Abstract
Erik J. Olsson and David Westlund have recently argued that the standard belief revision representation of an epistemic state is defective.1 In order to adequately model an epistemic state one needs, in addition to a belief set (or corpus, or theory, i.e. a set closed under deduction) \(\underline{\textrm K}\) and (say) an entrenchment relation E, a research agenda \(\underline{\textrm A}\), i.e. a set of questions satisfying certain corpus-relative preconditions (hence called \(\underline{\textrm K}\)-questions) the agent would like to have answers to. Informally, the preconditions guarantee that the set of potential answers represent a partition of possible expansions of \(\underline{\textrm K}\), hence are equivalent to well-behaved sets of alternative hypotheses.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
The original theory is presented in Olsson and Westlund (2006).
- 2.
There are many examples of logical analysis of questions, which will not be reviewed here. Our exposition is grounded in Hintikka’s account (especially Hintikka et al., 1999, and Hintikka, 2003), since his analysis pays systematic attention to both the epistemic and strategic aspects of questioning.
- 3.
It should be noted that Olsson and Westlund’s theory is taken as a first step toward a more detailed account of interrogative contraction, a topic which will not be addressed here. This kind of contraction is intended to model ‘pure’ use of contraction, while our proposal addresses the topic or ‘regular’ contraction as an analytic step of revision. The relevance of our proposal to interrogative contraction is yet to evaluate, and may be a subject for further research.
- 4.
Consider an research agenda to find linear models to explain some phenomenon. Should the agenda include recommendation to proceed through randomized experiments, or using some sampling method? If the former, should one randomize simultaneously or one variable at a time? If the later, how should one sample? (Thanks to an anonymous referee for raising these questions.)
- 5.
See Hintikka (1987b) for an analysis of probabilistic reasoning and statistical tests in the interrogative model framework. Standards of acceptance are discussed in Hintikka (2007a), where their context- and subject-dependency is related to decision-theoretic aspects of inquiry, i.e. the role of conclusions in decision-making.
- 6.
Since exclusive disjunction is not associative, any n-ary ⊻ should be treated as a distinct operator. We will nevertheless use it as is, relying on the readers’ logical acumen (or interpretative charity) to restore proper use and notation if they find our choices inappropriate.
- 7.
Finiteness is to be understood as holding up to logical equivalence between potential answers, since there can be infinitely many ‘equivalent’ ways to ask the ‘same’ question. Question equivalence is detailed p. 20, n. 42.
- 8.
Since a question can be raised only if its presupposition is known (or believed) to hold, interrogative strategies involve sometimes elaborate plans to establish the presupposition of a given question (see also n. 14).
- 9.
See van Fraassen (1980, p. 138); as van Fraassen adds in note, the reply is attributed to Sir Charles Napier (1782–1853), Commander-in-chief of India, after whom the city of Napier, New Zealand, is named.
- 10.
Strictly speaking, obtaining an answer is a mere deductive move if the question was: ‘Do you have Sind?’.
- 11.
Recanati considers that the reply ‘clearly […] provides an affirmative answer’ (see Recanati, 2001).
- 12.
This is the solution adopted by Hintikka, in contrast to others, e.g. Van Fraassen (1980).
- 13.
The latter category is meant to include conclusiveness conditions which are deductive modulo suitable background knowledge, as well as those which require some additional assumption (maybe some form of abductive reasoning) as illustrated in Recanati’s example.
- 14.
See in particular Hintikka (1987a), which includes a simple introduction to interrogative games from the perspective of Socratic questioning. As an example, the well-known Fallacy of Many Questions is a case where a question Q is asked though its presupposition has not been established: I cannot (rightfully) ask whether you have stopped beating your dog if I cannot take for granted a ‘positive’ answer to the question whether you (ever) beat it. This is not much a fallacy than it is an illegitimate ‘move’ in an ‘interrogative game’. This re-interpretation of fallacies dates back to R. Robinson (1971), who addresses (in a very spirited manner) the so-called Fallacy of Begging the Question. Its fruitfulness has been defended by Hintikka (1997) in contrast with other approaches.
- 15.
