Classical Conditioning Formulas: Classical Mathematics of Conditioning: Extinction, Spontaneous Recovery, and Advanced Processes

This paper presents a formal mathematical theory of classical conditioning that reinterprets the fundamental principles of behavior, including acquisition, extinction, spontaneous recovery, stimulus generalization, stimulus discrimination, and higher-order conditioning, as a coherent system of symbolic equations. The aim is to bridge empirical psychology and quantitative modeling by expressing learning mechanisms in precise mathematical terms. Within this framework, unconditioned and conditioned stimuli are represented as variables whose interactions over time determine associative strength, learning rates, and the probability of conditioned responses. To demonstrate the structure and application of the model, a Pavlovian conditioning example is presented in which a neutral sound stimulus (bell) becomes associated with an unconditioned stimulus (food), resulting in the development of a conditioned response. The proposed equations describe the progressive accumulation of associative strength during acquisition, its reduction through extinction, and its partial restoration through spontaneous recovery, thereby capturing key experimental patterns observed in classical conditioning research. Furthermore, the model enables detailed examination of influential parameters such as stimulus intensity, reinforcement frequency, and temporal relationships between stimuli. Beyond behavioral psychology, the mathematical conditioning framework has broad interdisciplinary applicability. In neuroscience, it can contribute to modeling synaptic plasticity and reward prediction processes; in artificial intelligence, it can inform reinforcement learning algorithms and adaptive agent design; in education, it offers quantitative insights into habit formation and feedback-based learning; and in behavioral economics, it provides a basis for simulating consumer learning, preference development, and decision-making under uncertainty. By translating established behavioral principles into a scalable, predictive, and computationally rigorous framework, this work advances the integration of psychology with computational and systems sciences and contributes toward the development of unified theories of learning that are theoretically coherent, empirically grounded, and computationally efficient.