Human interpretation and reasoning about conditionals

Transcription

1 Human interpretation and reasoning about conditionals Niki Pfeifer 1 Munich Center for Mathematical Philosophy Language and Cognition Ludwig-Maximilians-Universität München 1 FWF-project Mental probability logic

2 Outline Mental probability logic

3 Outline Mental probability logic Assertability/acceptability vs. truth conditions Classification of tasks involving conditionals

4 Outline Mental probability logic Assertability/acceptability vs. truth conditions Classification of tasks involving conditionals 2 experiments on Aristotle s thesis Motivation: Clarifying interpretation and scope ambiguity in reasoning about conditionals

5 Mental probability logic (Pfeifer & Kleiter, 2005, 2009a, 2009b, 2010) competence uncertain indicative If A, then B is interpreted as P(B A) B A is partially truth-functional (void, if A is false and undefined if A is a logical contradiction) premises are assessed by point values, intervals or second order probability distributions uncertainty is transmitted deductively from the premises to the conclusion coherence based probability logic framework

6 Coherence de Finetti, and {Coletti, Gilio, Lad, Regazzini, Scozzafava, Walley,...} degrees of belief complete algebra is not required conditional probability, P(B A), is primitive zero probabilities are exploited to reduce the complexity imprecision bridges to possibility, DS-belief functions, fuzzy sets, default reasoning,... provides semantics for System P (Gilio, 2002)

7 How people interpret indicative conditionals Prominent psychological predictions of the interpretation of indicative conditionals: Conjunction Material conditional Conditional event Material Conjunction Conditional conditional event A B A B A B B A s 1 true true true true true s 2 true false false false false s 3 false true true false void s 4 false false true false void

8 How people interpret indicative conditionals Prominent psychological predictions of the interpretation of indicative conditionals: Conjunction Material conditional Conditional event Material Conjunction Conditional conditional event A B A B A B B A s 1 true true true true true s 2 true false false false false s 3 false true true false void s 4 false false true false void

9 Assertability/acceptability vs. truth conditions (Douven, in press) Conditionals do not have truth conditions (e.g., Adams, 1975) Adams Thesis: Indicative conditionals are not truth-functional and their acceptability/assertability is equal to P(B A).

10 Assertability/acceptability vs. truth conditions (Douven, in press) Conditionals do not have truth conditions (e.g., Adams, 1975) Adams Thesis: Indicative conditionals are not truth-functional and their acceptability/assertability is equal to P(B A). Advantages: avoiding the paradoxes of the material conditional, intuitively plausible Disadvantages: conditionals are restricted to simple conditionals (no iteration, no nesting except quasi-conjunction)

11 Assertability/acceptability vs. truth conditions (Douven, in press) Conditionals have truth conditions. Are the truth conditions truth functional? yes material conditional (Lewis, 1976; Jackson, 1979; Grice, 1975) A B is assertable/acceptable iff (1) P(A B) is high, and (2) P(A B A) is high (and close to P(A B)).

12 Assertability/acceptability vs. truth conditions (Douven, in press) Conditionals have truth conditions. Are the truth conditions truth functional? yes material conditional (Lewis, 1976; Jackson, 1979; Grice, 1975) A B is assertable/acceptable iff (1) P(A B) is high, and (2) P(A B A) is high (and close to P(A B)). Jackson (1987, p. 31) notes that these conditions imply P(B A) is high.

13 Assertability/acceptability vs. truth conditions (Douven, in press) Conditionals have truth conditions. Are the truth conditions truth functional? yes material conditional (Lewis, 1976; Jackson, 1979; Grice, 1975) A B is assertable/acceptable iff (1) P(A B) is high, and (2) P(A B A) is high (and close to P(A B)). Jackson (1987, p. 31) notes that these conditions imply P(B A) is high. no possible worlds (e.g., Stalnaker, 1968)

14 Reasoning tasks from a probability-logical point of view Arguments involving conditionals If to if System P Complement: P(A B) = x P(A B) =? If elimination Conditional syllogisms (Suppression tasks) P(A) = x P(A B) = y P(B) =?

