2.1. Inquiry as an interrogative game

Hintikka's Interrogative Model of Inquiry (imi) models inquiry in analogy with a two-player game Inquirer plays against Nature. Inquirer aims at establishing whether a given hypothesis follows from her assumptions (premises), using analytic (deductive) and interrogative moves. Information obtained through answers supplements premises, and a question can be asked when its presupposition has been established analytically (presuppositions are discussed § 2.2). Whether the game-theoretic analogy can be strengthened is however unclear. Inquirer is an algorithmic player, while classical players of game theory are not; there are no explicit constraints on her preferences, while Nature has a ‘flat’ preference ordering, which makes equilibria hard to define.Footnote ²

Game theory is however relevant to cases pitting agents with well-defined preferences, goals, etc., against each other. Since sources may sometimes prefer not to answer truthfully, Inquirer may have to rely on ‘manipulative’ strategies to acquire trustworthy information. Games with knowledge manipulator are studied in Parikh et al. (Reference Parikh, Taşdcmir and Witzel2011), but assume classical, aka partitional knowledge structures (see § 3.2). Source players may lack such knowledge, when Inquirer can affect the representation of the game being played. Nonetheless, both cases may eventually be subsumed under a general theory of games with knowledge manipulator.

In what follows, we will focus on special cases, where the following condition holds:

Cases where Cond. 1 holds reveal the constraints that should be imposed on interactive models of inquiry games, in particular w.r.t. uncertainty. The condition remains general, having several contextual parameters: ‘follow from’ may be interpreted as deductive, statistical, etc., consequence; ‘reliable’ requires a standard of reliability (which can be more or less stringent); and ‘negatively affected’ corresponds to both the amount of deviation tolerated, and the link between the question and the deviation. Notice that it holds under Hintikka's interpretation: when reliability of Nature's answers depends on experimental constraints, reliability can be modeled as a probability for the answer to be true, given the way it is obtained.

In the examples we will examine, sources are strategic agents (with ‘non-flat’ preferences for outcomes). Under the simplifying assumption that sources are not mistaken (or delusional), reliability can be equated with truthfulness. In this case, a source is affected negatively by some action iff that action triggers a (conditional) preference for lying (or abstaining to answer) over being sincere. This preference is conditional on certain information being inferred following the utterance of an interrogative.

2.2. Pragmatic inferences

The distinction between the surface (linguistic) form of an interrogative, and the question it expresses is common since Hamblin (Reference Hamblin1958). A question is characterized (up to logical equivalence) by a set of mutually exclusive answers, which we denote Q (with subscripts if needed). The presupposition of a whether-question (the only kind needed in this paper) is the disjunction of its answers.Footnote ³

Let us say that X utters an interrogative expressing a question Q as a request for information (rfi) if: (a) X accepts the presupposition (or presuppositions, if multiple) of Q; (b) X does not know which answer (element of Q) holds; and: (c) X wants to know which answer (in Q) holds. Other possible uses of interrogatives are rhetorical or control questions, where (b) and (c) fail. Notice that control questions can be used to obtain information about a source's knowledge, but then the information sought for is not the one explicitly ‘requested.’

Let us also say that circumstances are normal for some type of interrogative, when this type dominates, i.e. when most occurrences of interrogatives belong to it. A certain vagueness is unavoidable: medical diagnosis and criminal inquiry are both normal circumstances for rfi, and courtroom cross-examination, for rhetorical questions; yet once an experienced MD or criminal investigator has formed a hypothesis, her interrogatives express control questions, without change of context. Likewise, an expert attorney or defense lawyer may use ‘surprise’ questions to reveal some new piece of information.

Normal circumstances for rfi (or rfi-normal) warrant inferences, from what is assumed in context, to conclusions about the utterer's epistemic state, the state of Nature, or both. In particular, given the definition of rfi, inferences to the following attitudes are warranted in all rfi-normal contexts:

Inferences to (I1)–(I3) are pragmatic, i.e. context-dependent. As conclusions, they follow deductively from assumptions about the context, rather than from the semantic content of the message. And they are defeasible, if contextual assumptions are revised, and deduction blocked – e.g. when it becomes clear that a criminal investigator or an MD has made her mind, and is now testing her hypothesis with control questions. Alternatively, (I1)–(I3), rephrased as first-person statements from X, may be viewed as conventional implicatures: other implicatures are possible (if the interrogative expresses a rhetorical or control question) and normality is then equivalent to a ‘default’ convention.

