The Paradoxes · where logic breaks

The knots in reason

Logic's destructive engine. Each paradox is a confrontation: an innocent construction, the contradiction it forces, and the rival escapes: stratify, allow gaps, or accept the contradiction.

The Liar مفارقةُ الكاذب

Consider the sentence that says of itself only that it is false.

The knot $L \leftrightarrow \neg True (┌ L ┐)$
If it is true, it is false; if false, true. It can be neither, consistently.
Ways out
- Tarski Stratify truth into a hierarchy of languages; none defines its own truth.
- Kripke Allow truth-value gaps; the Liar is simply ungrounded and never settles.
- Priest Accept it: the Liar is a true contradiction (dialetheism).
Alfred Tarski Philosophical Logic Tarski’s Undefinability
Russell’s Paradox مفارقةُ رسل

Form the set of all sets that are not members of themselves.

The knot $R = {x : x \in / x} \Rightarrow (R \in R \leftrightarrow R \in / R)$
R contains itself exactly when it does not: naive set theory collapses.
Ways out
- Type theory Forbid a set from being applied to itself; stratify by type.
- ZFC Replace unrestricted comprehension with the axiom of separation.
Bertrand Russell Set Theory & Foundations
The Sorites (Heap) مفارقةُ الكُومة

One grain is not a heap. Adding a single grain never turns a non-heap into a heap.

The knot $\neg H (1) \land \forall n (\neg H (n) \to \neg H (n + 1)) \Rightarrow \forall n \neg H (n)$
By induction, no number of grains is ever a heap, yet heaps plainly exist.
Ways out
- Fuzzy logic Truth comes in degrees; “heap” is gradually, not sharply, true.
- Supervaluationism There is no precise threshold, yet classical logic is preserved.
Non-Classical Logics
Curry’s Paradox مفارقةُ كَري

Consider the sentence: “If this very sentence is true, then everything is.”

The knot $C \leftrightarrow (C \to ⊥) \Rightarrow ⊥$
From it alone, by truth and modus ponens, anything whatsoever follows.
Ways out
- Substructural logic Reject the contraction rule that drives the derivation.
- Restrict truth Deny naive, unrestricted truth-introduction.
Non-Classical Logics
Berry’s Paradox مفارقةُ بيري

Name “the least integer not nameable in fewer than twenty syllables,” in nineteen syllables.

The knot $“the least n not definable in < k words” yet defines it in < k$
The phrase defines, in few words, a number it declares undefinable in few words.
Ways out
- Tarski Definability is not itself definable in the language; no such predicate exists.
Mathematical Logic Tarski’s Undefinability
Grelling–Nelson مفارقةُ غريلنغ–نلسون

Call a word “heterological” if it does not describe itself. Is “heterological” heterological?

The knot $Het (w) \leftrightarrow \neg w (w) \Rightarrow (Het (“Het”) \leftrightarrow \neg Het (“Het”))$
“Heterological” describes itself exactly when it does not: Russell, in words.
Ways out
- Stratification Bar predicates from applying to themselves.
Philosophical Logic

The Liar مفارقةُ الكاذب

Russell’s Paradox مفارقةُ رسل

The Sorites (Heap) مفارقةُ الكُومة

Curry’s Paradox مفارقةُ كَري

Berry’s Paradox مفارقةُ بيري

Grelling–Nelson مفارقةُ غريلنغ–نلسون