Asking a yes-or-no–question about some vague term, for example, may need some ‘contextual standard’ to be fixed. This problem is studied in David Lewis’ paper (1979).
- 16.
If Inquirer were allowed to take a guess (that is, if she asks questions which are not answerable), the game would proceed with imperfect information, since Inquirer would not know in which state she is. But she would still have perfect recall, since in a given play of the game, she could re-use any information obtained during that play.
- 17.
- 18.
For an alternate interpretation, giving a more ‘active’ role to the left column player, see n. 19.
- 19.
This interpretation of Beth Tableaux is often put forward by Hintikka, and is closely related to ∃loise vs. ∀belard games of Game-Theoretic Semantics (gts) (Harris, 1994, is an early attempt at a gts for interrogative tableaux). Another interpretation, in the tradition of Lorenzen, Lorenz and Rahman, is to see it as a game where the right-column player, Proponent, tries to defend a thesis against criticism of the left-hand player, Opponent. Both player can attack and defend statements according to certain particle rules (governing the use of logical constants and quantifiers) and structural rules (governing the overall conduct of the game). Proponent has a winning strategy if she can have the last word, whatever Opponent may do, using only information he has previously conceded (and logical moves). For the latter interpretation, see Rahman and Keiff (2005), and for its relation to the former, see Rahman and Tulenheimo (2007), where some kind of interrogative moves are introduced in dialogues.
- 20.
Addition of this rule defines, which is equivalent to the cut rule, defines what Hintikka, Halonen and Mutanen name extended interrogative logic (see Hintikka et al., 1999, p. 53), while the unextended interrogative logic is the cut-free (analytical) version obtained if Inquirer is not allowed to ask yes-or-no–questions.
- 21.
See Rahman and Keiff (2005) for such a rule in the context of proof games (called formal dialogues).
- 22.
For a more complete presentation, see Hintikka et al. (1999), which also includes various correspondence results between ‘interrogative’ reasoning and deductive reasoning, as well as some metatheorems we will use in this paper. In particular, it shows that the problem of finding the ‘best’ interrogative strategy reduces (in the case of ‘pure discovery’) to the problem of finding the best deductive strategy (at least for unextended interrogative logic) which, in view of the semi-decidability of first-order logic, is not solvable computationally.
- 23.
In Hintikka et al. (1999, sec. 8) the problem of reasoning with uncertain answers addressed briefly, but no explicit connection is proposed with non-monotonic logics. A connection is made with probabilistic reasoning, and especially the problem of cognitive fallacies, later developed in Hintikka (2004).
- 24.
For example, Isaac Levi (1991, p. 71) writes that: ‘The task of constructing potential answers to a question is the task of abduction in the sense of Peirce’. Hintikka (1999, p. 104) agrees with Levi to the extend that ‘from a strategic viewpoint, in that the choice of the set of alternative answers amounts to the choice of questions to be asked’, but insists that: ‘in abduction one may prefer one possible conjecture over others’ that is, favor one answer over others. o&w explicitly refer to Peirce (in Isaac Levi’s interpretation), and consider that a set of potential answers being considered as ‘given’ (by abduction) is a ‘methodological decision that has to [their] knowledge never been questioned’ (Olsson and Westlund, 2006, p. 179, n. 3).
- 25.
- 26.
This may be a way to understand what is meant by acceptance ‘as an interrogation’, or at least a sensible interpretation. Let’s for the moment simply say that what we’re aiming at is to formally reconstruct one possible understanding of Peirce’s idea.
- 27.
Assume that one wants to investigate the consequences of adding hypothesis a i to the belief set \(\underline{\textrm K}\), and forms the expansion \(\underline{\textrm K}+a_i\). Let furthermore \((a_i \leftrightarrow (b_1 \land b_2))\in \underline{\textrm K}\). Assume now that one wants to investigate the consequences of hypothesis a j , where a j is a ‘rival’ to a i , and then wants to restore the initial state before expansion by a i . Since epistemic economy recommends that contraction by a i only removes b 1 or b 2, but not both, it follows that either the question whether b 1 or the question whether b 2 will remain settled: then ‘purely hypothetical’ reasoning is not likely to be adequately modelled. Another option is to define a question-relative contraction, and such a solution was outilned in Olsson and Westlund (2006), yet not fully articulated.
