arXiv subjects

James Walsh

Publications and source records attributed to James Walsh.

At least 19 recordsLinked to original sources

arXiv2026

Stalnaker's logical problem of conditionals is unsolvable

The logical problem of conditionals, as conceived by Stalnaker, amounts to axiomatizing a particular semantics for conditionals which utilizes selection functions that take propositions (i.e., sets of worlds) as arguments. While the sentential form of this semantics is recursively axiomatizable, we prove that its enrichment with first-order quantifiers is not---that is, we show that Stalnaker's logical problem of conditionals is unsolvable in the language with first-order quantifiers. We demonstrate this by showing how to interpret arithmetic in the logic. In the conclusion, we discuss the implications of this result for the study of conditional logic.

math.LO↗

arXiv2026

Zero-Shot Satellite Image Retrieval through Joint Embeddings: Application to Crisis Response

Semantic search of Earth observation archives remains challenging. Visual foundation models such as CLAY produce rich embeddings of satellite imagery but lack the natural-language grounding needed for intuitive query, and full contrastive training of a remote-sensing CLIP-style model requires paired data and compute that are unavailable at global scale. To allow natural language querying at global scales, we present GeoQuery, a zero-shot retrieval system that sidesteps data and compute constraints through a two-stage semantic and visual search, leveraging a natural language embedding of a subset (proxy) of global data. Rather than training a joint encoder, we generate language descriptions for a 100k proxy subset of global Sentinel-2 tiles and optimise the description-generation prompt so that distances in the resulting text-embedding space correlate with distances in the frozen CLAY visual-embedding space. Queries are resolved in two stages, with a text-similarity search over the proxy subset followed by a visual nearest-neighbour search over worldwide CLAY embeddings On 76 disaster-location queries covering UK floods, US wildfires, and US droughts, GeoQuery achieves 31.6\% accuracy within 50\,km, with the strongest performance on floods (50\% within 50\,km) where terrain features are well captured by RGB embeddings. Deployed within a crisis response system called \ECHO{}, GeoQuery identified vulnerable areas during Brisbane's 2025 Cyclone Alfred, with downstream flood simulations reproducing historical patterns. Prompt-aligned proxies offer a practical bridge between EO foundation models and operational retrieval when full contrastive training is out of reach.

cs.CV↗

arXiv2026

Incompleteness in Quantified Conditional Logic

Stalnaker and Thomason famously proved that the conditional logic \textsf{C2} with first-order quantifiers is complete with respect to a selection function semantics. However, the selection functions used in this completeness result take formulas, rather than propositions (i.e., sets of worlds), as arguments. Yet Stalnaker has repeatedly emphasized the philosophical importance of viewing selection functions as functions on propositions, and many of the applications of his theory require this. Can their completeness result be extended to a selection function semantics in which the functions take propositions as arguments? We prove the answer is negative: Their logic is frame incomplete. Moreover, this result is invariant with respect to many choice points regarding the semantics, such as variable vs.~constant domains or whether to include an identity or existence predicate. We conclude by discussing some of the important and difficult questions for the philosophical and logical study of conditionals that our results raise.

math.LO↗

arXiv2025

Descending sequences in reflection hierarchies

There is no recursively enumerable sequence of sufficiently strong 2-consistent r.e. theories such that each proves the $2$-consistency of the next. Montalb\'an and Shavrukov independently asked whether this result generalizes to $0'$-recursive sequences. We consider a general version of this problem: For arbitrary $n$, for which complexity classes $\Gamma$ are there $\Gamma$-definable sequences of $n$-consistent r.e. theories each of which proves the $n$-consistency of the next? The answer to this question depends not only on $n$ and $\Gamma$ but also on the manner in which sequences are encoded in arithmetic. We provide positive answers for certain encodings and negative answers for others.

math.LO↗

arXiv2025

Reasoning With a Star: A Heliophysics Dataset and Benchmark for Agentic Scientific Reasoning

Scientific reasoning through Large Language Models in heliophysics involves more than just recalling facts: it requires incorporating physical assumptions, maintaining consistent units, and providing clear scientific formats through coordinated approaches. To address these challenges, we present Reasoning With a Star, a newly contributed heliophysics dataset applicable to reasoning; we also provide an initial benchmarking approach. Our data are constructed from National Aeronautics and Space Administration & University Corporation for Atmospheric Research Living With a Star summer school problem sets and compiled into a readily consumable question-and-answer structure with question contexts, reasoning steps, expected answer type, ground-truth targets, format hints, and metadata. A programmatic grader checks the predictions using unit-aware numerical tolerance, symbolic equivalence, and schema validation. We benchmark a single-shot baseline and four multi-agent patterns, finding that decomposing workflows through systems engineering principles outperforms direct prompting on problems requiring deductive reasoning rather than pure inductive recall.