15 Reasoning tasks from a probability-logical point of view Arguments involving conditionals If to if System P Complement: P(A B) = x P(A B) =? If introduction Probabilistic truth table task P(A B) = x 1 P(A B) = x 2 P( A B) = x 3 P( A B) = x 4 P(A B) =? If elimination Conditional syllogisms (Suppression tasks) P(A) = x P(A B) = y P(B) =?

16 Reasoning tasks from a probability-logical point of view Arguments involving conditionals If to if System P Complement: P(A B) = x P(A B) =? If introduction Probabilistic truth table task P(A B) = x 1 P(A B) = x 2 P( A B) = x 3 P( A B) = x 4 P(A B) =? If elimination Conditional syllogisms (Suppression tasks) P(A) = x P(A B) = y P(B) =?

17 Reasoning tasks from a probability-logical point of view Arguments involving conditionals If to if System P Complement: P(A B) = x P(A B) =? If introduction Probabilistic truth table task P(A B) = x 1 P(A B) = x 2 P( A B) = x 3 P( A B) = x 4 P(A B) =? If elimination Conditional syllogisms (Suppression tasks) P(A) = x P(A B) = y P(B) =? If the side shows a square, then the side shows black (Fugard, Pfeifer, Mayerhofer, & Kleiter, in press)

18 Reasoning tasks from a probability-logical point of view Arguments involving conditionals If to if System P Complement: P(A B) = x P(A B) =? If introduction Probabilistic truth table task Paradoxes of, e.g.: P(B) = x P(A B) =? If elimination Conditional syllogisms (Suppression tasks) P(A) = x P(A B) = y P(B) =?

19 Reasoning tasks from a probability-logical point of view Arguments involving conditionals If to if System P Complement: P(A B) = x P(A B) =? If introduction Probabilistic truth table task Paradoxes of, e.g.: P(B) = x P(A B) =? If elimination Conditional syllogisms (Suppression tasks) P(A) = x P(A B) = y P(B) =? P(B) = x 0 P(A B) x (conjunction) P(B) = x x P(A B) 1 (mat. cond.) P(B) = x 0 P(B A) 1 (cond. event)

20 Full vs. incomplete probabilistic knowledge Full probabilistic knowledge: e.g., probabilistic truth table task: The probability value of A B is precise Incomplete probabilistic knowledge: e.g., probabilistic modus ponens: from P(B A) = x and P(A) = y infer xy P(B) xy +1 y

21 Full vs. incomplete probabilistic knowledge Full probabilistic knowledge: e.g., probabilistic truth table task: The probability value of A B is precise Incomplete probabilistic knowledge: e.g., probabilistic modus ponens: from P(B A) = x and P(A) = y infer xy P(B) xy +1 y Sentences that are purely determined by their logical form: FO-Logic Probability logic Tautology the only coherent value: 1 Contradiction the only coherent value: 0 Sentences that are not purely determined by their logical form: FO-Logic Probability logic Contingent sentences the unit interval, [0, 1], may be coherent

22 Aristotle s Thesis

23 Aristotle s Thesis AT #1: ( A A) AT #2: (A A)

24 Aristotle s Thesis AT #1: ( A A) ( A A) AT #2: (A A) (A A)

25 Aristotle s Thesis AT #1: ( A A) ( A A) A A A AT #2: (A A) (A A) A A A

26 Aristotle s Thesis: Probability logical predictions AT #1: ( A A) P( ( A A)) = P( A)

27 Aristotle s Thesis: Probability logical predictions AT #1: ( A A) P( ( A A)) = P( A) P( ( A A)) = 1

28 Aristotle s Thesis: Probability logical predictions AT #1: ( A A) P( ( A A)) = P( A) P( ( A A)) = 1 P(A A) = 0, its negation: P( A A) = 1

