[Paper Review] Theoremizing Yablo's Paradox

Ahmad Karimi, Saeed Salehi|arXiv (Cornell University)|Jun 1, 2014

Logic, programming, and type systems6 references3 citations

TL;DR

This paper formalizes Yablo's non-self-referential paradox as a theorem in Linear Temporal Logic (LTL), demonstrating that certain modal operators—specifically those expressing infinite backward quantification over truth values—cannot have fixed points. By translating the paradox into temporal logic formulas involving 'always', 'eventually', and 'next' operators, the authors prove that statements like $\Box(\varphi \leftrightarrow \Circle\diamondsuit\Box\neg\varphi)$ are unsatisfiable, thus establishing Yablo’s paradox as a new class of mathematical theorems in modal logic.

ABSTRACT

To counter a general belief that all the paradoxes stem from a kind of circularity (or involve some self--reference, or use a diagonal argument) Stephen Yablo designed a paradox in 1993 that seemingly avoided self--reference. We turn Yablo's paradox, the most challenging paradox in the recent years, into a genuine mathematical theorem in Linear Temporal Logic (LTL). Indeed, Yablo's paradox comes in several varieties; and he showed in 2004 that there are other versions that are equally paradoxical. Formalizing these versions of Yablo's paradox, we prove some theorems in LTL. This is the first time that Yablo's paradox(es) become new(ly discovered) theorems in mathematics and logic.

Motivation & Objective

• To resolve the longstanding philosophical and logical challenge of whether paradoxes can exist without self-reference by formalizing Yablo’s paradox in a rigorous logical framework.
• To demonstrate that Yablo’s paradox, traditionally seen as a semantic or epistemic puzzle, can be transformed into a genuine mathematical theorem in Linear Temporal Logic (LTL).
• To show that certain temporal operators—particularly those involving infinite chains of truth and falsity—cannot support fixed-point sentences, thereby proving their logical impossibility.
• To extend the reach of paradox-to-theorem conversion beyond classical examples like Russell’s or Liar’s paradox by applying it to a novel, non-self-referential paradox.
• To establish a formal bridge between paradoxical reasoning and provable theorems in modal and temporal logic, enriching the logical foundations of computability and proof theory.

Proposed method

• Formalizing Yablo’s paradox using a sequence of sentences $\mathcal{Y}_n$ where each $\mathcal{Y}_n$ asserts that all subsequent sentences are false, i.e., $\mathcal{Y}_n \leftrightarrow \forall k>n\ (\mathcal{Y}_k \text{ is not true})$.
• Translating the paradox into Linear Temporal Logic (LTL) by expressing the infinite chain of truth-value dependencies using modal operators: $\Box$ (always), $\diamondsuit$ (eventually), and $\Circle$ (next).
• Defining key temporal formulas such as $\varphi \leftrightarrow \Circle\diamondsuit\Box\neg\varphi$ to represent the 'almost always' and 'infinitely often' variants of Yablo’s paradox.
• Using Kripke semantics to analyze the satisfiability of these formulas, showing that assuming a model where $\Box(\varphi \leftrightarrow \Circle\diamondsuit\Box\neg\varphi)$ holds leads to a contradiction.
• Applying proof by contradiction: assuming a state where $\varphi$ is true leads to a future state where $\varphi$ must be false, and vice versa, violating consistency.
• Leveraging duality and substitution (e.g., replacing $\varphi$ with $\neg\varphi$) to derive additional theorems from the core result, such as $\neg\Box(\varphi \leftrightarrow \Circle\Box\diamondsuit\neg\varphi)$.

Experimental results

Research questions

• RQ1 Can Yablo’s non-self-referential paradox be formalized as a valid theorem in a well-defined logical system?
• RQ2 Do temporal operators expressing infinite backward quantification over truth values admit fixed points in Linear Temporal Logic?
• RQ3 Is there a formal way to capture the infinite regress structure of Yablo’s paradox using modal logic, and if so, what are its logical consequences?
• RQ4 Can the paradoxical nature of Yablo’s construction be transformed into a provable inconsistency in LTL, thus turning a semantic paradox into a syntactic theorem?
• RQ5 What are the implications of such a transformation for the foundations of logic, particularly regarding self-reference, diagonalization, and fixed-point theorems?

Key findings

• The formula $\Box(\varphi \leftrightarrow \Circle\diamondsuit\Box\neg\varphi)$ is unsatisfiable in Linear Temporal Logic, meaning no model can satisfy it, thus proving a fixed-point impossibility.
• The paper establishes $\text{LTL} \models \neg\Box(\varphi \leftrightarrow \Circle\diamondsuit\Box\neg\varphi)$ as a valid theorem, formally capturing the 'almost always' version of Yablo’s paradox.
• By duality and substitution, the result extends to $\text{LTL} \models \neg\Box(\varphi \leftrightarrow \Circle\Box\diamondsuit\neg\varphi)$, showing the same inconsistency holds for the 'infinitely often' variant.
• The proof technique relies on deriving a contradiction from assuming a state where $\varphi$ is true, which forces a later state where $\varphi$ is false, and then a later one where it must be true again, violating consistency.
• The authors show that while many operators in LTL do have fixed points, the specific structure of Yablo’s paradox—based on infinite chains of truth-value dependencies—prevents such fixed points from existing.
• This work marks the first time that Yablo’s paradox has been transformed into a new, non-tautological mathematical theorem in formal logic, rather than just a semantic puzzle or a tool for proving existing results.