Much progress has been made in semi-supervised learning (SSL) by combining\nmethods that exploit different aspects of the data distribution, e.g.\nconsistency regularisation relies on properties of $p(x)$, whereas entropy\nminimisation pertains to the label distribution $p(y|x)$. Focusing on the\nlatter, we present a probabilistic model for discriminative SSL, that mirrors\nits classical generative counterpart. Under the assumption $y|x$ is\ndeterministic, the prior over latent variables becomes discrete. We show that\nseveral well-known SSL methods can be interpreted as approximating this prior,\nand can be improved upon. We extend the discriminative model to neuro-symbolic\nSSL, where label features satisfy logical rules, by showing such rules relate\ndirectly to the above prior, thus justifying a family of methods that link\nstatistical learning and logical reasoning, and unifying them with regular SSL.\n