(I1)–(I3) are context-type-dependent, conditional on the context being perceived as rfi-normal; they are context-token-independent, justified to the same extend in all rfi-normal contexts. By contrast, further context-token-dependent conclusions can be based on specific knowledge about X, such as:

Assume now that, for some ϕ, ϕ is among the potential answers of a question of interest to X. Then either X has obtained that answer (and knows that ϕ ), or has not (and does not know that ϕ ). Since X can safely be assumed to know whether she has received or not such an answer, she immediately knows that she knows, or that she does not know, the answer, and her knowledge (or lack thereof) is introspective – at least with respect to the topic of the question.

Formally, introspective knowledge is captured by partitional information structures. Two possible states of Nature belong to the same cell of the partition iff the agent cannot tell them apart. The resulting equivalence relation (‘belonging to the same cell as’) is therefore one of epistemic indiscernibility between possible states of Nature. Formal models are either semantic or probabilistic, but technical details are irrelevant to our exposition (interested readers should refer to van Ditmarsch et al. (Reference van Ditmarsch, van der Hoek and Kooi2007) and Osborne and Rubinstein (1994: ch. 5) for, resp., semantic and probabilistic models). The partition can be refined by incoming information, i.e. make some possible state of Nature discernible (we will return to that later). In what follows, we use introspective and partitional synonymously. Partitional knowledge is assumed for (ideally) rational players in game theory, who are eo ipso introspective (see e.g. Osborne and Rubinstein 1994: ch. 5; and Binmore and Shin Reference Binmore, Shin, Bicchieri and Chiara1992; for a critical assessment).

(D1)–(D3) may facilitate some hypotheses, or abductive inferences, which exceed the available information. Often identified with abduction, inference to the best explanation (ibe) occurs when one hypothesis, out of a known partition, is inferred from evidence that fall short of warranting (together with other background knowledge) its deduction. Being a variety of ‘jumping to conclusion,’ ibe is highly relevant in classical games where partitional knowledge is assumed. We take abduction to be distinct, and to comprise the very specification of the relevant partition. Publicly asking a question can facilitate both types of inferences.

Given (I1)–(I3), and assuming introspective knowledge (w.r.t. the topic of the question), further conclusions may be inferred:

Inference to (I4) requires hearers (in a given context) to trust X about the topic of her question. Furthermore, the difference between knowledge and belief can then be neglected, if ‘X knows that ϕ’ expresses unanimous judgment of hearers in the context (although an omniscient outside observer might disagree).

Only (I1), (I2), (D1) and (D1) can be formalized in (standard) epistemic logic, although operators capturing an agent's pursuing certain questions can be added for (I3) (see e.g. Enqvist Reference Enqvist2009). Inferences to (I4), (I5) and (I6) depend only on (I1) and (I2), and thus can be expressed in epistemic logic. (I3) and (D3) require a more comprehensive approach, as in game theory, where they are factored in probabilities assigned by a player X to some actions of a player Y, conditional on Y's uttering a given interrogative.

2.3. Context-dependency of pragmatic inferences

Let us say that some information is carried by a question (in a rfi-normal context) whenever (deductive) inference to it is warranted by either (I1)–(I6) or (D1)–(D3); and that it is carried in general in rfi-normal contexts, if inference to it is warranted in all rfi-normal contexts (when assuming the utterer to be reliable and introspective). Let us also say that some information is ‘trivial’ if the proposition that expresses this information holds in all the states of Nature – e.g. all the conditionals of the form (D1)–(D3) where the consequent is a tautology.Footnote ⁴

(I4) follows from (I1), together with the assumption of utterer's reliability, which is itself a special case of (D1). Inferences to factual information require premises of the type (D1)–(D3), where the consequent is factual, none of which is available in general, in the above sense. Assuming utterer's reliability, one obtains:

Some epistemic information can be recovered in rfi-normal contexts, in addition to (I1)–(I3), but most of it is trivial in an extended sense, i.e. holds about every agent with the appropriate type of knowledge. Moreover, besides (I3), it seems unlikely that any other strategic information about X can be inferred in general, as expressed in the following remark:

Information expressed by (D3), can be translated as the conditional probability that X will undertake certain actions, given that X has asked a question. Rem. 3 does not prevent one from inferring that there is some action that will be taken when the answer is obtained, but otherwise bars any more specific conclusion. Notice also that the action can be, simply, to accept the answer when received (equivalently: to infer from some testimony that some answer holds, that it holds).

