We demonstrate that the key components of cognitive architectures (declarative and procedural memory) and their key capabilities (learning, memory retrieval, probability judgment, and utility estimation) can be implemented as algebraic operations on vectors and tensors in a high-dimensional space using a distributional semantics model. High-dimensional vector spaces underlie the success of modern machine learning techniques based on deep learning. However, while neural networks have an impressive ability to process data to find patterns, they do not typically model high-level cognition, and it is often unclear how they work. Symbolic cognitive architectures can capture the complexities of high-level cognition and provide human-readable, explainable models, but scale poorly to naturalistic, non-symbolic, or big data. Vector-symbolic architectures, where symbols are represented as vectors, bridge the gap between the two approaches. We posit that cognitive architectures, if implemented in a vector-space model, represent a useful, explanatory model of the internal representations of otherwise opaque neural architectures. Our proposed model, Holographic Declarative Memory (HDM), is a vector-space model based on distributional semantics. HDM accounts for primacy and recency effects in free recall, the fan effect in recognition, probability judgments, and human performance on an iterated decision task. HDM provides a flexible, scalable alternative to symbolic cognitive architectures at a level of description that bridges symbolic, quantum, and neural models of cognition.