Daniel Quigley

Daniel Quigley

The holiday gift exchange game is a familiar social institution with nontrivial strategic structure. We provide a formal treatment of the game's mechanics, defining the state space, action sets, and the recursive structure of stealing chains; we prove termination and derive an algorithm for counting distinct game trajectories, which grow far faster than the space of possible final allocations. Beyond the base mechanics, we introduce a decorated model incorporating partial information, social costs, and adaptive strategies grounded in discrete choice theory and the frustration-aggression literature. A full factorial simulation of 240,000 games yields three findings of note: implicit social costs are the dominant regulator of aggression, reducing stealing by 27--48\% and outweighing both uncertainty and strategic sophistication; partial information, contrary to expectation, slightly increases stealing through asymmetric uncertainty; correlated valuations amplify every behavioral effect, so that consensus about gift quality, rather than the features themselves, is what intensifies competition. The first-player advantage is robust across all conditions.

Daniel Quigley

Formal semantics and distributional semantics are distinct approaches to linguistic meaning: the former models meaning as reference via model-theoretic structures; the latter represents meaning as vectors in high-dimensional spaces shaped by usage. This paper proves that these frameworks are structurally compatible for intensional semantics. We establish that Kripke-style intensional models embed injectively into vector spaces, with semantic functions lifting to (multi)linear maps that preserve composition. The construction accommodates multiple index sorts (worlds, times, locations) via a compound index space, representing intensions as linear operators. Modal operators are derived algebraically: accessibility relations become linear operators, and modal conditions reduce to threshold checks on accumulated values. For uncountable index domains, we develop a measure-theoretic generalization in which necessity becomes truth almost everywhere and possibility becomes truth on a set of positive measure, a non-classical logic natural for continuous parameters.

Daniel Quigley, Eric Maynard

The symbol grounding problem asks how tokens like cat can be about cats, as opposed to mere shapes manipulated in a calculus. We recast grounding from a binary judgment into an audit across desiderata, each indexed by an evaluation tuple (context, meaning type, threat model, reference distribution): authenticity (mechanisms reside inside the agent and, for strong claims, were acquired through learning or evolution); preservation (atomic meanings remain intact); faithfulness, both correlational (realized meanings match intended ones) and etiological (internal mechanisms causally contribute to success); robustness (graceful degradation under declared perturbations); compositionality (the whole is built systematically from the parts). We apply this framework to four grounding modes (symbolic; referential; vectorial; relational) and three case studies: model-theoretic semantics achieves exact composition but lacks etiological warrant; large language models show correlational fit and local robustness for linguistic tasks, yet lack selection-for-success on world tasks without grounded interaction; human language meets the desiderata under strong authenticity through evolutionary and developmental acquisition. By operationalizing a philosophical inquiry about representation, we equip philosophers of science, computer scientists, linguists, and mathematicians with a common language and technical framework for systematic investigation of grounding and meaning.