cs.AI↗

arXiv2025

Ranking theories via encoded $\beta$-models

Ranking theories according to their strength is a recurring motif in mathematical logic. We introduce a new ranking of arbitrary (not necessarily recursively axiomatized) theories in terms of the encoding power of their $\beta$-models: $T\prec_\beta U$ if every $\beta$-model of $U$ contains a countable coded $\beta$-model of $T$. The restriction of $\prec_\beta$ to theories with $\beta$-models is well-founded. We establish fundamental properties of the attendant ranking. First, though there are continuum-many theories, every theory has countable $\prec_\beta$-rank. Second, the $\prec_\beta$-ranks of $\mathcal{L}_\in$ theories are cofinal in $\omega_1$. Third, assuming $V=L$, the $\prec_\beta$-ranks of $\mathcal{L}_2$ theories are cofinal in $\omega_1$. Finally, $\delta^1_2$ is the supremum of the $\prec_\beta$-ranks of finitely axiomatized theories.

math.LO↗

arXiv2024

A Foundation Model for the Solar Dynamics Observatory

SDO-FM is a foundation model using data from NASA's Solar Dynamics Observatory (SDO) spacecraft; integrating three separate instruments to encapsulate the Sun's complex physical interactions into a multi-modal embedding space. This model can be used to streamline scientific investigations involving SDO by making the enormous datasets more computationally accessible for heliophysics research and enable investigations that require instrument fusion. We discuss four key components: an ingestion pipeline to create machine learning ready datasets, the model architecture and training approach, resultant embeddings and fine-tunable models, and finally downstream fine-tuned applications. A key component of this effort has been to include subject matter specialists at each stage of development; reviewing the scientific value and providing guidance for model architecture, dataset, and training paradigm decisions. This paper marks release of our pretrained models and embedding datasets, available to the community on Hugging Face and sdofm.org.

astro-ph.SR↗

arXiv2024

A classification of incompleteness statements

For which choices of $X,Y,Z\in\{\Sigma^1_1,\Pi^1_1\}$ does no sufficiently strong $X$-sound and $Y$-definable extension theory prove its own $Z$-soundness? We give a complete answer, thereby delimiting the generalizations of G\"odel's second incompleteness theorem that hold within second-order arithmetic.

math.LO↗

arXiv2024

Modal definability in Kripke's theory of truth

In Outline of a Theory of Truth, Kripke introduces some of the central concepts of the logical study of truth and paradox. He informally defines some of these -- such as groundedness and paradoxicality -- using modal locutions. We introduce a modal language for regimenting these informal definitions. Though groundedness and paradoxicality are expressible in the modal language, we prove that intrinsicality -- which Kripke emphasizes but does not define modally -- is not. We characterize the modally definable relations and completely axiomatize the modal semantics.

math.LO↗

arXiv2023

Generalized ordinal analysis and reflection principles in set theory

It is widely claimed that the natural axiom systems$\unicode{x2013}$including the large cardinal axioms$\unicode{x2013}$form a well-ordered hierarchy. Yet, as is well-known, it is possible to exhibit non-linearity and ill-foundedness by means of \emph{ad hoc} constructions. In this paper we formulate notions of proof-theoretic strength based on set-theoretic reflection principles. We prove that they coincide with orderings on theories given by the generalized ordinal analysis of Pohlers. Accordingly, these notions of proof-theoretic strength engender genuinely well-ordered hierarchies. The reflection principles considered in this paper are formulated relative to G\"odel's constructible universe; we conclude with generalizations to other inner models.

math.LO↗

arXiv2023

High-Cadence Thermospheric Density Estimation enabled by Machine Learning on Solar Imagery

Accurate estimation of thermospheric density is critical for precise modeling of satellite drag forces in low Earth orbit (LEO). Improving this estimation is crucial to tasks such as state estimation, collision avoidance, and re-entry calculations. The largest source of uncertainty in determining thermospheric density is modeling the effects of space weather driven by solar and geomagnetic activity. Current operational models rely on ground-based proxy indices which imperfectly correlate with the complexity of solar outputs and geomagnetic responses. In this work, we directly incorporate NASA's Solar Dynamics Observatory (SDO) extreme ultraviolet (EUV) spectral images into a neural thermospheric density model to determine whether the predictive performance of the model is increased by using space-based EUV imagery data instead of, or in addition to, the ground-based proxy indices. We demonstrate that EUV imagery can enable predictions with much higher temporal resolution and replace ground-based proxies while significantly increasing performance relative to current operational models. Our method paves the way for assimilating EUV image data into operational thermospheric density forecasting models for use in LEO satellite navigation processes.

physics.space-ph↗

arXiv2023

A theory satisfying a strong version of Tennenbaum's theorem

We answer a question of Pakhomov by showing that there is a consistent, c.e. theory $T$ such that no theory which is definitionally equivalent to $T$ has a computable model. A key tool in our proof is the model-theoretic notion of mutual algebraicity.

