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Intentic Semantics for Potentialist Truthmaking

Paul Gorbow

TL;DR

This work develops a semantic framework for potentialist truthmaking in first-order logic via intentic states, a structured notion of partial models distinguishing non-hypothetical from hypothetical reasoning. It defines a recursive truthmaking semantics $MT(u)$ over intentic states and introduces two extension orders, $u riangleleft_ hd v$ and $u riangleleft_F v$, yielding a Linnebo-style bi-modal semantics that is sound and complete with respect to standard natural deduction in the non-hypothetical fragment. It further analyzes the complexity of the non-hypothetical logic, proposing decidability conjectures for relational $ extsf{PA}$ and schematic proof-search procedures that rely on finiteness of constants and a strong subformula discipline. Finally, it aligns the resulting structure with Linnebo's potentialism, showing that the two accessibility relations satisfy the required frame conditions, thereby providing a rigorous, finitary bridge between potentialist semantics and classical proof theory.

Abstract

This draft introduces the technical machinery of a semantic framework for potentialist truthmaking based on our innovation of intentic states, which are structured partial models accounting for our distinction between non-hypothetical and hypothetical reasoning. The framework is developed for first-order logic in a purely relational language and is compatible with both classical and intuitionistic settings. Truthmaking is defined via a recursive construction over intentic states, yielding a semantic consequence relation that is shown to be sound and complete with respect to standard natural deduction. The resulting structure supports two natural extension relations, corresponding to truthmaking growth and hypothetical refinement, which are shown to satisfy the axioms governing Linnebo's bi-modal potentialist semantics. Moreover, we investigate the computational properties of the non-hypothetical fragment of natural deduction. Motivated by proof-theoretic and semantic considerations, we formulate a conjecture that non-hypothetical logic is decidable over Peano Arithmetic in a purely relational axiomatization, and more ambitiously over any fixed Peano Arithmetic theorem taken as an additional axiom. A schematic proof-search procedure is drafted to support this conjecture, identifying structural sources of finiteness. While preliminary, this analysis suggests a strong subformula discipline for normal non-hypothetical proofs and provides a proof-theoretic foundation for future work.

Intentic Semantics for Potentialist Truthmaking

TL;DR

This work develops a semantic framework for potentialist truthmaking in first-order logic via intentic states, a structured notion of partial models distinguishing non-hypothetical from hypothetical reasoning. It defines a recursive truthmaking semantics over intentic states and introduces two extension orders, and , yielding a Linnebo-style bi-modal semantics that is sound and complete with respect to standard natural deduction in the non-hypothetical fragment. It further analyzes the complexity of the non-hypothetical logic, proposing decidability conjectures for relational and schematic proof-search procedures that rely on finiteness of constants and a strong subformula discipline. Finally, it aligns the resulting structure with Linnebo's potentialism, showing that the two accessibility relations satisfy the required frame conditions, thereby providing a rigorous, finitary bridge between potentialist semantics and classical proof theory.

Abstract

This draft introduces the technical machinery of a semantic framework for potentialist truthmaking based on our innovation of intentic states, which are structured partial models accounting for our distinction between non-hypothetical and hypothetical reasoning. The framework is developed for first-order logic in a purely relational language and is compatible with both classical and intuitionistic settings. Truthmaking is defined via a recursive construction over intentic states, yielding a semantic consequence relation that is shown to be sound and complete with respect to standard natural deduction. The resulting structure supports two natural extension relations, corresponding to truthmaking growth and hypothetical refinement, which are shown to satisfy the axioms governing Linnebo's bi-modal potentialist semantics. Moreover, we investigate the computational properties of the non-hypothetical fragment of natural deduction. Motivated by proof-theoretic and semantic considerations, we formulate a conjecture that non-hypothetical logic is decidable over Peano Arithmetic in a purely relational axiomatization, and more ambitiously over any fixed Peano Arithmetic theorem taken as an additional axiom. A schematic proof-search procedure is drafted to support this conjecture, identifying structural sources of finiteness. While preliminary, this analysis suggests a strong subformula discipline for normal non-hypothetical proofs and provides a proof-theoretic foundation for future work.
Paper Structure (7 sections, 7 theorems, 9 equations, 1 figure, 4 algorithms)

This paper contains 7 sections, 7 theorems, 9 equations, 1 figure, 4 algorithms.

Table of Contents

  1. Complete Intentic Semantics for Predicate Logic
  2. Non-Hypothetical Logic
  3. Intentic States and their Truthmaking Semantics
  4. Soundness and Completeness
  5. Complexity of non-hypothetical logic
  6. Comparison and Discussion
  7. Relation to Linnebo's Potentialism

Key Result

Lemma 3

Let $u, v, w$ be intentic states over $\mathcal{L}$. If $u \leq_F v$ and $v \leq_F w$, then $u \leq_F w$. Moreover, if $u \leq_F v$, then $u \leq_\Vdash v$.

Figures (1)

  • Figure 2.1: Axioms of Peano Arithmetic in a relational signature

Theorems & Definitions (17)

  • Definition 1
  • Definition 2
  • Lemma 3
  • proof
  • Definition 4
  • Proposition 5
  • proof
  • Theorem 6: Soundness
  • proof
  • Lemma 7
  • ...and 7 more