ict2012 birkbeck — people inductively reason causality by calculating the probability of de...

People inductively reason causality by calculating the probability of

DeFinetti’s biconditonal event –

The pARIs rule, rarity assumption, and equiprobability

Junki Yokokawa and Tatsuji TakahashiJul 6th, 2012

Birkbeck, Univ. of London, UK

1

Summary★ Our intuition for generative causality from co-

occurrence data is the probability of biconditional event (or defective biconditional).

★ In causal induction, biconditional event focuses on rare events and neglects abundant events, in the uncertain world.★ pARIs: proportion of assumed-to-be rare instances

★ Biconditional event is turning out to have strong normative nature and theoretical grounds, so possibly will be proven to be normative as well.

2

Overview★ DeFinetti's biconditional event and new paradigm

psychology of reasoning.★ biconditional event in causal induction:

★ pARIs (proportion of assumed-to-be rare instances)

★ Meta-analysis to confirm the validity of pARIs★ Three experiments to give candidate rationales to

pARIs★ Theoretical background and connections to other

areas, such as:★ Developmental study of conditionals by Gauffroy and

Barouillet (2009),

★ Amos Tversky's study of similarity (1977), and so on

3

toc

★ conditional and biconditional event★ biconditional event in causal induction: pARIs

(proportion of assumed-to-be rare instances)

★ Meta-analysis★ Three experiments★ Theoretical background

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5

de Finetti's conditional event

★ The probability of "if p then q" is the conditional probability P(q|p).★ It neglects not-p cases.

★ "q|p" is itself a (conditional) event.

V: void case

material conditional conditional biconditionalconditional event event event

p q p⊃q q|p p|q p⟛qT T T T T TT F F F V FF T T V F FF F T V V V

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Causal induction

★ Diagnostic:★ Example: we often want to know the cause of a health

problem.

★ I sometimes have stiff shoulders and a headache. What's the cause? How about coffee?★ How frequently I got a headache after having a cup of

coffee? ...

8

Causal induction experiment

Stimulus presentation: a pair of two kinds of pictures illustrating the presence and absence of cause and effect, at left and right, respectively

Response: participants evaluate the causal intensity they felt from 0 to 100, using a slider

E ¬EC a b

¬C c d9

Causal (intensity) induction

★ Here, not structure but intensity★ Two phases of causal induction (Hattori & Oaksford 2007)

★ Phase 1: observational (statistical)

★ Phase 2: interventional (experimental)

★ We focus on causal induction of the phase 1 for generative cause because preventive causes are confusing and hard to treat especially in the observation phase (Hattori & Oaksford, 2007).

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Causal Induction★ Here we study about the intensity.★ Recent studies emphasize the structure (Bayes

network topology) rather than the intensity (node weight)

★ But not about "structure vs. intensity" or one's ascendancy.

★ Many problems about intensity remain untouched.★ Why ∆P doesn't fit the data?

★ Structure and intensity, mutual relationship.★ In an unknown situation, intensity is what matters

since structure is not known.11

The pARIs rule★ The frequency information of rare instances

conveys more information than abundant instances (rational analysis and rarity assumption, see esp. McKenzie 2007).

★ Because of the frame problem-like aspect, the d-cell information can be unreliable (depends strongly on how we frame and count).

★ Hence we calculate the causal intensity only by the proportion of assumed-to-be rare instances (pARIs)★ named after pCI: proportion of confirmatory

instances, White 2003.

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Rarity assumption★ We assume the effect in focus and the candidate

cause to be rare: P(C) and P(E) to be small.★ Originally in Oaksford & Chater, 1994,

★ then in Hattori & Oaksford, 2007, McKenzie 2007, in the study of causal induction

★ C and E to take small proportion in U.