The special case of yes-or-no questions is of interest because: (a) their presuppositions are instances of the excluded middle, so they can always be used in an interrogative game; and: (b) the inferential role played by arbitrary questions can always be played by yes-or-no questions (Yes-No Theorem, Hintikka et al. Reference Hintikka, Halonen and Mutanen1999).Footnote ⁵ Point (a) has the following consequence:

Rem. 4 translates formally as follows: w.r.t. partitional knowledge, no tautology can induce a refinement of the partition of indiscernible states of Nature. As remarked at the end of the last section, this does not preclude refinement through abduction or ibe. Knowledge of the form (D1) is semantically and probabilistically irrelevant—every tautology is true in every state-description, and has unit probability. However becoming aware of some instance of the excluded middle may have an impact on algorithmic players, who actually have to process information.

3.1. Strategic questioning and answering

Finding questioning strategies in circumstances where Cond. 1 holds is a routine task for pollsters and social science investigators. By Rem. 4, yes-or-no questions are particularly relevant for studying strategic information in these contexts, as shown in the following example:

Contextual parameters of Cond. 1 are of special importance in Ex. 5: if X relies on statistical correlations, some ϕ _i may follow from Y's answers (together with background information) with high probability, rather than deductively. How Y's reliability is assessed w.r.t. sincerity and accuracy, depends upon which indicators X relies on, but control questions play a critical role (see below). How Y is affected by a question depends upon Y's evaluation of the impact of his answer on X's judgment about his qualification.

Y may be tempted to answer strategically – anticipate X's possible conclusions, and choose the answer with expected positive impact whenever possible. Y also has to maintain coherence between his answers, a task that strategic answering makes cognitively more complex.Footnote ⁶Y has thus some cognitive incentive to answer sincerely (however he values sincerity), which is even stronger when the set of answers is large enough.

This suggests that X's best strategy to minimize the risk of strategic answering is to use knowledge Y is unlikely to possess, in order to relate Y's answers to ϕ ₁, … , ϕ _n. Questions with arcane connections (less likely to be known by Y) to relevant qualification, can also play the role of control questions. Making hypotheses, rather than relying on knowledge, to anticipate X's use of answers, may be judged by Y cognitively too costly, also causing delay in Y's answers that can ‘give away’ strategic answers. Hence, relying on information unlikely to be known to Y maximizes the likelihood of sincere answers.Footnote ⁷

X's best strategy is therefore to rely on little known correlations between each ϕ _i (or its negation) and potential answers, so that Y will be unlikely to evaluate their impact. Given that difficulty and the additional cognitive stress incurred by maintaining consistent lies, X expects Y to adopt a strategy where he answers sincerely under uncertainty. If Y is furthermore able to anticipate that X will use control questions, then his best option is to answer sincerely even when his answers may have a negative impact: this impact may be offset by positive impact of other (sincere) answers, while the negative impact of being perceived as a liar cannot.

3.2. Game models with imperfect information

Ex. 5 can be modeled as an extensive game with imperfect information, where players play in turn, and some moves are undisclosed, so that some players remain uncertain as to which course the game has actually followed. Players of such games consider, given their current information, several distinct histories (possible courses of the game) as indiscernible; and actions available to them may have different consequences in distinct indiscernible histories. The indiscernibility relation for player X induces a partition of histories, in jargon, the information set of X, of which players in classical game theory are assumed to be fully aware, i.e. to have introspective knowledge.

Players' information can be modeled by (subjective) probability distributions: indiscernible histories at one stage for some player X have the same marginal probability at that stage, although they can have distinct probabilities conditional on some event (including moves from some player Y). One player's moves carry some (nontrivial) information, w.r.t. the information set of another player, insofar as it modifies (through conditionalization) the relative probabilities of histories indiscernible before that stage. This information is strategic, because it alters expectations about future histories and other players' moves.

