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The treatment of uncertainty in climate-change science is dominated by the far-reaching influence of the 'frequentist' tradition in statistics, which interprets uncertainty in terms of sampling statistics and emphasizes p-values and statistical significance. This is the normative standard in the journals where most climate-change science is published. Yet a sampling distribution is not always meaningful (there is only one planet Earth). Moreover, scientific statements about climate change are hypotheses, and the frequentist tradition has no way of expressing the uncertainty of a hypothesis. As a result, in climate-change science, there is generally a disconnect between physical reasoning and statistical practice. This paper explores how the frequentist statistical methods used in climate-change science can be embedded within the more general framework of probability theory, which is based on very simple logical principles. In this way, the physical reasoning represented in scientific hypotheses, which underpins climate-change science, can be brought into statistical practice in a transparent and logically rigorous way. The principles are illustrated through three examples of controversial scientific topics: the alleged global warming hiatus, Arctic-midlatitude linkages, and extreme event attribution. These examples show how the principles can be applied, in order to develop better scientific practice. "La théorie des probabilités n'est que le bon sens reduit au calcul." (Pierre-Simon Laplace, Essai Philosophiques sur les Probabilités, 1819). "It is sometimes considered a paradox that the answer depends not only on the observations but on the question; it should be a platitude." (Harold Jeffreys, Theory of Probability, 1st edition, 1939). [ABSTRACT FROM AUTHOR]