C Eb ca

d

U

extreme rarity

!P = P (E|C)! P (E|¬C) (1)

!P = P (E|C)! P (E|¬C) (2)

!P = P (E|C)! P (E|¬C) =ad! bc

(a+ b)(c+ d)(3)

!P = P (E|C)! P (E|¬C) =ad! bc

(a+ b)(c+ d)(4)

PowerPC =!P

1! P (E|¬C)(5)

PowerPC =!P

1! P (E|¬C)(6)

PowerPC =!P

1! P (E|¬C)=

ad! bc

(a+ b)d(7)

PowerPC =!P

1! P (E|¬C)=

ad! bc

(a+ b)d(8)

H =!P (E|C)P (C|E) (9)

H =!P (E|C)P (C|E) (10)

H =!

P (E|C)P (C|E) =a!

(a+ b)(a+ c)(11)

H =!

P (E|C)P (C|E) =a!

(a+ b)(a+ c)(12)

limd!"

! =!P (E|C)P (C|E) = H (13)

limd!"

! =!P (E|C)P (C|E) = H (14)

1

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The pARIs rule★ C and E are both assumed to be rare (P(C) and

P(E) low)★ pARIs = proportion of assumed-to-be rare

instances (a, b, and c).

C Eb ca d

U

pARIs = P(C iff E) = P(C and E | C or E)P(C and E | C or E)P(C and E | C or E)

=P(C and E)

=a

=P(C or E)

=a+b+c

E -EC a b-C c d

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Data-fit of pARIs and PowerPC

0.0 0.2 0.4 0.6 0.8 1.00

20

40

60

80

100

Model prediction

Humanrating

AS95

0.0 0.2 0.4 0.6 0.8 1.00

20

40

60

80

100

Model prediction

Humanrating

BCC03exp1generative

0.0 0.2 0.4 0.6 0.8 1.00

20

40

60

80

100

Model prediction

Humanrating

BCC03exp3

0.0 0.2 0.4 0.6 0.8 1.00

20

40

60

80

100

Model prediction

Humanrating

H03

0.0 0.2 0.4 0.6 0.8 1.00

20

40

60

80

100

Model prediction

Humanrating

H06

0.0 0.2 0.4 0.6 0.8 1.00

20

40

60

80

100

Model prediction

Humanrating

LS00exp123

0.0 0.2 0.4 0.6 0.8 1.00

20

40

60

80

100

Model prediction

Humanrating

W03JEPexp2

0.0 0.2 0.4 0.6 0.8 1.00

20

40

60

80

100

Model prediction

Humanrating

W03JEPexp6

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Meta-analysis★ Fit with experiments (the same as Hattori & Oaksford, 2007)

★ pARIs fits the data set with the lowest correlation r < 0.89, the highest average correlation in almost all the data, and the smallest average error.

experiment \ model pARIs DFH PowerPC ∆P Phi P(E|C) P(C|E) pCIAS95 0.94 0.95 0.95 0.88 0.89 0.91 0.76 0.87

BCC03: exp1 0.98 0.97 0.89 0.92 0.91 0.82 0.51 0.92BCC03: exp3 0.99 0.99 0.98 0.93 0.93 0.95 0.88 0.93

H03 0.99 0.98 -0.09 0.01 0.70 -0.01 0.98 0.40H06 0.97 0.96 0.74 0.71 0.71 0.89 0.58 0.70

LS00 0.93 0.95 0.86 0.83 0.84 0.58 0.34 0.83W03.2 0.90 0.85 0.44 0.29 0.55 0.47 0.18 0.77W03.6 0.93 0.90 0.46 0.46 0.46 0.77 0.56 0.54

average r 0.95 0.94 0.65 0.63 0.75 0.67 0.60 0.75average error 11.97 18.48 33.39 24.30 27.18 27.78 24.75 29.93

Values other than in error row are correlation coefficient r.

best　next best　bad　otherwise

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-1.75

0

1.75

3.50

5.25

7.00

0.94 0.95 0.95 0.88 0.89 0.91 0.76 0.87

0.98 0.97 0.89 0.92 0.91 0.82 0.51 0.92

0.99 0.99 0.98 0.93 0.93 0.950.88

0.93

0.99 0.98

-0.09

0.01 0.70

-0.01

0.98 0.40

0.97 0.96

0.74 0.710.71

0.89 0.58 0.70

0.93 0.95

0.86 0.830.84

0.58 0.34 0.83

0.90 0.85

0.44 0.290.55

0.47 0.180.77

pARIs DFH PowerPC ΔP Phi P(E|C) P(C|E) pCI

Cor

AS95 BCC03exp1 BCC03exp3 H03 H06 LS00 W03.2

0

75

150

225

300

correlation

error19




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Experiments★ Experiment 1.1★ To test the validity of rarity assumption in ordinary