- 28.
There are other important motivations, but they require the discussion of interrogative strategies, and we will wait to have introduced and discussed this notion before returning to this question.
- 29.
29Cf. Olsson and Westlund (2006, p. 172).
- 30.
See Olsson and Westlund (2006 p. 172), proof given n. 10.
- 31.
By classical logic, \((a\veebar\neg a)\in\underline{\textrm K}\) whenever \(\underline{\textrm K}\) is consistent. (If is inconsistent, then \(\underline{\textrm K}=\L\), hence \(\{a, \neg a\}\cap\underline{\textrm K}ne\varnothing\) and \(\{a, \neg a\}\) is trivially settled.) If \(a\in \underline{\textrm K}\), then \(\{a, \neg a\}\) is not a \(\underline{\textrm K}\)-question since \(\{a\} \subset \{a, \neg a\}\), but then it is settled. The same holds with \(\neg a\).
- 32.
The theorem is proved in Hintikka et al. (1999, p. 55), and its philosophical consequences are developed in Hintikka (2007c). A formal version for corpora and agendas is given in Genot (2009). It requires the use of the equivalent of the cut rule of proof theory (see n. 20). Hence, for any procedure using yes-or-no–question–such as those we will introduce for question update—the question whether the procedure admits of cut elimination or not will be of considerable interest (and in our case, a topic of further research).
- 33.
33It is easily proved by contraposition: assume that \(\{a, \neg a\}\) is not a \(\underline{\textrm K} \div b\)-question. Then either \(\underline{\textrm K} \div b\) entails a or \(\neg a\). Assume the former: since, by Inclusion postulate for contraction, \(\underline{\textrm K} \div b\subseteq \underline{\textrm K}\), we have (by Closure) \(a\in\underline{\textrm K}\). Hence Q is not a \(\underline{\textrm K}\)-question. A symmetric conclusion follows from the assumption that \(\neg a\in\underline{\textrm K} \div b\).
- 34.
34By definition of a \(\underline{\textrm K}\)-question and of a \(\underline{\textrm K}\)-agenda, and construction of \({|{\scriptstyle ^Y_N }\text{-}Q|}\), if \(Q\in Ag(\underline{\textrm K})\), then \(\|{\scriptstyle^Y_N}\text{-}Q\| \subseteq \text{\upshape Q}_{\underline{\textrm K}}\), which, combined with (5), yields that \(\|{\scriptstyle^Y_N}\text{-}Q\|\subseteq\text{\upshape Q}_{\underline{\textrm K} \div a}\) as desired.
- 35.
Obviously, not every point of the map may be taken an alternative, on pain of having not only an infinite set thereof, but an uncountable one as well. Moreover, in a situations like Alice’s, the set of alternative locations usually will not even be fully specified (at least, not given a full attention) before some step of the search has been reached. The analogy with yes-or-no–questions will be conspicuous to everybody who as shared Alice’s predicament, since a good way to find where one is, is first to divide the map in two, with some imaginary line meeting the point one comes from, or using grid of the map (if there is one) and try to find in which half one is, using some landmark. Once this is done, one can proceed to further divisions, and will eventually identify some area. The set of areas, though specified at the outset (by the grid of the map, or some imaginary equivalent one could ‘picture’), is indeed used in what is equivalent to yes-or-no–questions.
- 36.
Each square delineated by the map grid may be associated with the different visual patterns of landmarks one could see standing somewhere in the corresponding area. Usually some orientation (aligning the map, and oneself, to the North) is used to narrow down the overwhelming number of alternate possibilities that may be associated to each square, depending on the precise point one stands at, the direction one faces, etc. It is then clear that, though one may have some way to define a set of alternatives, in many cases, one will not proceed to actually define it before taking some steps to narrow down the range of alternatives. One will rather proceed, as described above (see n. 35), with a sequence of questions, each excluding at least one more (set of) alternatives (with respect to some method to list or determine them).
- 37.