A complete strategy for a player is a function defined for each position where this player can move, and returning as value an action for that player. Planning one requires a player to know in advance all the possible courses the game may follow, which yields knowledge of other players' possible strategies. Strategies may recommend different actions, at some positions, for histories indiscernible at that position; strategies that do not are called uniform. In the absence of uniform strategy, players can choose randomly, using probabilities of histories as weights.Footnote ⁸

In Ex. 5, whether Y answers sincerely or not is undisclosed to X, as are undisclosed to Y conclusions X draws from his answers. Whether Y can anticipate X's conclusions depends on sharing X's background information. Yet, Y has introspective knowledge (or ignorance) of the effect of his answers: if he knows (does not know) whether the answer will be favorable, neutral or detrimental, he knows that he does (does not).

This knowledge structure obtains even if X and Y fall short of being idealized players of classical game theory. One cannot realistically assume Y to be able to anticipate all the questions X may address him, nor to assume that X has a complete strategy anticipating all combinations of answers from Y (although using multiple-choice questions permits it in principle). Even so, X and Y still have introspective knowledge (or ignorance) of the relevant aspects of the game: for Y, of the possible effect of each answers; for X, of Y's sincerity; and for both, of the possible outcomes of the interview, depending on X's assessment.

Subsequently, even an algorithmic approach to Ex. 5 would analyze it as a case of reasoning under uncertainty, with either well-defined or ambiguous probability distributions over a known partition of possible states of Nature (possibly including information about other players' strategies), of which players have introspective knowledge or ignorance. Hence, some objections against partitional knowledge grounded in algorithmic complexity are not as relevant here as elsewhere.Footnote ⁹

3.3. Sherlock Holmes and the dress

Consider the following example, from the short story Silver Blaze (Conan Doyle Reference Conan Doyle1960: 335–50).

Holmes suspects that Straker died accidentally from a blow delivered by Silver Blaze, while he was attempting to wound the horse, impairing his performance in a forthcoming race where Silver Blaze is the favorite. Straker would have used the inside knowledge to bet with a guarantee of high returns. The following hypothesis:

would thus explain Straker's scheme, providing a motive to foster a source of additional income. We will not try to explain how Holmes comes to consider (H), for explaining hypothesis formation is hazardous, especially in a fictional case. Yet, once the hypothesis is considered for critical examination, Holmes's reasoning can be rendered as purely deductive, from the following premises:

(P₁)–(P₃) are consequences of some facts that Holmes takes for granted at the beginning of his inquiry, and of inferences based on them.Footnote ¹⁰ They are not sufficient to conclude to (H), but they would be if (as Holmes clearly expects) it turns out that Straker's widow does not own the dove-colored silk dress (proof left to the reader). Holmes eventually obtains the information he needs, immediately upon being introduced to Mrs Straker, as shown in the following dialogue:

We propose to construct Ex. 6 as a case where Cond. 1 holds, with contextual parameters set as follows: conclusion (H) has to follow deductively from Holmes' background knowledge (P₁)–(P₃) together with answers from the widow; the widow is considered a reliable source w.r.t. to her ownership of the dress; and the mere possibility that she could foresee Holmes's use of her answer to establish (H), suffices to consider an answer as unreliable.

This latter provision applies in particular to the question expressed by the following interrogative:

We take (Q₁) to be a borderline case of control question, i.e. that Holmes, before having to choose between it or some other, has already made his mind about (H), and expects a negative answer to it. This would suffice to obtain (H) deductively from (P₁)–(P₃), while a positive answer would leave the question undecided (with appraisal based on the same premises). But the framing of (Q₁) may alter the widow's willingness to answer truthfully, if the answer is indeed negative.

Indeed, if Straker's widow does not own the dress, but has not given any thought about it, she could find (Q₁) surprising, unless Holmes has evidence that her late husband had bought such a dress, and then guess (P₁). Neither (P₂) nor (P₃) need to be guessed, if they belong to (or follow from) the widow's background knowledge, and become relevant only in presence of (P₁). If Holmes assumes that they do, and that the widow furthermore prefers that her late husband's indiscretions remain unknown, he will anticipate from her a preference to lie if addressed with Q₁, if the answer is negative.

Pending a way to tell truth from lie, this would make a positive answer valueless, for Holmes would not know whether to treat it as a lie, or to revise his beliefs in accordance to it. Holmes's strategy, as displayed in the above dialog, shares a resemblance with the one recommended for X in Ex. 5. Yet Straker's widow's problem is not the same as Y's: Y's uncertainty pertains to which alternative obtains, within the range known to him. By contrast, Holmes's success can be explained as rational, if the widow lacks partitional knowledge of Holmes's possible conclusions, as established in the next section.

