Kognitywistyka: tworzenie pojęć i rozumowanie Inferencyjna Logika Pytań: scenariusze erotetyczne*

Kognitywistyka: tworzenie pojęć i rozumowanie Inferencyjna Logika Pytań: scenariusze erotetyczne* Mariusz Urbański Instytut Psychologii UAM Mariusz.Urbanski@.edu.pl * er tema (gr.) pytanie

Skąd to? Dzisiejsza opowieść również bazuje na pracach Andrzeja Wiśniewskiego [1; 10 14]. kognitywistyka, rok V (IP UAM) K:TPiR 2 / 33

Internal Question Processing [1] Questions are not only asked and then (possibly) answered, but they are also internally processed. Internal question processing (IQP) is a cognitive phenomenon which is distinct from question answering. When a question is internally processed, the immediate outcome need not be an answer to this question: an inference performed on a question can lead to another question, which may be send by an agent either to itself or to a certain external source of information and then answered, but can also be processed further in an analogous way. The relevant transformations of questions usually facilitate question answering and problem-solving. Sometimes, however, they finally result in a plausible answer/solution to a question/problem. kognitywistyka, rok V (IP UAM) K:TPiR 3 / 33

Implikowanie Pytanie Q implikuje pytanie Q 1 w oparciu o zbiór d-formuł X = {A 1,..., A n,...} (symbolicznie: Q, A 1,... A n,... Im Q 1 )wtw 1 dla każdego A dq: X {A} = dq 1 oraz 2 dla każdego B dq 1 istnieje niepusty podzbiór właściwy Y zbioru dq, taki że X {B} = Y Jeśli X =, piszemy: Q Im Q 1. kognitywistyka, rok V (IP UAM) K:TPiR 4 / 33

Q: Kto ukradł ciasteczka? D 1 : Ciasteczka ukradł któryś z dworzan Królowej Kier, zaproszonych na popołudniową herbatkę. kognitywistyka, rok V (IP UAM) K:TPiR 5 / 33

Q: Kto ukradł ciasteczka? D 1 : Ciasteczka ukradł któryś z dworzan Królowej Kier, zaproszonych na popołudniową herbatkę. Q 1 : Którzy dworzanie Królowej Kier zostali zaproszeni na popołudniową herbatkę? Q, D 1 Im Q 1 ponieważ: 1 dla każdego A dq: X {A} = dq 1 oraz 2 dla każdego B dq 1 istnieje niepusty podzbiór właściwy Y zbioru dq, taki że X {B} = Y kognitywistyka, rok V (IP UAM) K:TPiR 5 / 33

? ± A, B?(A#B) Im? ± A, B (gdzie # jest dowolnym z,,, )? ± A, B Im?A? ± A, B Im?B? ± A, B, A Im?B przypomnijmy:? ± A, B oznacza binarne pytanie koniunkcyjne postaci:?{a B, A B, A B, A B}?A oznacza pytanie rozstrzygnięcia postaci:?{a, A} kognitywistyka, rok V (IP UAM) K:TPiR 6 / 33

Decomposition Principle [9] Decompose an initial problem (IP) into simpler subproblems (SPs) in such a way that solutions to SPs can be assembled into an overall solution to IP. kognitywistyka, rok V (IP UAM) K:TPiR 7 / 33

Historia kanoniczna [12] Q Where did Andrew W. leave for: Paris, London, Kiev, or Moscow? D 1 Andrew W. left for Paris or London iff he departed in the morning. D 2 Andrew W. left for Kiev or Moscow iff he departed in the evening. D 3 If Andrew W. took a train, then he did not leave for London or Moscow. D 4 If Andrew W. left for Paris or Kiev, then he took a train. kognitywistyka, rok V (IP UAM) K:TPiR 8 / 33

Q Where did Andrew W. leave for: Paris, London, Kiev, or Moscow? D 1 Andrew W. left for Paris or London iff he departed in the morning. D 2 Andrew W. left for Kiev or Moscow iff he departed in the evening. D 3 If Andrew W. took a train, then he did not leave for London or Moscow. D 4 If Andrew W. left for Paris or Kiev, then he took a train. Q 1 When did Andrew W. depart: in the morning, or in the evening? kognitywistyka, rok V (IP UAM) K:TPiR 9 / 33

