1. CONDITIONALS PROBABILITIES AND ENTHYMEMES Mike Oaksford, Birkbeck College, U of London

2. Introduction  “...our position is not an imperialist one. Rather it is that while people may well be capable of assessing deductive correctness when explicitly asked to, this is rarely, if ever, their focus of interest in evaluating an argument.” (Oaksford & Hahn, 2007)  Goal: To provide a method of assessing people’s assessment of inductive strength using probabilistic information provided in the task instructions.

3. Introduction  Motivation (Oberauer, 2006)  One configural response not predicted = DA, MT  Suppose P(p) = P(q) =.2 and P(q|p) =.5, then P(MP) = P(AC) =.5 and P(MT) = P(DA) =.875 Using Oberauer’s Best-Fit Threshold (.58), a DA, MT configrual response is therefore predicted But in non-probabilistic binary theories, MT without MP is impossible  Some Evidence  Some non-probabilistic theories argue selection task performed by drawing conditional inferences But in one condition of Oaksford & Wakefield (2003), ¬p (DA) and ¬q (MT) card selections dominated

4. Enthymematic Reasoning  Enthymemes  Conditional reasoning usually relies on an enthymematic basis, S (Rescher, 2007), for example: All men are mortal,  Socrates is mortal (Minor premise = S) Socrates is a man,  Socrates is mortal (Major premise = S) “But it is obviously true that most people never engage in explicit non-enthymematic formal reasoning” Dennett (1998)

5. Natural Enthymemes  S = major premise  This case seems more natural We rarely state the universally quantified, indicative conditionals, or law-like relations, that underpin most of our inferences [Stalnaker’s methodological policies to change our beliefs] (Speculation) Corpus analysis: Conditionals are the rarest logical term (BNC, OEC; carrying out textual analysis )  Law-like relations Open indicative conditionals (Stalnaker); but more likely, birds fly, bears have four legs etc.

6. Enthymemes and Context  Enthymematic Basis (S) Derives From:  Prior beliefs (cognitive context; old information)  Prior discourse (discourse context; what you have just been told; new information)  Deictic context (current surroundings)  All present in Psych Experiments manipulating probs  Prior beliefs Always present but active to different degrees dependent on instructions  Prior discourse Information about P(p), P(q), and P(q|p) in instructions  BUT not usually enthymematic (but see Liu, 2003) Include the Assertion of the conditional premise: “if p then q”

7. Probability Logic:Assert “if p then q” = P(q|p) is high (I) C2 shows that the components of the deictic context/prior discourse (II) can conflict with the assertion of the conditional (as well prior belief (III)/logic conflicts) Conflicts between Context and Assertion C1 P(W|S,C1) =.8 C2 P(W|S,C2) =.2 “if S then W” Then I have said something true (in C1) “if S then W” Then I have said something false (in C2)

8. Discourse Examples  You believe: (III) Most migrant workers have jobs  “Most migrant workers on the Estate were unemployed (II). We talked to Aleksy who had lived on the Estate for two years.” (Employed or Unemployed?) II vs III  “Most migrant workers on the Estate were unemployed. We talked to Alan a local councillor, who said that in this area if you were a migrant worker you were unemployed (I). We also talked to Aleksy who was a migrant worker.” (Employed or Unemployed?) II vs III but I and II consistent  “Most migrant workers on the Estate were unemployed. We talked to Alan a local councillor, who said that in this area if you were a migrant worker you were in work. We also talked to Aleksy who was a migrant worker.” (Employed or Unemployed?) II vs III, I vs II, I and III consistent.

9. Conflict Resolution  Conflicts can arise because:  (I) The assertion of the conditional is inconsistent with (II) The probabilities given in the instructions  (I) or (II) or both, are inconsistent with (III) Prior beliefs  Principles of Conflict Resolution  Unless accessing (III) is mandatory most recently encountered information is given priority: I > II > III (But Fundamental Computational Bias; and enthymemes (Stanovich, 1998))  C2|- “if S then W”: Ps may assume they are being asked to suppose, counterfactually, that C2 is like C1, OR “if S then W” is a cue that they are actually in C1 and not C2

10. Probabilistic Approach  Inferential Asymmetries  MP vs the Rest (DA, AC, & MT) All but MP require mandatory access to prior knowledge of P 0 (p) and P 0 (q) (at least to derive point values)

11. A Test of the Probabilistic Approach  Can DA, MT response (or probabilistic equivalent) be elicited using II (instructions)?  Problems P 0 (q|p, C) =.5 in II. Conflicts with I, i.e., asserting the conditional means P 0 (q|p) is high P 0 (p|C) =.2 in II. Conflicts with prior knowledge, III, because best fits show people’s priors are P 0 (p|C) ≈.5 < P 0 (q|C), i.e., the context is one in which p is not uncommon (NB For DA, AC, MT accessing III is mandatory)  Solutions II vs I: Use Enthymemes (minus conditional)(I) > II > III II vs III: Use Blank Predicates(I) > II > (III)

12. A Fair Test?  Enthymemes  Non-probabilistic theories: Enthymemes are simply a matter of where the conditional premise comes from Having it asserted, retrieved from LTM, or built up from prior discourse makes no difference to the underlying mental representation So whether “All men are mortal” is presented or not, we should still endorse “Socrates is mortal” on being told “Socrates is a man” BECAUSE “All men are mortal” retrieved from LTM  Blank Predicates  These should not access prior knowledge, apart from facts about superordinate categories For example, “Shreebles,” a made up bird of whose colour we are ignorant, may still be assumed to be invariant with respect to colour in virtue of being a bird

13. Experiment  If swan then white, x is a swan,  x is white IIIII I IIIII I III II III II  (I) P(p) = P(q) =.1 and P(q|p) =.5  If shreeble then blue, x is a shreeble,  x is blue  x is a swan,  x is white  x is a shreeble,  x is blue

14. Design  4 Inference (MP, DA, AC, MT) × 2 (Inf Type: Complete/Incomplete) × 2 (Predicate: Blank/Familiar) Mixed Design  N = 68 (17 assigned randomly to each Inf Type/Predicate group)  Probability Instructions (Prior Discourse):  “I want you to imagine that you are helping Mr A, who is employed at a bird sanctuary, ringing all birds there. Mr A tells you that there are 1000 birds in the sanctuary. 10% of all the birds are white. Moreover, 10% of all the birds are swans and half of all the swans are white.”  If you turn the key, your car starts. I now tell you that: Your car starts. How likely is this conclusion?The key has been turned______ (0-100)

15. Comp = Major Premise, Incomp = No Major Premise (Enthymeme) Familiar Predicates (Prior Knowledge) *

16. Comp = Major Premise, Incomp = No Major Premise (Enthymeme) Blank Predicates (No Prior Knowledge) ** *

17. Using 0.58 threshold (Oberauer, 2006) calculated number of DAMT responses  2 (1, N = 68) = 10.29, p <.001 Proportion DAMT Response (0.58 Threshold)

18. DAMT Index = (DA + MT) – (MP + AC) (like Pollard Indices in Selecton task) Need to do model fits for individual Ps DAMT Index (on Thresholded Data)

19. Conclusions  Problems?  Task Demands? Prob instructions present in all 4 conditions, mainly affected Enthymeme/Blank  Build up discourse model, like an MM and annotated? Why MT but not MP?  Inductive Strength  Ps will use from information in prior discourse, as long as no prior knowledge, no explicit conditional takes priority  Most information used enthymematically: birds fly, bears have four legs, polar bears are white etc. Theories of reasoning must account for enthymematic reasoning as well, if not better, than explicit reasoning