29 Aristotle s Thesis: Probability logical predictions AT #1: ( A A) P( ( A A)) = P( A) P( ( A A)) = 1 P(A A) = 0, its negation: P( A A) = 1 AT #2: (A A) P( (A A)) = P(A) P( (A A)) = 1 P( A A) = 0, its negation: P( A A) = P(A A) = 1

30 Experiment 1: Abstract version, Aristotle s Thesis #1 The letter A denotes a sentence, like It is raining. There are sentences, where you can infer only on the basis of their logical form, whether they are guaranteed to be false or guaranteed to be true. For example: A and not-a is guaranteed to be false. A or not-a is guaranteed to be true. There are sentences, where you cannot infer only on the basis of their logical form, whether they are true or false. The sentence A ( It is raining. ), for example, can be true but it can just as well be false: this depends upon whether it is actually raining. Evaluate the following sentence (please tick exactly one alternative): It is not the case, that: If not-a, then A. The sentence in the box is guaranteed to be false The sentence in the box is guaranteed to be true One cannot infer whether the sentence is true or false

31 Experiment 1: Abstract version, Aristotle s Thesis #2 The letter A denotes a sentence, like It is raining. There are sentences, where you can infer only on the basis of their logical form, whether they are guaranteed to be false or guaranteed to be true. For example: A and not-a is guaranteed to be false. A or not-a is guaranteed to be true. There are sentences, where you cannot infer only on the basis of their logical form, whether they are true or false. The sentence A ( It is raining. ), for example, can be true but it can just as well be false: this depends upon whether it is actually raining. Evaluate the following sentence (please tick exactly one alternative): It is not the case, that: If A, then not-a. The sentence in the box is guaranteed to be false The sentence in the box is guaranteed to be true One cannot infer whether the sentence is true or false

32 Experiment 1: Concrete version, Aristotle s Thesis #1 There is either a dog or a cat behind the door, but not both. There are sentences, where you can infer only on the basis of their logical form, whether they are guaranteed to be false or guaranteed to be true. For example: There is a dog and a cat behind the door is guaranteed to be false. There is either a dog or a cat behind the door is guaranteed to be true. There are sentences, where you cannot infer only on the basis of their logical form, whether they are true or false. The sentence There is a dog behind the door., for example, can be true but it can just as well be false: this depends upon whether there is in fact a dog behind the door. Evaluate the following sentence (please tick exactly one alternative): It is not the case, that: If there is a dog behind the door, then there is a cat behind the door. The sentence in the box is guaranteed to be false The sentence in the box is guaranteed to be true One cannot infer whether the sentence is true or false

33 Experiment 1: Concrete version, Aristotle s Thesis #2 There is either a dog or a cat behind the door, but not both. There are sentences, where you can infer only on the basis of their logical form, whether they are guaranteed to be false or guaranteed to be true. For example: There is a dog and a cat behind the door is guaranteed to be false. There is either a dog or a cat behind the door is guaranteed to be true. There are sentences, where you cannot infer only on the basis of their logical form, whether they are true or false. The sentence There is a dog behind the door., for example, can be true but it can just as well be false: this depends upon whether there is in fact a dog behind the door. Evaluate the following sentence (please tick exactly one alternative): It is not the case, that: If there is a cat behind the door, then there is a dog behind the door. The sentence in the box is guaranteed to be false The sentence in the box is guaranteed to be true One cannot infer whether the sentence is true or false

34 Experiment 1: Participant s evaluation of the tasks Is the task clear and comprehensible? (please tick) very clear very unclear How sure are you, that your solution is correct? very sure very uncertain How difficult is the task for you? very easy very difficult

35 Experiment 1: Participant s evaluation of the tasks Is the task clear and comprehensible? (please tick) M = 35% (SD = 28%) X very clear How sure are you, that your solution is correct? very sure M = 44% (SD = 27%) X How difficult is the task for you? very easy very unclear very uncertain M = 61% (SD = 22%) X very difficult