Q Where did Andrew W. leave for: Paris, London, Kiev, or Moscow? D 1 Andrew W. left for Paris or London iff he departed in the morning. D 2 Andrew W. left for Kiev or Moscow iff he departed in the evening. D 3 If Andrew W. took a train, then he did not leave for London or Moscow. D 4 If Andrew W. left for Paris or Kiev, then he took a train. Q 1 When did Andrew W. depart: in the morning, or in the evening? A 1 Andrew W. departed in the morning. D 5 Andrew W. left for Paris or London. D 6 If Andrew W. took a train, then he did not leave for London. D 7 If Andrew W. did not take a train, then he did not leave for Paris. kognitywistyka, rok V (IP UAM) K:TPiR 10 / 33

Q Where did Andrew W. leave for: Paris, London, Kiev, or Moscow? D 1 Andrew W. left for Paris or London iff he departed in the morning. D 2 Andrew W. left for Kiev or Moscow iff he departed in the evening. D 3 If Andrew W. took a train, then he did not leave for London or Moscow. D 4 If Andrew W. left for Paris or Kiev, then he took a train. Q 1 When did Andrew W. depart: in the morning, or in the evening? A 1 Andrew W. departed in the morning. D 5 Andrew W. left for Paris or London. D 6 If Andrew W. took a train, then he did not leave for London. D 7 If Andrew W. did not take a train, then he did not leave for Paris. Q 2 Did Andrew W. take a train? kognitywistyka, rok V (IP UAM) K:TPiR 11 / 33

Q Where did Andrew W. leave for: Paris, London, Kiev, or Moscow? D 1 Andrew W. left for Paris or London iff he departed in the morning. D 2 Andrew W. left for Kiev or Moscow iff he departed in the evening. D 3 If Andrew W. took a train, then he did not leave for London or Moscow. D 4 If Andrew W. left for Paris or Kiev, then he took a train. Q 1 When did Andrew W. depart: in the morning, or in the evening? A 1 Andrew W. departed in the morning. D 5 Andrew W. left for Paris or London. D 6 If Andrew W. took a train, then he did not leave for London. D 7 If Andrew W. did not take a train, then he did not leave for Paris. Q 2 Did Andrew W. take a train? A 2 Andrew W. did not take a train. D 8 Andrew W. left for London. kognitywistyka, rok V (IP UAM) K:TPiR 12 / 33

Q Where did Andrew W. leave for: Paris, London, Kiev, or Moscow? D 1 Andrew W. left for Paris or London iff he departed in the morning. D 2 Andrew W. left for Kiev or Moscow iff he departed in the evening. D 3 If Andrew W. took a train, then he did not leave for London or Moscow. D 4 If Andrew W. left for Paris or Kiev, then he took a train. Q 1 When did Andrew W. depart: in the morning, or in the evening? A 1 Andrew W. departed in the morning. D 5 Andrew W. left for Paris or London. D 6 If Andrew W. took a train, then he did not leave for London. D 7 If Andrew W. did not take a train, then he did not leave for Paris. Q 2 Did Andrew W. take a train? A 2 Andrew W. took a train. D 8 Andrew W. left for Paris. kognitywistyka, rok V (IP UAM) K:TPiR 13 / 33

Q Where did Andrew W. leave for: Paris, London, Kiev, or Moscow? D 1 Andrew W. left for Paris or London iff he departed in the morning. D 2 Andrew W. left for Kiev or Moscow iff he departed in the evening. D 3 If Andrew W. took a train, then he did not leave for London or Moscow. D 4 If Andrew W. left for Paris or Kiev, then he took a train. Q 1 When did Andrew W. depart: in the morning, or in the evening? A 1 Andrew W. departed in the evening. D 5 Andrew W. left for Kiev or Moscow. D 6 If Andrew W. took a train, then he did not leave for Moscow. D 7 If Andrew W. did not take a train, then he did not leave for Kiev. Q 2 Did Andrew W. take a train? A 2 Andrew W. took a train. D 8 Andrew W. left for Kiev. kognitywistyka, rok V (IP UAM) K:TPiR 14 / 33

Q Where did Andrew W. leave for: Paris, London, Kiev, or Moscow? D 1 Andrew W. left for Paris or London iff he departed in the morning. D 2 Andrew W. left for Kiev or Moscow iff he departed in the evening. D 3 If Andrew W. took a train, then he did not leave for London or Moscow. D 4 If Andrew W. left for Paris or Kiev, then he took a train. Q 1 When did Andrew W. depart: in the morning, or in the evening? A 1 Andrew W. departed in the evening. D 5 Andrew W. left for Kiev or Moscow. D 6 If Andrew W. took a train, then he did not leave for Moscow. D 7 If Andrew W. did not take a train, then he did not leave for Kiev. Q 2 Did Andrew W. take a train? A 2 Andrew W. did not take a train. D 8 Andrew W. left for Moscow. kognitywistyka, rok V (IP UAM) K:TPiR 15 / 33

