes

    • A series of paradoxes that challenge notions of motion and infinity, such as Achilles and the tortoise.
    • Demonstrates that dividing a distance into infinite parts leads to contradictions in reaching a destination.
    • Raises questions about continuity, limits, and the nature of space and time in formal logic.