2x2 causal induction

★ Experiment 1.2★ To test the validity of rarity assumption in 3x2 causal

induction★ Difference in the cognition between rare events (a, b, and

c-type) and non-rare d-type event, people just vaguely recognize and memorize the occurrence of d-type events.

★ Experiment 2★ Rarity and presence-absence (yes-no)

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Experiment 1.1: c and d in 2x2 table

★ 27 undergraduates, 9 stimuli.★ p: to give artificial diet to your

horse, q: your horse gets ill.★ After the presentation of

(a,b,c,d), participants are asked the causal intensity and then the frequency of c- and d-type event.

stim. a b c d1 1 9 1 92 1 9 5 53 1 9 9 14 5 5 1 95 5 5 5 56 5 5 9 17 9 1 1 98 9 1 5 59 9 1 9 1

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Result of exp. 1.1

★ Participants' estimation of c and d occurrence was basically faithful.

★ d is estimated moderately than the real stimuli.

stim. a b c d1 1 9 1 92 1 9 5 53 1 9 9 14 5 5 1 95 5 5 5 56 5 5 9 17 9 1 1 98 9 1 5 59 9 1 9 1

02468

10

1 2 3 4 5 6 7 8 9

c cell

real c estimated c

0

3

5

8

10

1 2 3 4 5 6 7 8 9

d cell

real d estimated d

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Experiment 1.2:c and d in 3x2 table

★ 54 undergraduates, 2 stimuli.

★ As a medical scientist, p: to give a medicine (three types, p1, p2 and p3) to a patient q: the patient develops antibodies against a virus.

★ After the presentation of six kinds of events, participants are asked the causal intensity of p1 to q and p2 to q, and then the frequency of c- and d-type event.

stimulus A q not-qp1 6 4p2 9 1p3 2 8

stimulus B q not-qp1 5 5p2 8 2p3 1 9

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Experiment 1.2:c and d in 3x2 table

★ Each participant estimates the intensity of causal relationship from p1 to q.

★ Then asked the value of c, as "How often q happened in the absence of p1?." The given value of c is 9+2=11.

stimulus A q not-qp1 6 4p2 9 1p3 2 8

a b

c dfocus

+ +

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Exp. 1.2: Result

0

3

7

10

13

1 2 3 4

c cell

real c estimated c

0

4

7

11

14

1 2 3 4

d cell

real d estimated d

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Theoretical background of biconditional event and pARIs

★ Angelo Gilio is studying biconditional event as with the notion of quasi-conjunction.

★ There are some equivalent indices.

★ Possibly it contributes to the maximization of information acquisition as in Wason selection task by Oaksford & Chater 1994.

★ Computer simulations shows that pARIs is very efficient in inferring the correlation of the population from a small sample set, with the highest reliability and precision.

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Simulation

★ pARIs enables the reliable and precise grasp of population correlation with a very small sample

0"

0.1"

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1" 2" 3" 4" 5" 6" 7" 8" 9" 10" 11" 12" 13" 14" 15" 16" 17" 18" 19" 20" 21" 22" 23" 24" 25" 26" 27" 28" 29"

pARIs"

DFH"

Delta"P"

Phi"

PowerPC"

Correlation of the population is 0.2

0"

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1" 2" 3" 4" 5" 6" 7" 8" 9" 10" 11" 12" 13" 14" 15" 16" 17" 18" 19" 20" 21" 22" 23" 24" 25" 26" 27" 28" 29" 30" 31"

pARIs"

DFH"

Delta"P"

Phi"

PowerPC"

mean value through MC sim.

sd value29

Indices equivalent to the probability of biconditional event

★ Psychology★ Tversky index of similarity, Tversky (1977)

★ Asymmetric similarity measure comparing a variant to a prototype. Also in: Gregson (1975) and Sjöberg (1972)

★ Mathematics, machine learning and statistics: ★ Probable equivalence, or the probabilistic

indentity of two sets A and B, P(A=B) by Kosko (2004)

★ Tanimoto similarity coefficient

★ Jaccard similarity measure

30

Tversky indexPsychological Review

J Copyright © 1977 C_? by the American Psychological Association, Inc.