?{p, q, r, s} p q t r s u w (q s) p r w?{t, u} t p q w q w p?w w q p w p q u r s w s w r?w w s r w r s kognitywistyka, rok V (IP UAM) K:TPiR 16 / 33

Scenariusze erotetyczne An e-scenario for a question Q relative to a set of d-wffs X is a finite tree Φ such that: 1. the nodes of Φ are (occurrences of) questions and d-wffs; they are called e-nodes and d-nodes, respectively; 2. Q is the root of Φ; 3. each leaf of Φ is a direct answer to Q; 4. dq X = ; 5. each d-node of Φ: 1 is an element of X, or 2 is a direct answer to an e-node of Φ different from the root Q, or 3 is entailed by (a set of) d-nodes which precede the d-node in Φ; kognitywistyka, rok V (IP UAM) K:TPiR 17 / 33

Scenariusze erotetyczne, c. d. 6. for each e-node Q of Φ different from the root Q: 1 dq dq and 2 Q Im Q for some e-node Q of Φ which precedes Q in Φ, or 3 Q, A 1,..., A n Im Q for some e-node Q and some d-nodes A 1,..., A n of Φ that precede Q in Φ; 7. each d-node has at most one immediate successor; 8. an immediate successor of an e-node different from the root Q is either a direct answer to the e-node, or exactly one e-node; 9. if the immediate successor of an e-node Q is not an e-node, then each direct answer to Q is an immediate successor of Q. kognitywistyka, rok V (IP UAM) K:TPiR 18 / 33

Wampiry, zombie i bankomaty [8] On a certain island the inhabitants have been bewitched by some kind of magic. Half of them turned into zombies, the other half turned into vampires. The zombies and the vampires of this island do not behave like the conventional ones (if any): they move about and talk in as lively a fashion as do the humans, and the vampires even prefer drinking strong mocca over anything else. It s just that the zombies of this island always lie and the vampires of this island always tell the truth. What is also important, both vampires and zombies never miss a reasonable opportunity to tell the truth or to lie, respectively. Thus they always do their best to answer questions addressed to them. A native named Eugene has been suspected of an attempt to break in an ATM near the police station. The case has been assigned to Inspector Negombo (a vampire) of local police force. His first task was to establish if the accused is a vampire or a zombie. Inspector Negombo was clever enough to determine that Eugene is a vampire on the basis of the suspected s answer to a single question. What was Negombo s question? kognitywistyka, rok V (IP UAM) K:TPiR 19 / 33

Wampiry, zombie i bankomaty dane 1. Every native is either a vampire or a zombie. 2. Every native who is a vampire utters true sentences. 3. Every native who utters a true sentence is a vampire. 4. Eugene is a native of the island. kognitywistyka, rok V (IP UAM) K:TPiR 20 / 33

Wampiry, zombie i bankomaty dane 1. Every native is either a vampire or a zombie. 2. Every native who is a vampire utters true sentences. 3. Every native who utters a true sentence is a vampire. 4. Eugene is a native of the island. Poręczniej: 2. For every native x, if x gives back a true answer to a posed question, then x is a vampire. 3. For every native x, if x is a vampire, then x gives back a true answer to a posed question. kognitywistyka, rok V (IP UAM) K:TPiR 20 / 33

Wampiry, zombie i bankomaty robaczki Let V (x), Z(x), N(x) stand for expressions: x is a vampire, x is a zombie, x is a native of the island respectively, let U(x, A i,?{a 1,..., A n}) stand for x gives back an answer A i to the question?{a 1,..., A n}, and let the constant a represent Eugene. T stands for the lack of factual knowledge. Assume that R stands for the formula V (a) V (a).?{v (a), Z(a), T} x(n(x) V (x) Z(x)) x(u(x, R,?R) V (x)) x(v (x) U(x, R,?R)) N(a)?V (a)?u(a, R,?R) U(a, R,?R) V (a) U(a, R,?R) V (a) Z(a) kognitywistyka, rok V (IP UAM) K:TPiR 21 / 33

