Is–ought problem
Overview

Hume discusses the problem in book III, part I, section I of his book, A Treatise of Human Nature (1739):

In every system of morality, which I have hitherto met with, I have always remarked, that the author proceeds for some time in the ordinary way of reasoning, and establishes the being of a God, or makes observations concerning human affairs; when of a sudden I am surprised to find, that instead of the usual copulations of propositions, is, and is not, I meet with no proposition that is not connected with an ought, or an ought not. This change is imperceptible; but is, however, of the last consequence. For as this ought, or ought not, expresses some new relation or affirmation, it's necessary that it should be observed and explained; and at the same time that a reason should be given, for what seems altogether inconceivable, how this new relation can be a deduction from others, which are entirely different from it. But as authors do not commonly use this precaution, I shall presume to recommend it to the readers; and am persuaded, that this small attention would subvert all the vulgar systems of morality, and let us see, that the distinction of vice and virtue is not founded merely on the relations of objects, nor is perceived by reason.

Hume calls for caution against such inferences in the absence of any explanation of how the ought-statements follow from the is-statements. But how exactly can an "ought" be derived from an "is"? The question, prompted by Hume's small paragraph, has become one of the central questions of ethical theory, and Hume is usually assigned the position that such a derivation is impossible.

In modern times, "Hume's law" often denotes the informal thesis that, if a reasoner only has access to non-moral factual premises, the reasoner cannot logically infer the truth of moral statements; or, more broadly, that one cannot infer evaluative statements (including aesthetic statements) from non-evaluative statements. An alternative definition of Hume's law is that "If P implies Q, and Q is moral, then P is moral". This interpretation -driven definition avoids a loophole with the principle of explosion. Other versions state that the is–ought gap can technically be formally bridged without a moral premise, but only in ways that are formally "vacuous" or "irrelevant", and that provide no "guidance". For example, one can infer from "The Sun is yellow" that "Either the Sun is yellow, or it is wrong to murder". But this provides no relevant moral guidance; absent a contradiction, one cannot deductively infer that "it is wrong to murder" solely from non-moral premises alone, adherents argue.
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