V O L U M E 84 N U M B E R 4 J U L Y 1 9 7 7

Features of Similarity

Amos TverskyHebrew UniversityJerusalem, Israel

The metric and dimensional assumptions that underlie the geometric represen-tation of similarity are questioned on both theoretical and empirical grounds.A new set-theoretical approach to similarity is developed in which objects arerepresented as collections of features, and similarity is described as a feature-matching process. Specifically, a set of qualitative assumptions is shown toimply the contrast model, which expresses the similarity between objects as alinear combination of the measures of their common and distinctive features.Several predictions of the contrast model are tested in studies of similarity withboth semantic and perceptual stimuli. The model is used to uncover, analyze,and explain a variety of empirical phenomena such as the role of common anddistinctive features, the relations between judgments of similarity and differ-ence, the presence of asymmetric similarities, and the effects of context onjudgments of similarity. The contrast model generalizes standard representa-tions of similarity data in terms of clusters and trees. It is also used to analyzethe relations of prototypicality and family resemblance.

Similarity plays a fundamental role in errors of substitution, and correlation betweentheories of knowledge and behavior. It serves occurrences. Analyses of these data attempt toas an organizing principle by which individuals explain the observed similarity relations andclassify objects, form concepts, and make gen- to capture the underlying structure of the ob-eralizations. Indeed, the concept of similarity jects under study.is ubiquitous in psychological theory. It under- The theoretical analysis of similarity rela-lies the accounts of stimulus and response tions has been dominated by geometricgeneralization in learning, it is employed to models. These models represent objects asexplain errors in memory and pattern recogni- points in some coordinate space such that thetion, and it is central to the analysis of con- observed dissimilarities between objects cor-notative meaning. respond to the metric distances between the

Similarity or dissimilarity data appear in respective points. Practically all analyses ofdifferent forms: ratings of pairs, sorting of proximity data have been metric in nature,objects, communality between associations, although some (e.g., hierarchical clustering)

yield tree-like structures rather than dimen-This paper benefited from fruitful discussions with sionallv. organized spaces. However, most

Y. Cohen, I. Gati, D. Kahneman, L. Sjeberg, and theoretical and empirical analyses of similarityS. Sattath. assume that objects can be adequately repre-

Requests for reprints should be sent to Amos Tversky, , , . . . ,. , iDepartment of Psychology, Hebrew University, sented as points m some coordinate space andJerusalem, Israel. that dissimilarity behaves like a metric dis-

327

330 AMOS TVERSKY

A-B

APIB

B - A

Figure 1. A graphical illustration of the relation betweentwo feature sets.

lection of features is viewed as a product of aprior process of extraction and compilation.

Second, the term, feature usually denotes thevalue of a binary variable (e.g., voiced vs.voiceless consonants) or the value of a nominalvariable (e.g., eye color). Feature representa-tions, however, are not restricted to binary ornominal variables; they are also applicable toordinal or cardinal variables (i.e., dimensions).A series of tones that differ only in loudness,for example, could be represented as a sequenceof nested sets where the feature set associatedwith each tone is included in the feature setsassociated with louder tones. Such a represen-tation is isomorphic to a directional unidimen-sional structure. A nondirectional unidimen-sional structure (e.g., a series of tones thatdiffer only in pitch) could be represented by achain of overlapping sets. The set-theoreticalrepresentation of qualitative and quantitativedimensions has been investigated by Restle(1959).

