Key Logical Paradoxes

Why This Matters

Paradoxes aren't just clever puzzles—they're stress tests for logical systems. When you encounter a paradox in Formal Logic I, you're seeing exactly where our rules of inference, truth conditions, and definitions break down. These breakdowns forced logicians to develop more rigorous foundations, from axiomatic set theory to multi-valued logics. Understanding why each paradox creates a contradiction teaches you more about logical structure than memorizing a hundred valid argument forms.

You're being tested on your ability to identify the mechanism behind each contradiction: Is it self-reference? Vagueness? Unrestricted set formation? Don't just memorize the paradoxes—know what principle each one challenges and how logicians have responded. When an exam question asks you to analyze a novel paradox, you'll need to recognize which family it belongs to and what logical tools address it.

---

Self-Reference Paradoxes

These paradoxes arise when a statement or definition refers back to itself, creating a loop that generates contradiction. The core mechanism is that self-reference allows a statement to assert something about its own truth value or membership conditions.

Liar Paradox

• "This statement is false"—the canonical example of semantic self-reference creating logical explosion
• Truth-value assignment fails because assuming true yields false, and assuming false yields true (a genuine dialetheic situation)
• Challenges bivalence and motivates developments like Tarski's hierarchy of metalanguages and truth predicates

Epimenides Paradox

• Historical origin of liar-type reasoning—Epimenides the Cretan declares "All Cretans are liars"
• Weaker than the Liar because it's only paradoxical if Epimenides is the only Cretan or all other Cretans always lie
• Demonstrates contingent self-reference where context determines whether contradiction actually arises

Curry's Paradox

• Conditional self-reference—"If this statement is true, then $P$" proves any arbitrary proposition $P$
• Exploits material implication and the principle that a conditional with a false antecedent is vacuously true
• More dangerous than the Liar because it doesn't just create contradiction—it trivializes the entire logical system

---

Set-Theoretic Paradoxes

These paradoxes expose problems with unrestricted set formation—the naive assumption that any definable collection forms a legitimate set. The mechanism is that certain definitions generate sets whose membership conditions are self-contradictory.

Russell's Paradox

• The set $R$ of all sets that don't contain themselves—does $R \in R$ or $R \notin R$?
• Destroyed naive set theory and forced development of axiomatic systems like ZFC with restricted comprehension
• The most historically significant paradox in foundations of mathematics—know this one cold for any exam

Barber Paradox

• Informal version of Russell's—a barber shaves exactly those who don't shave themselves; who shaves the barber?
• Resolution differs from Russell's because we can simply deny such a barber exists (no logical system collapses)
• Useful for teaching the structure of self-referential set membership without technical notation

---

Semantic and Definability Paradoxes

These paradoxes arise from self-referential definitions involving language, naming, or description. The mechanism involves definitions that reference the very system of definition itself.

Berry Paradox

• "The smallest integer not definable in fewer than eleven words"—but that phrase is a definition in fewer than eleven words
• Targets definability rather than truth, showing that "definable" can't be consistently applied within a language
• Connects to Gödel's work on the limits of formal systems and undefinability of truth

Grelling–Nelson Paradox

• Autological adjectives describe themselves ("short" is short, "English" is English); heterological ones don't ("long" isn't long)
• Is "heterological" heterological? If yes, it describes itself, so it's autological; if no, it doesn't describe itself, so it's heterological
• Parallel structure to Russell's but operates in natural language rather than set theory

---

Vagueness Paradoxes

These paradoxes exploit the absence of sharp boundaries in ordinary predicates. The mechanism is that tolerance principles ("small differences don't matter") combined with transitivity lead to absurd conclusions.

Sorites Paradox

• The heap problem—one grain isn't a heap; adding one grain can't create a heap; therefore no collection is a heap
• Challenges classical logic's law of excluded middle by showing some predicates resist binary true/false assignment
• Motivates fuzzy logic and supervaluationism as alternatives that handle borderline cases

---

Epistemic and Modal Paradoxes

These paradoxes involve knowledge, expectation, and reasoning about what agents can know or predict. The mechanism typically involves self-defeating predictions or knowledge that undermines itself.

Unexpected Hanging Paradox

• A prisoner is told he'll be hanged unexpectedly one day next week—he reasons through backward induction that it's impossible
• When he's hanged on Wednesday, he's genuinely surprised—his "proof" of impossibility was flawed
• Challenges assumptions about knowledge and surprise and reveals subtleties in epistemic reasoning

Zeno's Paradoxes

• Achilles and the Tortoise—Achilles must complete infinitely many tasks to catch up, seemingly impossible
• Dichotomy paradox—to travel any distance, you must first travel half, then half of that, ad infinitum
• Resolved by calculus and limits showing infinite series can have finite sums—but still philosophically debated regarding the metaphysics of motion

---

Quick Reference Table

---

Self-Check Questions

1. Which two paradoxes share the same logical structure but differ in whether they threaten formal systems? What makes one "merely impossible" and the other "unavoidable"?

2. If asked to identify a paradox that challenges bivalence through vagueness rather than self-reference, which paradox would you choose, and what makes it structurally different from the Liar?

3. Compare Curry's Paradox and the Liar Paradox: both involve self-reference, but why is Curry's considered more dangerous to logical systems?

4. An FRQ asks you to explain why restricting set comprehension axioms solves Russell's Paradox but wouldn't help with the Sorites Paradox. How would you answer?

5. Which paradoxes would you cite as evidence that natural language contains features that resist formalization? What specific mechanism do they exploit?