arXiv subjects

Thomas Macaulay Ferguson

Publications and source records attributed to Thomas Macaulay Ferguson.

3 recordsLinked to original sources

arXiv2026

Hyperformalism for Relevant Modal Logics

The property of hyperformalism has proven to be a powerful tool in the analysis of relevant logics, revealing that increasingly weak relevant logics are closed under increasingly strong classes of non-uniform substitutions. In such substitutions, two instances of the same atom may be treated independently in virtue of syntactic features of their appearances in a complex. In this work, we extend the scope of hyperformalism to relevant modal logics by considering MPos-hyperformalism, that is, a property in which relevant modal logics are closed under substitutions in which nesting within the scope of modal operators is taken into account. We prove that the weak relevant modal logic B-Box is MPos-hyperformal and investigate the classes of non-uniform substitutions under which several extensions are closed. We then consider corresponding refinements of the variable sharing property that hold of such logics. We conclude by introducing a modal logic K-MPos that constitutes the largest MPos-hyperformal sublogic of the classical modal logic K and provide soundness and completeness results.

cs.LO↗

arXiv2026

Sound and Complete Neurosymbolic Reasoning with LLM-Grounded Interpretations

Large language models (LLMs) have demonstrated impressive capabilities in natural language understanding and generation, but exhibit problems with logical consistency in their output. How can we harness LLMs' broad-coverage parametric knowledge in formal reasoning despite their inconsistency? We present a method for directly integrating an LLM into the interpretation function of the formal semantics for a paraconsistent logic. We evaluate the method empirically using datasets derived from the short-form factuality benchmarks GPQA and SimpleQA, showing that bilateral factuality evaluation improves macro-F1 over a unilateral baseline by roughly 6 percentage points on both benchmarks (at the cost of reduced coverage, as abstention is triggered on inconsistent or uncertain cases). We further describe a proof-of-concept tableau reasoner implementing the method, and apply it to a medication-safety knowledge base of 228 asserted and 712 inferred statements: the system detects 92 gluts corresponding to medically significant errors (e.g., opioids inferred as non-addictive, beta-blockers inferred as safe in asthma) while remaining satisfiable, demonstrating that contradictions are localized rather than causing logical explosion. Unlike prior work, our method offers a theoretical framework with a practical implementation for neurosymbolic reasoning that leverages an LLM's knowledge while preserving the underlying logic's soundness and completeness properties.

cs.AI↗

arXiv2024

Topics, Non-Uniform Substitutions, and Variable Sharing

The family of relevant logics can be faceted by a hierarchy of increasingly fine-grained variable sharing properties -- requiring that in valid entailments $A\to B$, some atom must appear in both $A$ and $B$ with some additional condition (e.g., with the same sign or nested within the same number of conditionals). In this paper, we consider an incredibly strong variable sharing property of lericone relevance that takes into account the path of negations and conditionals in which an atom appears in the parse trees of the antecedent and consequent. We show that this property of lericone relevance holds of the relevant logic $\mathbf{BM}$ (and that a related property of faithful lericone relevance holds of $\mathbf{B}$) and characterize the largest fragments of classical logic with these properties. Along the way, we consider the consequences for lericone relevance for the theory of subject-matter, for Logan's notion of hyperformalism, and for the very definition of a relevant logic itself.

math.LO↗