Let s(a,b) be a measure of the similarity ofa to b denned for all distinct a, b in A. Thescale s is treated as an ordinal measure ofsimilarity. That is, s(a,b) > s(c,d) means thata is more similar to b than c is to d. Thepresent theory is based on the followingassumptions.

1. Matching:s(a,b) = F(AH B, A - B, B - A).

The similarity of a to b is expressed as afunction F of three arguments: AHB, thefeatures that are common to both a and b;A — B, the features that belong to a but notto b; B — A, the features that belong to b but

not to a. A schematic illustration of thesecomponents is presented in Figure 1.

2. Monotonicity:s(a,b) > s(a,c)

whenever, A - B C A - C ,

andB - A C C - A.

Moreover, the inequality is strict whenevereither inclusion is proper.

That is, similarity increases with additionof common features and/or deletion of distinc-tive features (i.e., features that belong to oneobject but not to the other). The monotonicityaxiom can be readily illustrated with blockletters if we identify their features with thecomponent (straight) lines. Under this as-sumption, E should be more similar to F thanto I because E and F have more commonfeatures than E and I. Furthermore, I shouldbe more similar to F than to E because I andF have fewer distinctive features than I and E.

Any function F satisfying Assumptions 1and 2 is called a matching function. It measuresthe degree to which two objects—viewed assets of features—match each other. In thepresent theory, the assessment of similarity isdescribed as a feature-matching process. It isformulated, therefore, in terms of the set-theoretical notion of a matching functionrather than in terms of the geometric conceptof distance.

In order to determine the functional formof the matching function, additional assump-tions about the similarity ordering are intro-duced. The major assumption of the theory(independence) is presented next; the remain-ing assumptions and the proof of the represen-tation theorem are presented in the Appendix.Readers who are less interested in formaltheory can skim or skip the following para-graphs up to the discussion of the representa-tion theorem.

Let $ denote the set of all features associatedwith the objects of A, and let X,Y,Z,... etc.denote collections of features (i.e., subsets of$). The expression F(X,Y,Z) is defined when-ever there exists a, b in A such that A C\ B = X,

FEATURES OF SIMILARITY 333

matching function of interest is the ratio model,_

. , - f (AnB)+af(A-B)+^f(B-A)'« , /3>0,

where similarity is normalized so that S liesbetween 0 and 1. The ratio model generalizesseveral set-theoretical models of similarityproposed in the literature. If a = ft = 1, S (a,b)reduces to f (A H B)/f (A (J B) (see Gregson,1975, and Sjoberg, 1972). If « = ft = i S(a,b)equals 2f(AH B)/(f(A) + f(B)) (see Eisler &Ekman, 1959). If a = 1 and ft = 0, S(a,b) re-duces to f (AH B)/f (A) (see Bush & Hosteller,1951). The present framework, therefore, en-compasses a wide variety of similarity modelsthat differ in the form of the matching functionF and in the weights assigned to its arguments.

In order to apply and test the present theoryin any particular domain, some assumptionsabout the respective feature structure must bemade. If the features associated with eachobject are explicitly specified, we can test theaxioms of the theory directly and scale thefeatures according to the contrast model. Thisapproach, however, is generally limited tostimuli (e.g., schematic faces, letters, stringsof symbols) that are constructed from a fixedfeature set. If the features associated with theobjects under study cannot be readily speci-fied, as is often the case with natural stimuli,we can still test several predictions of thecontrast model ̂ which involve only generalqualitative assumptions about the featurestructure of the objects. Both approaches wereemployed in a series of experiments conductedby Itamar Gati and the present author. Thefollowing three sections review and discuss ourmain findings, focusing primarily on the testof qualitative predictions. A more detailed de-scription of the stimuli and the data are pre-sented in Tversky and Gati (in press).

Asymmetry and FocusAccording to the present analysis, similarity

is not necessarily a symmetric relation. Indeed,it follows readily (from either the contrast orthe ratio model) thats(a,b) = s(b,a) iff of (A - B) + 0f (B - A)

= of (B - A) + /3f (A - B)iff (a - /9)f (A - B) = (a - 0)f (B - A).

