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Smullyan's truth and provability

Taishi Kurahashi, Kohei Tominaga

TL;DR

The paper formalizes Smullyan's truth-provability framework by defining Smullyan models, analyzes how the naming prefixes n and r interact with core theorems, and proves a network of equivalences and non-implications among key properties. It isolates four central properties—T-Tarski, F-Tarski, T-Tarski^+, and F-Tarski^+—and shows how FPT and related results tie these properties together. By constructing arithmetic-based realizations, namely M_N and M_PA, the work connects the Smullyan framework to standard truth and provability in arithmetic, yielding interpretations of fixed-point phenomena, undefinability in N, and Gödel-type incompleteness for arithmetically definable theories. The results provide a clear, modular bridge between the simple symbolic framework and deep results in mathematical logic, with potential extensions to Rosser-style arguments in future work.

Abstract

We revisit Smullyan's paper ``Truth and Provability'' (2013) for three purposes. First, we introduce the notion of Smullyan models to give a precise definition for Smullyan's framework discussed in that paper. Second, we clarify the relationship between three theorems proved by Smullyan and other newly introduced properties for Smullyan models in terms of both implications and non-implications. Third, we construct two Smullyan models based on arithmetical ideas and show the correspondence between the properties of these Smullyan models and those concerning truth and provability in arithmetic.

Smullyan's truth and provability

TL;DR

The paper formalizes Smullyan's truth-provability framework by defining Smullyan models, analyzes how the naming prefixes n and r interact with core theorems, and proves a network of equivalences and non-implications among key properties. It isolates four central properties—T-Tarski, F-Tarski, T-Tarski^+, and F-Tarski^+—and shows how FPT and related results tie these properties together. By constructing arithmetic-based realizations, namely M_N and M_PA, the work connects the Smullyan framework to standard truth and provability in arithmetic, yielding interpretations of fixed-point phenomena, undefinability in N, and Gödel-type incompleteness for arithmetically definable theories. The results provide a clear, modular bridge between the simple symbolic framework and deep results in mathematical logic, with potential extensions to Rosser-style arguments in future work.

Abstract

We revisit Smullyan's paper ``Truth and Provability'' (2013) for three purposes. First, we introduce the notion of Smullyan models to give a precise definition for Smullyan's framework discussed in that paper. Second, we clarify the relationship between three theorems proved by Smullyan and other newly introduced properties for Smullyan models in terms of both implications and non-implications. Third, we construct two Smullyan models based on arithmetical ideas and show the correspondence between the properties of these Smullyan models and those concerning truth and provability in arithmetic.

Paper Structure

This paper contains 10 sections, 31 theorems, 28 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Smullyan models and Smullyan's theorems
  3. Properties of Smullyan models
  4. Equivalences between the properties
  5. Non-implications
  6. Smullyan models based on arithmetic
  7. Arithmetic-based Smullyan frames
  8. The $\mathbb{N}$-based Smullyan model $M_{\mathbb{N}}$
  9. $\mathsf{PA}$-based Smullyan model $M_{\mathsf{PA}}$
  10. Concluding remarks

Key Result

Theorem 1

Every predicate has a fixed point.

Figures (2)

  • Figure 1: Implications between the properties
  • Figure 2: Implications between the properties

Theorems & Definitions (67)

  • Theorem : Theorem F
  • Theorem : Theorem T
  • Theorem : Theorem G
  • Definition 2.1: Smullyan models
  • Definition 2.2: Predicates and sentences
  • Lemma 2.3
  • proof
  • Definition 2.4
  • Definition 2.5: $\mathsf{n}$-Smullyan models, $\mathsf{r}$-Smullyan models, and $\mathsf{n} \mathsf{r}$-Smullyan models
  • Definition 2.6
  • ...and 57 more