Twierdzenie o złotej ścieżce Jeśli wyjściowe pytanie Q scenariusza erotetycznego Φ jest trafne (czyli jeśli istnieje prawdziwa odpowiedź bezpośrednia na Q) i jeśli wszystkie przesłanki deklaratywne scenariusza Φ są prawdziwe, to dla co najmniej jedna ścieżka e scenariusza Φ: ścieżka e prowadzi do prawdziwej odpowiedzi bezpośredniej na Q oraz wszystkie d-formuły należące do ścieżki e są prawdziwe a wszystkie pytania należące do e są trafne. kognitywistyka, rok V (IP UAM) K:TPiR 22 / 33

Operacje na scenariuszach 1 wklejanie 2 skracanie kognitywistyka, rok V (IP UAM) K:TPiR 23 / 33

ILP a tabelowe metody dowodzenia 1 ILP a tabele syntetyczne: Każdą tabelę syntetyczną dla formuły A można rozszerzyć do scenariusza erotetycznego dla pytania?a w oparciu o zbiór pusty. 2 ILP a tabele analityczne: Jakie wnioskowania erotetyczne pojawiają się w konstrukcji tabel analitycznych? 3 Skąd bierze się różnica między erotetycznym tłem obu rodzajów tabel? kognitywistyka, rok V (IP UAM) K:TPiR 24 / 33

Źródła I [1] http://www.staff..edu.pl/~p_lup/aw_pliki/unilog/ [2] Bolotov, A., Łupkowski, P., Urbański, M. [2006]. Search and check. Problem solving by problem reduction, w: A. Cader et al. (red.), Artificial Intelligence and Soft Computing, Warszawa, s. 505-510. [3] Grobler, A. [2006]. Metodologia nauk. Wyd. Aureus, Wyd. Znak, Kraków. [4] Kuipers, T. A. F., Wiśniewski, A. [1994]. An Erotetic Approach to Explanation by Specification. Erkenntnis, 3(40):377 402. [5] Łupkowski, P. [w druku]. Test Turinga a logika pytań, Wyd. Naukowe UAM, Poznań. [6] Łupkowski. P. [w druku]. Cooperative answering and Inferential Erotetic Logic. [7] Urbański, M. [2002]. Tabele syntetyczne a logika pytań. Wydawnictwo UMCS, Lublin. kognitywistyka, rok V (IP UAM) K:TPiR 25 / 33

Źródła II [8] Urbański, M., Łupkowski, P. [w druku]. Erotetic search scenarios: Revealing interrogator s hidden agenda. [9] Urbański, M., Wiśniewski, A. [manuskrypt] Socratic Trees. [10] Wiśniewski, A. [1995]. The Posing of Questions: Logical Foundations of Erotetic Inferences. Kluwer Academic Publishers, Dordrecht Boston London. [11] Wiśniewski, A. [2001]. Questions and Inferences. Logique et Analyse, 173-174-175:5 43. [12] Wiśniewski, A. [2003]. Erotetic search scenarios. Synthese, 134(3):389 427. [13] Wiśniewski, A. [2004]. Erotetic search scenarios, problem-solving, and deduction. Logique et Analyse, 185 188:139 166. [14] Wiśniewski, A., Pogonowski, J. [w druku]. Interrogatives, Recursion, and Incompleteness. Journal of Logic and Computation. kognitywistyka, rok V (IP UAM) K:TPiR 26 / 33

Wampiry, zombie i bankomaty c. d. Having established if Eugene is a vampire or a zombie it would be quite simple to solve the problem: just ask Eugene if he really did try to break in the ATM. But this simple plan has been ruined by discovery that one premise on which Negombo s inferences were dependent is false: Eugene was not a native of the island. He came there as an immigrant from the nearby island, inhabited exclusively by humans, who are totally unpredictable as for the truth or falsity of what they tell. Moreover, Eugene refused to give any sort of testimony. A short investigation among Eugene s friends (all confirmed being vampires by Negombo s test) led Negombo to establish the following rules: kognitywistyka, rok V (IP UAM) K:TPiR 27 / 33