4.1. A reconstruction of Holmes's reasoning

We reconstruct Holmes reasoning, in Ex. 6, as applying a generalized version of the following principle, introduced by J. Harsanyi:

Because a player's ‘worst case’ strategy in a competitive game is called a maximin strategy, this principle is sometimes called Harsanyi maximin. Harsanyi proved that this principle suffices to solve competitive games (where players' preferences are opposed), under weaker conditions than usually assumed in classical games.Footnote ¹¹

Holmes cannot rationally expect that the situation will not turn into a competitive game if the widow understands his goals – he cannot rule out that the widow's preferences are in that respect strictly opposed to his. Hence, he can be viewed as applying a (suitably) ‘generalized Harsanyi Maximin’, or gh-maximin for short, that applies also to expectations about preferences.

Holmes could not however select this strategy in a game with classical (partitional) knowledge structures, where Holmes's and the widow's uncertainty would be modeled, resp., by partitions of her possible preferences, and of his possible conclusions.Footnote ¹² Consider, for sake of simplicity, Holmes's question as the following interrogative:

Holmes's strategy is to hope that the widow will take (Q₂) as a ‘control’ question (i.e. as implying that he believes they did), and that he wants to know her answer to decide whether his (feigned) belief that:

must be revised. Yet, if Holmes assumes Straker's widow to have a fine-grained enough partitional knowledge of his possible goals, he must also expect her to know that:

entails a negative answer to (Q₂), but also to (Q₁) (together with (P₁)–(P₃)). Hence, she can realize that from (R₁), one can derive both (H) and (H*). Reasoning under gh-maximin, he should act as if anticipating her to be aware of that possible conclusion; to have a (strict) preference for (H) not to be known; and to subsequently choose a uniform strategy facing uncertainty between (H) and (H*), namely to lie, preventing Holmes from deducing (H) from his background knowledge and her answers.

In other words, in a game with ‘classical’ uncertainty, Holmes should act as if expecting that the widow could guess that (P₁) holds whether he utters (Q₁) or (Q₂). He could still prefer (Q₂) over (Q₁), if he were expecting the widow more likely to react irrationally to the former than to the latter. Symmetrically, under conditions of ‘classical’ uncertainty, the widow's reply (R₁) must be irrational.

4.2. Inferences to goals

Failing to guess (P₁) upon hearing (Q₁), is failing to find an explanation for its utterance. This, in turn, is an abductive inference, to some conclusion of the form (D3). Abduction relates more to guesswork than to computation. Failure at guesswork amounts to cluelessness (when guessing is easy), or lack of insight (when it is hard), but not to irrationality. Players of classical games are assumed to have complete (partitional) knowledge of the possible courses of the game. Finding the ‘best’ explanation for another player's move is identifying the strategy it (most likely) belongs to. If there are several such strategies, it amounts to ibe rather than abduction (see § 2.3, p. 4). Compared to abduction, ibe is closer to computation: failing to find the best explanation (given one's background enumeration) may qualify as irrational, in the sense of failing to implement some normative acceptance rule.Footnote ¹³ Classical games collapse abductive reasoning on ibe, and cluelessness on irrationality.

Holmes should consider (R₁) irrational, if his reasoning (under gh-maximin) assumed the wife to have a partitional knowledge of his possible goals—it is contrary to the widow's ‘worst case’ preferences. Expecting (R₁) would be betting on a missed ibe step, preventing her from implementing her strategy when the best explanation is that Holmes is aiming at (H). But if Straker's widow has to guess first which are Holmes's possible uses for her potential replies, gh-maximin applies also to expectations about guesses. The widow could, upon hearing (Q₂), guess he is aiming at (H) instead of (H*) – i.e. guess (P₁), become aware of (P₂) and (P₃), and deduce (H) from it. Clearly, gh-maximin would recommend that Holmes make his goal harder to guess; and it is indeed a critical feature of his strategy.

X's goal in a given context, can be conjectured by some witness Y, on the basis of X's observed behavior. In the special case of a situation that can be modeled as a game, Y has to guess what X's strategy is, but X's moves depend on Y's, so that some aspects of X's strategy can never be observed. Without partitional knowledge of X's goals, and suitably defined probabilities over them, ascribing a goal to X is the conclusion of an ampliative inference. Classical game theory solves the problem by assuming Y to have the same representation of the situation as an omniscient modeler, so that Y can either deduce which X's actual goal is, or ‘bet’ on the most likely one.

