Intentic Semantics for Potentialist Truthmaking

TL;DR

This paper develops a semantic framework for potentialist truthmaking using intentic states, providing a sound and complete recursive construction for first-order logic, and explores the computational properties of non-hypothetical reasoning.

Contribution

It introduces intentic states for potentialist semantics, extending truthmaking to both classical and intuitionistic logic, and investigates decidability of non-hypothetical logic over Peano Arithmetic.

Findings

Semantic framework based on intentic states is sound and complete.

Supports truthmaking growth and hypothetical refinement relations.

Suggests decidability of non-hypothetical logic over Peano Arithmetic.

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.