For the notions of action profile, strategic games, etc., see Osborne and Rubinstein (1994, Chap. 2). In Alice’s case, the possibility to recognize the landmarks on the basis of the pictures on the small map is one of those factors (since two different buildings my have a similar pictorial representation), and ultimately depends on previous choices of the map designers. Relying on the map as a source of answers (rather than relying on some native) requires the map to be treated, abstractly, as a player of the game. Since distances and how they affect recognition (the well-known Cartesian example of a squared tower appearing round from afar is a case in point) has to be taken into account, but does not depend on some foreseeable ‘action’, reliability of the visual system may be modeled through a probability measure.
Some of these factors may or may not affect the ‘phrasing’ of the principal question. In particular, one can ask a question while knowing that one lacks the resources to obtain a complete answer, or even before knowing which other ‘players’ one will face. They may also influence the choice of the first hypothesis to test, and this is one way to understand Peirce’s notion of (epistemic) economy.
- 38.
The following matrix (borrowed from Osborne and Rubinstein 1994, p. 13) represent abstractly a two-player game where each player, Row and Column, has two possible actions: the set \(\{T,B\}\) for Row, and the set \(\{L,R\}\) for Column.
Each pair of value is a pair of utilities, the first for player Row, the second for player Column. Clearly, utilities are associated to combinations of actions of both players. For example, x 1 is the value for Row of the outcome of \((T,R)\) while its value for Column is x 2.
- 39.
This analogy can be pursued formally, and various correspondence result proved (see Genot 2009). However, we will limit our exposition to a mostly informal, and at most semi-formal, overview.
- 40.
Let \(Q=\{a_1,\dots,a_n\}\), and let’s assume that oracle will answer in the most direct and informative manner, that is either by some (strictly) partial answer: \((\neg a_i\land\dots\land\neg a_j)\) (with \(1\leq i,j \leq n\)), or some complete answer \(a_i\in Q\). Then the result of asking Q and receiving an answer will be either \(\underline{\textrm K}+a_i\) or \(\underline{\textrm K}+(\neg a_i\land\dots\land\neg a_j)\). The consequence is clearly that the result of asking every \(\{a_i, \neg a_i\}\in\|{\scriptstyle^Y_N}\text{-}Q\|\) will be the same if Oracle answers to each the same way as it answers to Q, that is iff the union of answers is equivalent either to \((\neg a_i\land\dots\land\neg a_j)\) or to a i , hence iff Oracle is insensitive to the order of questions (since union does not preserve ordering).
- 41.
Other assumptions, besides insensitivity to order and to formulation, may include the absence of a time-limit, the availability of all sources at any time (if Oracle is a ‘team’ player) at various stages of the game, etc. The reader is encouraged to imagine his or her own counterexamples, throwing in various constraints. Real-life inquiries (or crime novels and TV shows) abound of cases in which those assumptions cannot be made.
- 42.
Question equivalence is easily defined as two-way inclusion, with question inclusion defined as follows: a question Q includes a question Q’ iff any expansion which answers the first also answers the second, that is if for all \(a\in Q\), if \(a\in\text{Cn}(\underline{\textrm K}\cup \{d\})\) (for some d), there is a \(b\in Q'\) such that \(b\in\text{Cn}(\underline{\textrm K}\cup \{d\})\). In particular, since the condition for inclusion holds when \(d=a\) for some \(a\in Q\), it is equivalent (by the Deduction property of Cn) to the condition that any potential answer to Q entails some potential answer to Q’. As a consequence, two equivalent questions have (pairwise) equivalent potential answers (proof left to the reader)
- 43.
See Genot (2009) and Enquist and Olsson (2008). For the Normal Form Theorem see e.g. Smullyan (1968, p. 13). The result holds when using the procedure to obtain normal form up to disjunctions having as disjunct conjunctions of potential answers or their negations, without necessarily going up to disjunction of literals. Yet nothing prevents the definition (or the usefulness) of an atomic interrogative normal form. Enqvist and Olsson (2008) proposes a Topic Strategy for updating questions which allows to ‘break down’ initial potential answers and rearranging answer to obtain an new \(\underline{\textrm K}\)-question which falls somewhere in between the State Description Strategy (mentioned earlier n. 45) and the full atomic interrogative normal form.
- 44.
