Epistemology has given extensive attention to belief and very little to inquiry. The traditional analysis asks what it takes for an agent to know that p, treating knowledge as a relation to a declarative content. Yet the activity that actually generates knowledge is interrogative. A scientist designing an experiment, a physician ordering a test, and a learning algorithm selecting its next query are all doing something prior to believing: they are deciding what to ask. If rationality governs how we update on answers, it should also govern which questions we pose.

The difficulty is that questions seem to have the wrong logical type. A declarative sentence is true or false; a question is neither. Classical logic is built around truth preservation, so it offers no obvious account of what it means for one question to follow from another, or for a reply to count as an answer at all. For much of the twentieth century this was treated as a reason to leave interrogatives to pragmatics.

Formal semantics changed that. The work began with Hamblin and was refined by Groenendijk and Stokhof, and it gave questions a precise denotation: a partition of possible worlds. Once questions are partitions, they inherit a lattice structure and an entailment relation. Combined with Ramsey's subjective probabilities, they also acquire a measurable epistemic value. What follows develops that framework in three steps: the semantics of questions, the logic connecting them, and the decision-theoretic account of what makes a question worth asking.

Begin with a set W of possible worlds, each a maximally specific way things might be. A declarative proposition is a subset of W: the worlds where it is true. Groenendijk and Stokhof proposed that a question denotes an equivalence relation on W. Two worlds stand in the relation when they agree on the answer. For the polar question “Is it raining?”, two worlds are equivalent if it rains in both or in neither. For “Who came?”, they are equivalent if exactly the same individuals came.

Every equivalence relation induces a partition. A partition is a family of nonempty, pairwise disjoint subsets, called cells, whose union is the space of worlds under consideration. Each cell is a proposition, and each such proposition is a complete answer. Suppose the domain is {Ann, Bob}. Then “Who came?” divides W into four cells: both came, only Ann, only Bob, and neither. “Did Ann come?” divides it into two. A question, on this view, is a structured division of logical space.

Two properties make this definition attractive. The first is exhaustivity. A complete answer to “Who came?” specifies who came and, implicitly, who did not. The partition analysis predicts this, which is why replying “Ann came” when Bob also came feels misleading, not merely incomplete. The second is mutual exclusivity. Exactly one cell contains the actual world, so at each world every question has exactly one true complete answer.

The connection to epistemic logic is immediate. Let R be an agent's accessibility relation, so R(w) is the set of worlds compatible with what the agent knows at w. Let Q(w) be the cell of Q containing w. The agent knows whether, or knows who, relative to Q exactly when R(w) ⊆ Q(w). Knowing-wh thus reduces to knowing a proposition, but which proposition depends on the world. This explains why “knows who came” is not equivalent to knowing any single fixed sentence.

Information states fit the same picture. An agent's state is a set s ⊆ W, and a question restricted to s is the partition it induces on s. The question is resolved in s when s lies wholly inside one cell. It is open when s overlaps two or more cells. Inquiry, formally, is the process of shrinking s until every question on the agenda is resolved. Each answer received intersects the state with a cell and discards the worlds that answer excludes.

Takeaway

A question is not an absence of information but a structure imposed on it: it specifies in advance which distinctions between possibilities matter and which can be safely ignored.

With questions as partitions, entailment can be defined without appeal to truth. Q entails Q′ if and only if every complete answer to Q entails some complete answer to Q′. Formally, for each cell c of Q there is a cell c′ of Q′ with c ⊆ c′. Equivalently, Q is a refinement of Q′. “Who came?” entails “Did Ann come?” because each of its four cells sits inside one of the latter's two cells. Whoever answers the first has thereby answered the second.

Refinement is a partial order, and partitions ordered this way form a lattice. The meet of two questions, their coarsest common refinement, corresponds to conjunction. Asking “Did Ann come, and did Bob come?” yields the cells obtained by intersecting the two partitions' cells pairwise. The join, their finest common coarsening, captures what both questions settle in common. One extreme is the trivial question with a single cell, which is always already answered. The other is the identity partition, whose answer would settle everything.