math.LO↗

arXiv2022

Characterizations of ordinal analysis

Ordinal analysis is a research program wherein recursive ordinals are assigned to axiomatic theories. According to conventional wisdom, ordinal analysis measures the strength of theories. Yet what is the attendant notion of strength? In this paper we present abstract characterizations of ordinal analysis that address this question. First, we characterize ordinal analysis as a partition of $\Sigma^1_1$-definable and $\Pi^1_1$-sound theories, namely, the partition whereby two theories are equivalent if they have the same proof-theoretic ordinal. We show that no equivalence relation $\equiv$ is finer than the ordinal analysis partition if both: (1) $T\equiv U$ whenever $T$ and $U$ prove the same $\Pi^1_1$ sentences; (2) $T\equiv T+U$ for every set $U$ of true $\Sigma^1_1$ sentences. In fact, no such equivalence relation makes a single distinction that the ordinal analysis partition does not make. Second, we characterize ordinal analysis as an ordering on arithmetically-definable and $\Pi^1_1$-sound theories, namely, the ordering wherein $T< U$ if the proof-theoretic ordinal of $T$ is less than the proof-theoretic ordinal of $U$. The standard ways of measuring the strength of theories are consistency strength and inclusion of $\Pi^0_1$ theorems. We introduce analogues of these notions -- $\Pi^1_1$-reflection strength and inclusion of $\Pi^1_1$ theorems -- in the presence of an oracle for $\Sigma^1_1$ truths, and prove that they coincide with the ordering induced by ordinal analysis.

math.LO↗

arXiv2022

Evitable iterates of the consistency operator

Let's fix a reasonable subsystem $T$ of arithmetic; why are natural extensions of $T$ pre-well-ordered by consistency strength? In previous work, an approach to this question was proposed. The goal of this work was to classify the recursive functions that are monotone with respect to the Lindenabum algebra of $T$. According to an optimistic conjecture, roughly, every such function must be equivalent to an iterate $\mathsf{Con}_T^\alpha$ of the consistency operator in the limit. In previous work the author established the first case of this optimistic conjecture; roughly, every recursive monotone function is either as weak as the identity operator in the limit or as strong as $\mathsf{Con}_T$ in the limit. Yet in this note we prove that this optimistic conjecture fails already at the next step; there are recursive monotone functions that are neither as weak as $\mathsf{Con}_T$ in the limit nor as strong as $\mathsf{Con}_T^2$ in the limit. In fact, for every $\alpha$, we produce a function that is cofinally as strong as $\mathsf{Con}^\alpha_T$ yet cofinally as weak as $\mathsf{Con}_T$.

math.LO↗

arXiv2022

A robust proof-theoretic well-ordering

It is well-known that natural axiomatic theories are pre-well-ordered by logical strength, according to various characterizations of logical strength such as consistency strength and inclusion of $\Pi^0_1$ theorems. Though these notions of logical strength coincide for natural theories, they are not generally equivalent. We study analogues of these notions -- such as $\Pi^1_1$-reflection strength and inclusion of $\Pi^1_1$ theorems -- in the presence of an oracle for $\Sigma^1_1$ truths. In this context these notions coincide; moreover, we get genuine pre-well-orderings of axiomatic theories and may drop the non-mathematical quantification over "natural" theories.

math.LO↗

arXiv2021

A characterization of ordinal analysis

Ordinal analysis induces a partition of $\Sigma^1_1$-definable and $\Pi^1_1$-sound theories whereby two theories are equivalent if they have the same proof-theoretic ordinal. We show that no equivalence relation $\equiv$ is finer than the ordinal analysis partition if both: (1) $T\equiv U$ whenever $T$ and $U$ prove the same $\Pi^1_1$ sentences; (2) $T\equiv T+U$ for every set $U$ of true $\Sigma^1_1$ sentences. In fact, no such equivalence relation makes a single distinction that the ordinal analysis partition does not make.

math.LO↗

arXiv2021

An incompleteness theorem via ordinal analysis

We present an analogue of G\"{o}del's second incompleteness theorem for systems of second-order arithmetic. Whereas G\"{o}del showed that sufficiently strong theories that are $\Pi^0_1$-sound and $\Sigma^0_1$-definable do not prove their own $\Pi^0_1$-soundness, we prove that sufficiently strong theories that are $\Pi^1_1$-sound and $\Sigma^1_1$-definable do not prove their own $\Pi^1_1$-soundness. Our proof does not involve the construction of a self-referential sentence but rather relies on ordinal analysis.

math.LO↗

arXiv2021

Low Temperature Plasma for Biology, Hygiene, and Medicine: Perspective and Roadmap

Plasma, the fourth and most pervasive state of matter in the visible universe, is a fascinating medium that is connected to the beginning of our universe itself. Man-made plasmas are at the core of many technological advances that include the fabrication of semiconductor devices, which enabled the modern computer and communication revolutions. The introduction of low temperature, atmospheric pressure plasmas to the biomedical field has ushered a new revolution in the healthcare arena that promises to introduce plasma-based therapies to combat some thorny and long-standing medical challenges. This paper presents an overview of where research is at today and discusses innovative concepts and approaches to overcome present challenges and take the field to the next level. It is written by a team of experts who took an in-depth look at the various biomedical applications, made critical analysis, and proposed ideas and concepts that should help the research community focus their efforts on clear and practical steps necessary to keep the field advancing for decades to come.

physics.med-ph↗