Equivalence in the sense of n. 42 clearly holds, since for any \(a_i\in Q\):
$$a_i\leftrightarrow (\neg a_1\land\dots\land a_i\land\dots\land\neg a_n)\in\text{Cn}(\underline{\textrm K})$$Hence each element of Q is equivalent to exactly one element of Q’.
- 45.
- 46.
Consider a situation where a restaurant has been booked for the evening for some social event, with only one menu, and where a columnist of some celebrity magazine tries to know what the menu is, but cannot access directly the information. Assume that, for some reason, only the chef does know the whole menu, but the journalist can only ask the kitchen commis. Then asking (15) would be pointless, and he would have better results asking (14a) and (14b) to different commis—if the information he has already obtained allows him to restrict his range of questions to those two.
- 47.
If we assume that the order in which the questions are put to sources is irrelevant, then the family is a equivalence class: only the set of answers matters, not their sequence.
- 48.
It is easily checked that this strategy (or any other with another ordering) will indeed deliver an answer to \(Q= \{a_1,\ldots,a_n\}\), if all questions in \({|{\scriptstyle ^Y_N }\text{-}Q|}\) are answerable—hence if Q is, since we assumed our source not to be sensitive to the way questions are put. It is obviously still assumed that the set Q is finite, so the strategy will not guarantee an answer to, say an ‘existential’ question about an infinite domain.
- 49.
The procedure is detailed in Genot (2009).
- 50.
See Genot (2009).
- 51.
The phrase is taken here in the sense of blundered substitution, rather than in its legal sense (the former is common in Latin country, and acknowledged in English too, though according to the Oxford Concise Dictionary, 4th ed. 1950 rarely so used).
- 52.
Use of this ‘topological’ semantics is common since Grove’s seminal paper (1988), which has started a semantic undercurrent working with possible-worlds modelling within the agm-tradition. We choose it because of its relation with ordinary (relational) possible worlds semantics, but as is well-known, Grove proved its equivalence with other constructions, using entrenchments or selection functions.
- 53.
Or the set of possible outputs, in the case of relational approaches to brt (originated in Lindstrom and Rabinowicz 1991).
- 54.
G. Simenon, Maigret and the Pickpocket, quoted in Hintikka (2007a, p. 32).
- 55.
Belief revision theory has not been as much a target of Hintikka and his associates as non-monotonic logics have. However, on the one hand, they are “two faces of the same coin” (to borrow D. Makinson’s well known image), and on the other, the criticism addressed by Hintikka to the latter holds, mutatis mutandis against the former: “[…] in the ultimate epistemological perspective [ampliative logics] are but types of enthymemic reasoning, relying on tacit premises […]. An epistemologist’s primary task here is not to study the technicalities of such modes of reasoning, fascinating though they are in their own right. It is to uncover the tacit premises on which such euthymemic reasoning is in reality predicated on.” (Hintikka, 2007a, p. 21). The equivalent of “tacit premises” in brt are, if we are correct, the entrenchment, or ordering of ‘fall back theories’, of which one of the tasks of (formal) epistemology should be to study the formation.
- 56.
In a talk given in Lille, 23rd of January 2007, which provided the initial motivation of the present work.
References
Enqvist, Sebastian, and Erik J. Olsson. 2008. Contraction in interrogative belief revision. Unpublished manuscript.
Emmanuel J. Genot. 2009. Extensive questions. In Logic and its applications, third indian conference, ICLA 2009, Chennai, India, January 7–11, 2009. Proceedings. Berlin: Springer.
Grove, Adam. 1988. Two modellings for theory change. Journal of Philosophical Logic 17:157–170.
Harris, Stephen. 1994. GTS and interrogative taleaux. Synthese 99:329–343.
Hintikka, Jaakko. 1987a. The fallacy of fallacies. Argumentation 1(3):211–238.
Hintikka, Jaakko. 1987b. The interrogative approach to inquiry and probabilistic inference. Erkenntnis 26(3):429–442.
Hintikka, Jaakko. 1988. What is abduction? The fundamental problem of contemporary epistemology. Transactions of the Charles S. Peirce Society 34:503–533.
Hintikka, Jaakko. 1997. What was aristotle doing in his early logic, anyway? Synthese 113:241–249.
