Comprehensive Study Notes on Logical and Logical and Philosophical Paradoxes

Nature and Definition of Paradoxes

• A paradox is defined as a statement that leads to an apparently self-contradictory or logically unacceptable conclusion, despite beginning with sound reasoning from true premises.

• Paradoxes involve contradictory yet interrelated elements that occur simultaneously and persist over time.

• Value in logic: While some logical paradoxes are recognised as invalid arguments, they are valuable for promoting critical thinking. They have identified errors in definitions previously assumed to be rigorous, leading to the re-examination of axioms in mathematics and logic.

• Examples of paradoxes:

  • Russell’s Paradox: Questions whether a "list of all lists that do not contain themselves" would include itself. This showed that foundational attempts to link set theory with predicates or properties were flawed.

  • Ship of Theseus: A philosophical problem questioning if a ship remains the same after every single one of its wooden parts is replaced over time.

  • Visual Media: M.C. Escher created perspective-based paradoxes in drawings where walls are viewed as floors from different perspectives and staircases climb endlessly.

  • Common Usage: Often refers to ironic or unexpected statements, such as the observation that standing can be more tiring than walking.

Paradoxes of Material Implication

• These represent formulae that are true in classical logic but appear intuitively problematic. The root issue is a mismatch between natural language validity and formal interpretation dating back to George Boole’s algebraic logic.

• Truth-Functional Interpretation: In classical logic, material implication ($p \rightarrow q$) is defined as "it is not the case that $p$ is true and $q$ is false." This is equivalent to the expression $(\sim p \vee q)$.

• Example: "If it is raining, then I will bring an umbrella" is equivalent to saying "it is not raining, or I will bring an umbrella, or both."

• Formal Paradoxes of Implication:

  1. $(\sim p \wedge p) \rightarrow q$: Known as the paradox of entailment; a proposition and its negation imply any arbitrary $q$.

  2. $p \rightarrow (q \rightarrow p)$: If $p$ is true, it is implied by every proposition $q$.

  3. $\sim p \rightarrow (p \rightarrow q)$: If $p$ is false, it implies every arbitrary $q$. This is known as "explosion," and the statement $(p \rightarrow q)$ is described as being vacuously true.

  4. $p \rightarrow (q \vee \sim q)$: Because the disjunction of a proposition and its negation is always true, it is implied by every $p$.

  5. $(p \rightarrow q) \vee (q \rightarrow r)$: Given any three propositions, either $p$ implies $q$ or $q$ implies $r$. If $r$ is taken as $p$, it suggests that between two arbitrary propositions, one must imply the other even if they are contradictory. Example: "Nadia is in Barcelona implies Nadia is in Madrid, or Nadia is in Madrid implies Nadia is in Barcelona."

  6. $\sim (p \rightarrow q) \rightarrow (p \wedge \sim q)$: If $p$ does not imply $q$, then $p$ is true and $q$ is false. This is surprising because it dictates the truth values of the propositions based solely on the failure of implication.

• Foundational Problem: Material implication is considered true merely because an antecedent is false or a consequent is true. Consequently, "If the moon is made of green cheese, then the world is coming to an end" is technically true because the moon is not made of green cheese. It implies that any contradiction implies anything, and any tautology is implied by anything.

Russell’s Paradox and Set Theory

• The Barber Analogy: Consider a barber who shaves only those men who do not shave themselves. If the barber does not shave himself, he must shave himself by definition. If he does shave himself, he violates the rule that he only shaves those who do not shave themselves.

• Historical Context: Discovered by Bertrand Russell in 1901 and published in Principles of Mathematics (1903). It challenged Gottlob Frege’s attempt to found mathematics on symbolic logic.

• Set-Builder Notation:

  • Example: x = \{n: n \text{ is an integer and } 3 < n < 7\} results in the set ${4, 5, 6}$.

  • Example: $y = \{x: x \text{ is a male resident of the United States}\}$.

• The Paradoxical Set: Russell and Ernst Zermelo independently identified that the set definition $x = \{a: a \notin a\}$ leads to a contradiction. If $x$ is in $x$, then it must not be in $x$; if it is not in $x$, it must be in $x$.

• Proposed Solutions:

  • Russell’s Theory of Types: A hierarchy of objects (numbers, sets of numbers, sets of sets, etc.) to prevent the confusion of different set levels. This is used in computer science and philosophy.

  • Zermelo’s Axiom: Replaced the axiom "for every formula $A(x)$ there is a set $y = \{x: A(x)\}$" with "for every formula $A(x)$ and every set $b$ there is a set $y = \{x: x \in b \text{ and } A(x)\}$."

• Reference Note: Correspondence between Russell and Frege is documented in From Frege to Godel, a Source Book in Mathematical Logic, 1879-1931 by Jean van Heijenoort.

The Liar Paradox

• Statement: Occurs when a liar states they are lying, such as "I am lying" or "Everything I say is false."

• Logical Analysis: If the statement is true, then the speaker is lying, making the statement false. If the statement is false, then the speaker is indeed lying, making the statement true.

• Strengthened Version: (A) "This statement is false." Attempting to assign a binary truth value leads to an endless loop of contradiction.

• The Bivalence Issue: Some conclude the sentence is "neither true nor false," rejecting the principle of bivalence (the claim that every statement must be either true or false).

• Counter-Challenge: Rejection of bivalence leads to a second strengthened version: (B) "This statement is not true." If (B) is neither true nor false, then it is "not true," which is exactly what (B) claims, making it true and restarting the paradox.

Zeno’s Paradoxes of Motion

• Origin: Devised by Zeno of Elea (ca. 490–430 BC) to support Parmenides’ doctrine that motion and change are illusions and that plurality is mistaken.

• Methodology: These arguments are early examples of reductio ad absurdum (proof by contradiction) and a source of the dialectic method used by Socrates.

• Achilles and the Tortoise: Achilles gives a tortoise a 100-meter head start. To overtake the tortoise, Achilles must first reach the tortoise’s starting point. By the time he arrives, the tortoise has moved forward (e.g., 10 meters). Achilles must then reach that new point, but the tortoise has moved again. Zeno argues Achilles can never overtake the tortoise.

• The Dichotomy Paradox (Race Course Paradox): One must reach the halfway point of a distance before reaching the goal. To reach that half, one must first reach the quarter-mark, then the eighth-mark, and so on.

  • Sequence: $\{\dots , \frac{1}{16}, \frac{1}{8}, \frac{1}{4}, \frac{1}{2}, 1\}$.

  • Conclusion: One must complete an infinite number of tasks, which Zeno claims is impossible. Furthermore, as any finite distance can be halved, there is no "first distance" to run, meaning motion cannot even begin.

  • Alternative Interpretation: Henri Bergson proposed that motion, time, and distance are not actually divisible.

• The Arrow Paradox (Fletcher’s Paradox): At any single duration-less instant, a flying arrow is neither moving to where it is (it is already there) nor to where it is not (no time elapses for motion). If time is composed of instants where no motion occurs, motion itself is impossible. This paradox divides time into points rather than segments.