36 Experiment 1: Sample N = 141 all psychology students 91% third semester 78% female median age: 21 (1st Qu. = 20, 3rd Qu. =23)

37 Concrete (n=71) versus abstract (n=71) task material Frequency abstract concrete FALSE TRUE CANNOT INFER

38 Task features of Aristotle s Thesis If introduction inference from the empty premise set Only one variable: A Logical form is complex: two negations, one conditional

39 Task features of Aristotle s Thesis If introduction inference from the empty premise set Only one variable: A Logical form is complex: two negations, one conditional Scope ambiguity: negating the conditional: (A A) }{{} wide scope negating the consequent: (A A) }{{} narrow scope

40 Experiment 2: Design Between participants: Explicit (n 1 = 20) vs. implicit negation (n 2 = 20) Within participants: 12 Tasks Task Name Argument form 1 Aristotle s Thesis 1 (A A) 2 Negated Reflexivity (A A) 3 Aristotle s Thesis 2 ( A A) 4 Reflexivity A A 5 Contingent Arg. 1 A B 6 Contingent Arg. 2 (A B) Probabilistic truth-table tasks 11 Paradox 1 from B infer A B 12 Neg. Paradox 1 from B infer A B

41 Experiment 2: Predictions Argument form Scope wide narrow (A A) T CT T T (A A) F F CT CT ( A A) T CT T T A A T T T CT A B CT CT CT CT (A B) CT CT CT CT from B infer A B U H U from B infer A B U T L Note: CT=can t tell, T=true, F=false, U=uninformative conclusion probability, H=high conclusion probability, L=low conclusion probability

42 Experiment 2: Predictions against wide vs. narrow scope of Argument form Scope wide narrow (A A) T CT T T (A A) F F CT CT ( A A) T CT T T A A T T T CT A B CT CT CT CT (A B) CT CT CT CT from B infer A B U H U from B infer A B U T L Note: CT=can t tell, T=true, F=false, U=uninformative conclusion probability, H=high conclusion probability, L=low conclusion probability

43 Experiment 2: Aristotle s Thesis #1, explicit version [...] Hans expects to be visited by Thea and Ida. He is sitting in his room. Suddenly someone knocks at the door. Hans is absolutely certain, that either Thea or Ida is knocking.

44 Experiment 2: Aristotle s Thesis #1, explicit version [...] Hans expects to be visited by Thea and Ida. He is sitting in his room. Suddenly someone knocks at the door. Hans is absolutely certain, that either Thea or Ida is knocking. Evaluate the following sentence (please tick exactly one alternative): It is not the case, that: If Ida knocks, then Thea knocks. The sentence in the box is guaranteed to be false The sentence in the box is guaranteed to be true One cannot infer whether the sentence is true or false

45 Experiment 2: Aristotle s Thesis #1, implicit version [...] Hans expects to be visited by Thea and Ida. He is sitting in his room. Suddenly someone knocks at the door. Hans is absolutely certain, that either Thea or Ida is knocking. Evaluate the following sentence (please tick exactly one alternative): It is not the case, that: If Ida knocks, then Ida does not knock. The sentence in the box is guaranteed to be false The sentence in the box is guaranteed to be true One cannot infer whether the sentence is true or false

46 Experiment 2: Sample N = 40 no psychology students individual tested, 5 for participation 50% female median age: 22 (1st Qu. = 21, 3rd Qu. =23)

47 Experiment 2: Results Argument form Scope Responses wide narrow in percent T F CT (A A) T CT T T (A A) F F CT CT ( A A) T CT T T A A T T T CT A B CT CT CT CT (A B) CT CT CT CT from B infer A B U H U from B infer A B U T L Note: CT=can t tell, T=true, F=false, U=uninformative conclusion probability, H=high conclusion probability, L=low conclusion probability

