意图语义与潜在主义真理制造
Intentic Semantics for Potentialist Truthmaking
AI总结：
本文提出基于意图状态的潜在主义真理制造框架，结合非假设性推理与假设性推理，构建了可判定的逻辑结构并提供证明论基础。
摘要：
本文草案介绍了一个基于我们创新的意图状态的语义框架的技术 machinery，用于潜在主义真理制造。意图状态是结构化的部分模型，用于区分非假设性推理和假设性推理。该框架是为一阶逻辑在纯关系语言中开发的，并且与经典和直觉设置兼容。通过在意图状态上进行递归构造来定义真理制造，从而得到一个被证明在标准自然演绎中是正确且完整的语义推导关系。所得到的结构支持两种自然扩展关系，分别对应于真理制造增长和假设性细化，这些关系被证明满足Linnebo的双模潜在主义语义的公理。此外，我们研究了自然演绎非假设性片段的计算性质。受证明论和语义考虑的启发，我们提出一个猜想，即非假设性逻辑在纯关系公理化下对皮亚诺算术是可判定的，并更雄心地在任何固定的皮亚诺算术定理作为附加公理下也是可判定的。我们起草了一个示意性的证明搜索程序来支持这个猜想，识别出结构上的有限性来源。虽然初步，这种分析表明正常非假设性证明具有强的子公式纪律，并为未来的工作提供了证明论基础。
英文摘要：
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.
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