Holmes is aware that the widow can understand his question about the dress, as either relevant to his inquiry, or merely prompted by the circumstance of their meeting. Reasoning under gh-maximin rules out (Q₁), but his actual question is much safer. Although a bet on the widow's guessing powers, it is a rather safe bet. He seems to ask for a mere confirmation (not even mentioning the dress at first), and press her to change a reply that calls his memories into question. He can expect her to answer sincerely, as long as the question seems to be disconnected from his current inquiry – more or less courteously, depending on her dispositions towards him. If Holmes can bet that she will think his question unrelated to his inquiry, he can even disregard the possible goals she might assign him. It will only affect the manner of her response, not its content, and only the latter matters to him.

The widow's expectations and dispositions depend also on what she knows about Holmes'; on her character; or on whether she has other reasons to suspect that the dress might be connected to Holmes's inquiry. She does not know – unlike Watson, or us readers – that he never engages in pleasantries. Yet she might guess it, as she might be a pathological liar, or suspect her husband to have been unfaithful, etc. Disregarding each of these ‘worst-case’ scenarios is a bet, not however contrary to gh-maximin, if Holmes estimates the odds against him low enough.

Unlike the reader – for whom Holmes's goals are eventually made explicit, and who can reconstruct the parts of his reasoning that are not (as we just did) – Straker's widow lacks a fine-grained enough representation of Holmes's possible strategies, to choose her reply according to her best interest. Translated in game terms, Straker's widow does not know which game she is playing. More importantly, this knowledge is not introspective: she does not know that she does not know.

4.3. Unawareness and uncertainty

Holmes, in Ex. 6, is a knowledge manipulator, who manipulates the widow's knowledge about the game being played. She identifies an action in an inquiry game, as belonging to some other type of language game, as a result from a truncated perspective forced by Holmes's actions. She is unaware of the game being played, and maintaining this unawareness is critical for Holmes's strategy to be successful. As previously argued, this unawareness amounts to a form of uncertainty, distinct from the uncertainty that can accompany partitional knowledge about possible histories in a classical game (which implies full awareness of the game being played).

The widow's unawareness about the game is related to, although distinct from, her unawareness of (P₂) and (P₃). Both result from a failure of partitional knowledge. The question whether (P₂) holds, or whether (P₃) holds, are not relevant to the widow's current interests: these questions lie outside her current ‘range of attention.’ What is known can be semantically defined as what holds in all epistemic alternatives not eliminated by current evidence. In this sense the widow knows (P₂) or (P₃), but is unaware of it, hence has non-introspective knowledge.

In the same sense, she knows, also non-introspectively, that Holmes's question could be related to his current inquiry: if the question whether the dress can be relevant to understand the circumstances of Straker's death were raised, she would not need further evidence to answer that it could be – and this could turn her attention to (P₂) and (P₃).

That Holmes is aiming at establishing (H), or controlling (H*), are among the countless possible goals compatible (more or less remotely) with Holmes's behavior. Having a complete representation of the game being played is actually tantamount to have eliminated those countless possibilities, and kept the relevant one. Inference, from observed behavior to strategies (and goals) is ampliative (cf. supra). Similarly, inference from the possible games being played to the game actually played is also ampliative reasoning, as soon as a wide enough range of possible games is considered.

A formal treatment of games with unaware players has been proposed in Halpern and Chaves Rego (Reference Halpern and Chaves Rego2006). In addition to the (classical) representation of the ‘underlying game’ – viewed by an omniscient modeler – it associates to each player an awareness function, mapping positions of the underlying game to the representation the player has of it (at that position). Modeling Ex. 6 as a game with unawareness allows us to represent the effect of (Q₁) on the widow, as a change in the widow's awareness, in the representation Holmes has of the game.Footnote ¹⁴ This effect can encapsulate inferences to (P₁), (P₂) and (P₃), the first being ampliative, and the second and third deductive. An interrogative (semantically) equivalent to (Q₂) is not expected by Holmes to have the same effect on the widow's awareness.

Ex. 5 exemplifies a case where Y is aware of his unawareness (he knows there is something to know about the game that eludes him), and where X chooses his questions so as to affect Y's awareness as little as possible. In Ex. 6, the widow is unaware of her unawareness. Both X and Holmes are knowledge manipulators and awareness manipulators, affecting the outcome of the game by manipulating its representation.