Hintikka, Jaakko. 1999. Inquiry as inquiry: A logic of scientific discovery. Dordrecht: Kluwer.
Hintikka, Jaakko. 2003. A second generation epistemic logic and its general signifiance. In Knowledge contributors, eds. V.F. Hendricks, K.F. Jørgensen, and S.A. Pedersen, chapter 3, 33–56. Dordrecht: Kluwer, Reprinted in Jaakko Hintikka (2007d), chap. 3.
Hintikka, Jaakko. 2004. A fallacious fallacy?. Synthese 140:25–35. Reprinted in Jaakko Hintikka (2007d), chap. 9.
Hintikka, Jaakko. 2007a. Epistemology without belief and without knowledge. In Socratic epistemology, chapter 1, 11–37. Cambridge: Cambridge University Press.
Hintikka, Jaakko. 2007b. Logical explanations. In Socratic epistemology, chapter 7, 161–188. Cambridge: Cambridge University Press.
Hinitkka, Jaakko. 2007c. Presuppositions and other limitations of inquiry. In Socratic epistemology, chapter 4, 83–107. Cambridge: Cambridge University Press.
Hintikka, Jaakko. 2007d. Socratic epistemology. Cambridge: Cambridge University Press.
Hintikka, Jaakko, and Ilpo Halonen. 1999. Interpolation as explanation. Philosophy of Science 66(3):414–423.
Hintikka, Jaakko, Ilpo Halonen, and Arto Mutanen. 1999. Interrogative logic as a general theory of reasoning. In Inquiry as inquiry: A logic of scientific discovery, 47–90. Dordrecht: Kluwer.
Levi, Isaac. 1991. The fixation of belief and its undoing. Cambridge: Cambridge University Press.
Lewis, David K. 1979. Scorekeeping in a language game. Journal of Philosophical Logic 8(3):339–359.
Lindström, Sten, and Wlodek Rabinowicz. 1991. Epistemic entrenchment with incomparabilities and relational belief revision. In The logic of theory change, eds. A. Fuhrmann and M. Morreau, 93–126. Berlin: Springer.
Mackinson, David. 2005. Bridges from classical to nonmonotonic Logic, volume 5 of texts in computing. London: King’s College Publications.
Olsson, Erik J., and David Westlund. 2006. On the role of research agenda in epistemic change. Erkenntnis 65:165–183.
Olsson, Erik J. Belief revision and agenda management. This volume, XX–XX.
Osborne, Martin J., and Ariel Rubinstein. 1994. A course in game theory. Cambridge, MA: MIT Press.
Peirce, Charles S. 1940. The philosophy of Peirce: Selected writings. Routledge.
Rahman, Shahid, and Laurent Keiff. 2005. On how to be a dialogician. In Logic, thought and action, ed. D. Vanderveken, LEUS. Dordrecht: Kluwer.
Rahman, Shahid, and Tero Tulenheimo. 2007. From games to dialogues and back. In Logic, games and philosophy: Foundational perspectives, eds. Ondrej Majer, Ahti Pietarinen, and Tero Tulenheimo. Berlin: Springer.
Recanati, François. 2001. What is said. Synthese 128:75–91.
Robinson, Richard. 1971. Begging the question 1971. Analysis 31:113–117.
Smullyan, Raymond. 1968. First-order logic. Berlin: Spinger.
van Fraassen, Bas C. 1980. The scientific image. Oxford: Oxford University Press.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2010 Springer Science+Business Media B.V.
About this chapter
Cite this chapter
Genot, E. (2010). The Best of All PossibleWorlds: Where Interrogative Games Meet Research Agendas. In: Olsson, E., Enqvist, S. (eds) Belief Revision meets Philosophy of Science. Logic, Epistemology, and the Unity of Science, vol 21. Springer, Dordrecht. https://doi.org/10.1007/978-90-481-9609-8_10
Download citation
DOI: https://doi.org/10.1007/978-90-481-9609-8_10
Published:
Publisher Name: Springer, Dordrecht
Print ISBN: 978-90-481-9608-1
Online ISBN: 978-90-481-9609-8
eBook Packages: Humanities, Social Sciences and LawPhilosophy and Religion (R0)