Hence, s(a,b) = s(b,a) if either a = /8, orf(A - B) = f(B - A), which implies f (A) =f(B), provided feature additivity holds. Thus,symmetry holds whenever the objects are equalin measure (f(A) = f(B)) or the task is non-directional (a = /3). To interpret the lattercondition, compare the following two forms:

(i). Assess the degree to which a and b aresimilar to each other.

(ii). Assess the degree to which a is similartob.

In (i), the task is formulated in a nondirectionalfashion; hence it is expected that a = ft ands(a,b) = s(b,a). In (ii), on the other hand, thetask is directional, and hence a and /3 maydiffer and symmetry need not hold.

If s(a,b) is interpreted as the degree towhich a is similar to b, then a is the subjectof the comparison and b is the referent. Insuch a task, one naturally focuses on the sub-ject of the comparison. Hence, the features ofthe subject are weighted more heavily thanthe features of the referent (i.e., a > 0). Con-sequently, similarity is reduced more by thedistinctive features of the subject than by thedistinctive features of the referent. It followsreadily that whenever a > ft,

s(a,b) > s(b,a) iff f(B) > f(A).Thus, the focusing hypothesis (i.e., a > /3)implies that the direction of asymmetry isdetermined by the relative salience of thestimuli so that the less salient stimulus is moresimilar to the salient stimulus than vice versa.In particular, the variant is more similar tothe prototype than the prototype is to thevariant, because the prototype is generallymore salient than the variant.

Similarity of CountriesTwenty-one pairs of countries served as

stimuli. The pairs were constructed so that oneelement was more prominent than the other(e.g., Red China-North Vietnam, USA-Mexico,Belgium-Luxemburg). To verify this relation,we asked a group of 69 subjects2 to select in

2 The subjects in all out experiments were Israelicollege students, ages 18-28. The material was pre-sented in booklets and administered in a group setting.

31

Biconditional event

★ Developmental★ Merely transient in the

process of narrowing the scope, between conjunctive and conditional? (Gauffroy and Barouillet, 2009)

★ Probably there are theoretical reasons for the dominance of defective biconditional (biconditional event).

tween grade 9 and adults (p < .001). This age-related effect was strongly reduced for the strong rela-tions, F(3, 164) = 2.38, p = .07, g2

p ! :04, resulting in a significant age " type of relation interaction,F(3, 164) = 7.17, p < .001, g2

p ! :11. As in Experiment 1, this interaction was no longer significant whenadults were discarded from the analysis.

Response patterns analysis

We categorized the response patterns for the eight causal conditionals on the basis of the fourinterpretations already observed in Experiment 1, that is conjunctive, defective biconditional, defec-tive conditional and matching interpretations (Fig. 3). The matching pattern was produced only byyounger participants (third graders), explaining the age-related increase in ‘‘false” responses on thep :q case. First of all, as we predicted, conjunctive response patterns predominated in younger

0%

10%

20%

30%

40%

50%

60%

70%

80%

90%

3 6 9 adultsGrades

Weak

Conjunctive Def Bicond Def Cond

MPOther

0%

10%

20%

30%

40%

50%

60%

70%

80%

90%

3 6 9 adultsGrades

Strong

Conjunctive Def Bicond Def Cond

MPOther

Fig. 3. Percent of response patterns categorized as conjunctive, defective biconditional (Def Bicond), defective conditional (DefCond), matching (MP), and others as a function of grades for strong and weak causal conditionals in Experiment 2.

C. Gauffroy, P. Barrouillet / Developmental Review 29 (2009) 249–282 263

Gauffroy and Barouillet, 200932

Conclusion★ Our intuition for generative causality from co-

occurrence data is the probability of biconditional event (or defective biconditional).

★ In causal induction, biconditional event focuses on rare events and neglects abundant events, in the uncertain world.★ pARIs: proportion of assumed-to-be rare instances

★ Biconditional event is turning out to have strong normative nature and theoretical grounds, so possibly will be proven to be normative as well.

33

ict2012 birkbeck — people inductively reason causality by calculating the probability of de...

Spiritual