48 Experiment 2: Results Argument form Scope Responses wide narrow in percent T F CT (A A) T CT T T (A A) F F CT CT ( A A) T CT T T A A T T T CT A B CT CT CT CT (A B) CT CT CT CT from B infer A B U H U from B infer A B U T L Note: CT=can t tell, T=true, F=false, U=uninformative conclusion probability, H=high conclusion probability, L=low conclusion probability

49 Experiment 2: Results (consistency with interpretation) Scoring: Aristotle s thesis 1+2, (negated) reflexivity, (negated) paradox 1 (n 1+2 = 40, 6 tasks, min = 0, max = 6) Interpretation Mean SD Scope narrow wide

50 Concluding remarks New evidence against the material conditional interpretation Scope ambiguity of the negation of conditionals Most responses consistent with the conditional event interpretation

51 Concluding remarks New evidence against the material conditional interpretation Scope ambiguity of the negation of conditionals Most responses consistent with the conditional event interpretation Acceptability/assertability conditions are consistent with the conditional event interpretation Unclear, if truth conditions play any role in human reasoning about conditionals

52 Concluding remarks New evidence against the material conditional interpretation Scope ambiguity of the negation of conditionals Most responses consistent with the conditional event interpretation Acceptability/assertability conditions are consistent with the conditional event interpretation Unclear, if truth conditions play any role in human reasoning about conditionals An implication that satisfies Aristotle s theses and the Boethius theses (A B) (A B) (A B) (A B) is called a connexive implication (McCall, 1966; Angell, 2002).

53 References I Adams, E. W. (1975). The logic of conditionals. Dordrecht: Reidel. Douven, I. (in press). Indicative conditionals. In R. Pettigrew & L. Horsten (Eds.), Continuum companion to philosophical logic (pp. xx xx). London, New York: Continuum Press. Fugard, A. J. B., Pfeifer, N., Mayerhofer, B., & Kleiter, G. D. (in press). How people interpret conditionals: Shifts towards the conditional event. Journal of Experimental Psychology: Learning, Memory, and Cognition. Gilio, A. (2002). Probabilistic reasoning under coherence in System P. Annals of Mathematics and Artificial Intelligence, 34, Grice, H. P. (1975). Logic and conversation. In P. Cole & J. L. Morgan (Eds.), Syntax and semantics (Vol. 3: Speech acts.). New York: Seminar Press.

54 References II Jackson, F. (1979). On assertion and indicative conditionals. Philosophical Review, 88, (Reprint with postscript in (Jackson, 1991, ); the page references are to the reprint) Jackson, F. (1987). Conditionals. Oxford: Blackwell. Jackson, F. (Ed.). (1991). Conditionals. Oxford: Oxford University Press. Lewis, D. (1976). Probabilities of conditionals and conditional probabilities. Philosophical Review, 85, (Reprint with postscript in (Jackson, 1991, ); the page references are to the reprint) Pfeifer, N., & Kleiter, G. D. (2005). Towards a mental probability logic. Psychologica Belgica, 45(1), Pfeifer, N., & Kleiter, G. D. (2009a). Framing human inference by coherence based probability logic. Journal of Applied Logic, 7(2),

55 References III Pfeifer, N., & Kleiter, G. D. (2009b). Mental probability logic. Commentary on Oaksford & Chater: Bayesian rationality. Behavioral and Brain Sciences, 32, Pfeifer, N., & Kleiter, G. D. (2010). Uncertain deductive reasoning. In K. Manktelow, D. E. Over, & S. Elqayam (Eds.), The science of reasoning: A Festschrift for Jonathan St. BT Evans (pp ). Hove, UK: Psychology Press. Stalnaker, R. (1968). A theory of coditionals. In N. Rescher (Ed.), Studies in logical theory (pp ). Oxford: Blackwell. (Reprint in (Jackson, 1991, ); the page references are to the reprint)