5. If Eugene did run short of money, then he attempted to break in an ATM or borrowed some money from Eustace. 6. If Eugene didn t run short of money, then he went shopping or visited his favourite pub. 7. If Eugene attempted to break in an ATM or went shopping, then he has been seen in a local mall. 8. If Eugene borrowed some money from Eustace or visited his favourite bar, then he hasn t been seen in a local mall. On this basis Inspector Negombo devised the plan for further interrogation of Elyssa, the only of Eugene s friends able to describe the course of events of that particular day. At(x) stands for x attempted to break in an ATM, Sh(x) stands for x ran short of money, Br(x, y) stands for x borrowed money from y, Sp(x) stands for x went shopping, Pb(x) stands for x visited his favourite pub, Ml(x) stands for x has been seen in a local mall and c stands for Eustace. kognitywistyka, rok V (IP UAM) K:TPiR 28 / 33

?{At(a), At(a), T} Sh(a) At(a) Br(a, c) Sh(a) Sp(a) Pb(a) At(a) Sp(a) Ml(a) Br(a, c) Pb(a) Ml(a)?Ml(a) Ml(a) (Br(a, c) Pb(a)) Br(a, c) Pb(a)?Sh(a) Ml(a) (At(a) Sp(a)) At(a) Sh(a) At(a) Br(a, c) At(a) Sh(a) Sp(a) Pb(a) Sp(a) T kognitywistyka, rok V (IP UAM) K:TPiR 29 / 33

If he s guilty and if this is my lucky day, I ll send him to the court in two questions, Negombo said to himself. If he s been in the mall but didn t run short of money then my information is insufficient and I will need new evidence. If he hasn t been in the mall, he s innocent. Well, we ll see. He ordered one of his lieutnants to conduct an interrogation according to this plan (the reader may notice that both queries of this scenario, that is?ml(a) and?sh(a), might demand plans for investigation in the form of e-scenarios on themselves; such auxiliary e-scenarios can be incorporated into the main one by the embedding operation). The lieutnant soon reported the outcome: Elyssa answered first query with No. Eugene hasn t been in the mall. He s innocent! kognitywistyka, rok V (IP UAM) K:TPiR 30 / 33

However, it occured that Elyssa is a human, too. Unfortunately for Elyssa and Eugene, Negombo studied nonverbal behaviour of human liars and has been identified as a Truth Wizard, that is, a person who can identify deception with exceptional accuracy of more than 80%. Besides his natural talent he devised for himself a list of behaviours that help identifying lies with satisfactory precision. Negombo, highly suspicious as for the truth of what Elyssa testified, decided to try to kill two birds with one stone and possibly accuse her of false testimony. Negombo repeated Elyssa s interrogation, this time personally, having in mind the following agenda: L(x, A) stands for x lies saying A, S(x, B, A) stands for x expresses the set of behaviours B while saying A and b stands for Elyssa kognitywistyka, rok V (IP UAM) K:TPiR 31 / 33

?L(b, Ml(a)) U(b, Ml(a),?Ml(a)) x(s(x, B 1, A) L(x, A))... x(s(x, B n, A) L(x, A)) x( S(x, B 1, A)... S(x, B n, A) L(x, A))?{L(b, Ml(a)), L(b, Ml(a)), S(b, B 1, Ml(a))}?S(b, B 1, Ml(a)) S(b, B 1, Ml(a)) L(b, Ml(a)) S(b, B 1, Ml(a))...?S(b, B n, Ml(a)) S(b, B n, Ml(a)) L(b, Ml(a)) S(b, B n, Ml(a)) L(b, Ml(a)) kognitywistyka, rok V (IP UAM) K:TPiR 32 / 33

This scenario shows that the only question that Negombo should actually pose to Elyssa is Has Eugene been in the mall? although it is known what will the answer be. All the remaining questions play the role of milestones on Negombo s way of thinking in solving the initial problem. Notice that they are concerned not with the content of Elyssa statement but with the way she provided that statement. Elyssa repeated her previous testimony that Eugene has not been in the mall. But saying this she expressed a set of behaviours characteristic for a liar (say that they were microexpressions of her lips indicating disbelief in what she has been saying; an interested reader may have a look at the example of such microexpression on Paul Ekman s page: http://www.paulekman.com). On this basis Negombo determined that she is lying that Eugene has not been in the mall. To finish his investigation quickly he decided to employ ethically disputable means. He produced a fake witness (who testified that he has seen Eugene in the mall) and confronted Elyssa with him. Elyssa finally admitted that she was lying and that Eugene in fact has been in the mall on that particular day. Her statement has been recorded. The case has been sent to the court. kognitywistyka, rok V (IP UAM) K:TPiR 33 / 33