Presupposition receives a parallel treatment once partial equivalence relations are allowed. If a question's cells jointly cover only a subset D of W, then D is its presupposition: the question is defined only where D holds. On its singular reading, “Which student failed?” is defined only in worlds where exactly one student failed. Every answer, complete or partial, carries this proposition along. Asking the question in a state s is felicitous only if s ⊆ D.

Answerhood becomes graded. A proposition p is a partial answer to Q when it is incompatible with at least one cell and compatible with at least one other. It eliminates some complete answers without selecting a single one. “Bob did not come” is a partial answer to “Who came?”: it rules out two cells and leaves two open. Analogously, one question partly answers another when each of its complete answers eliminates at least one complete answer of the other.

The payoff is a logic of inquiry in which the structure of subquestions is explicit. A strategy for answering a large question can be represented as a sequence of smaller questions whose meet refines it. Decision trees in machine learning exploit exactly this structure. So do question-under-discussion models of discourse. In both, a complex query is decomposed into entailed subquestions, each tractable on its own, whose combined answers guarantee a resolution of the whole.

Takeaway

To ask one question is often to have already asked many others; interrogative entailment tells you which subproblems a complete answer will solve for free.

Partitions tell us what a question asks; probability tells us how much asking is worth. Following Ramsey, let P be an agent's credence function over W. Each cell c of Q then receives a probability P(c), and the question inherits a Shannon entropy, H(Q) = −Σ P(c) log P(c). Entropy measures the agent's expected surprise on learning the answer. Equivalently, it is the expected number of bits the answer conveys.

Entropy is monotone under refinement: if Q entails Q′, then H(Q) ≥ H(Q′). Finer questions carry more information in expectation. But raw informativeness is a poor measure of quality, because most bits are irrelevant to any given purpose. What matters is the reduction in uncertainty about some target. Take a hypothesis partition T. The value of asking Q is the mutual information I(T; Q) = H(T) − Σ P(c) H(T | c). This is the entropy of T now, minus its expected entropy after Q is answered.

This makes precise a familiar intuition about good diagnostic questions. Mutual information is bounded by H(Q), so a question cannot tell you more than its own answer contains. It is maximised when Q's cells divide the hypothesis space into roughly equiprobable pieces that discriminate between hypotheses. A question whose answer you are already nearly certain of is nearly worthless, however profound its subject matter.

Van Rooy extended the framework from information to decision. Suppose an agent faces actions with utility U(a, w). The value of Q is the expected utility of choosing optimally after learning its answer, minus the expected utility of choosing now. Formally, V(Q) = Σ P(c) · max over a of EU(a | c), minus max over a of EU(a). By Good's theorem, V(Q) ≥ 0 whenever the answer is free and updating proceeds by conditionalisation. Even so, a question with high entropy can have zero decision value if no answer would change what one does.

The two orderings connect through a result in the spirit of Blackwell. Q refines Q′ if and only if Q is at least as valuable as Q′ for every decision problem and every prior. Logical entailment between questions is thus exactly the ordering of uniform epistemic dominance. The logic developed in the previous section and the value theory developed here are one structure, viewed from two sides.

Takeaway

The worth of a question lies not in how much its answer says but in how much it could change what you believe or do; a perfectly precise question can be epistemically idle.

The traditional picture treated inquiry as a prelude to epistemology proper: a psychological activity that comes before the real business of justified belief. The partition framework shows that interrogatives have a logic as rigorous as that of declaratives. Questions have denotations, stand in entailment relations, carry presuppositions, and admit graded answers.

Adding subjective probability completes the picture. Entropy, mutual information and expected decision value give quantitative answers to the normative question of what to ask. The Blackwell correspondence shows that these measures respect the logical structure rather than floating free of it. Refinement is both what a question entails and what it guarantees.

For epistemology, the lesson is that rationality has an interrogative dimension. An agent who conditionalises flawlessly but asks poorly will learn slowly and act badly. Formal methods show that choosing questions is not a matter of curiosity alone. It is an optimisation problem over partitions of logical space, with a well-defined solution relative to one's credences and aims.