The psychology of indicative conditionals and conditional bets:

Further developments

David OverPsychology, Durham University

Collaborators at the Jean Nicod and Paris VIII:Jean Baratgin, Guy Politzer, Jean-Louis Stilgenbauer,

& Thomas Charreau

General outline

• Will recall two talks at ProbNet10, Salzburg, 27 Feb., 2010

• My talk last year, “Indicative conditionals and conditional bets”.

• Peter Milne, “Conditionals, conditional bets, and conditional events, or things fall apart”.

• At this meeting, I will introduce further studies of conditional bets and offer some reflections on the above talks.

New paradigm psychology of reasoning

• The new Bayesian / probabilistic paradigm in the psychology of reasoning rests on two broad types of experimental results.

• People judge that the probability of the indicative conditional “if A then B”, P(if A then B), is the conditional probability, P(B|A).

• People judge that “if A then B” is true when “A &B”, false when “A & not-B”, and neither true nor false when “not-A”.

The Ramsey test and the de Finetti table

• The new paradigm rests on the Ramsey test, which implies that P(if A then B) = P(B|A).

• It also rests on the de Finetti table, which implies that “if A then B” is true when “A &B”, false when “A & not-B”, and neither true nor false when “not-A”.

• Ramsey and de Finetti imply that indicative conditionals are closely related to conditional bets.

Relevant psychological work

• Recent psychological studies and theories that are relevant to this talk: Evans & Over, Douven & Verbrugge, Oaksford & Chater, Pfeifer & Kleiter , and Politzer, Over, & Baratgin.

The Ramsey test

Ramsey (1931): People could judge “if p then q” by “...adding p hypothetically to their stock of knowledge …” They would thus fix “...their degrees of belief in q given p…”, P(q|p).

In Ramsey’s original example, the two people were arguing about “if p then q”, and so there could be a winner and a loser in the debate.

Probabilistic evaluation task

Type of task first studied (Evans, Handley, & Over, 2003)

A pack contains cards which are either blue or yellow and have either a triangle or a circle printed on them. In total there are:

10 blue triangles

40 blue circles

40 yellow triangles

10 yellow circles

How likely is the following singular conditional to be true of a card

drawn at random from the pack?

“If the card is blue then it has a circle printed on it.”

Early experiments on the probability conditional (Evans et al., 2003)

A minority of people judged:

P(if A then B) = P(A & B)

The majority of people judged:

P(if A then B) = P(B|A)

Almost no one judged:

P(if A then B) = P(not-A or B)

“Causal” conditionals: Over, Hadjichristidis, Evans, Handley, & Sloman (2007)

Consider “causal” conditionals:

“If global warming continues, London will be flooded.”

Given such conditionals, participants judge that:

P(if G then F) = P(F|G)

Note that arguments about such conditionals can easily

slip into conditional bets: “If G then I bet that F”.

The “defective” truth table

The “defective” truth table discovered by Wason (1966):

A pack contains cards which are either blue or yellow and have either a triangle or a circle printed on them., e.g. there may be:

10 blue triangles

40 blue circles

40 yellow triangles

10 yellow circles

One card after another is to be drawn from the pack. For each

card, participants are asked to evaluate the singular conditional:

“If the card is blue then it has a circle printed on it.”

The results of experiments on the

“defective” truth table

“If the card is blue then it has a circle printed on it.”

Participants tend to say that a blue card with a circle

on it makes the above conditional true, a blue card

with a triangle on it makes the above false, and that

yellow cards are “irrelevant”.

The “defective” truth table should be called

the de Finetti table

“If the card is blue then it has a circle printed on it.”

De Finetti (1937) held that a blue circle card makes the

above conditional true, and a blue triangle card makes

it false, but a yellow card makes it “void”.

The de Finetti table for the conditional bet

“If the card is blue then it has a circle printed on it.”

Suppose Marie bets Pierre that the above conditionalholds. Marie will win the bet when the card is blue andhas a circle on it and lose the bet when the card is blueand has a triangle on it. The bet will be “void” whenthe card is yellow.

The (restricted) de Finetti table for indicative conditionals and conditional bets

T = true, F = false, W = win, L = lose

A B If A then B

T T T (W)

T F F (L)

F T Void

F F Void

Indicative conditionals and conditional bets

• Ramsey and de Finetti imply that there should be a close relation between the indicative conditional and the conditional bet.

• Politzer, Over, & Baratgin (2010) tried to test this relation.

This drawing represents chips

● ● ●

■ ■ ■ ■

A chip is chosen at random. Consider the following sentence:

If the chip is square then it is black.

What are the chances the sentence is true?

Politzer , Over, & Baratgin (2010): Indicative conditionals

This drawing represents chips

● ● ●

■ ■ ■ ■

A chip is chosen at random. Marie bets Pierre 1 euro that:

If the chip is square then it is black.

What are the chances that Marie wins her bet?

Politzer , Over, & Baratgin (2010): Conditional bets

The results of the betting experiment

Indicative conditionals and conditional bets are close

in people’s judgments, as both Ramsey and de Finetti

argued on theoretical and normative grounds.

The majority judged the chances that “if S then B”

is true and that “Marie wins her bet” to be P(B|S).

A minority judged the chances to be P(S & B). Almost

no one judged these chances to be P(not-S or B).

Fugard, Pfeifer, Mayerhofer, & Kleiter (2011) find that participants who interpret P(if A then B) as P(A & B) tend to shift to P(B|A) as they do more frequency tasks.

Gauffroy & Barrouillet (2009) found a developmental trend. Young children tend to interpret P(if A then B) as P(A & B), but shift to the dominant adult response of P(B|A) as they get older.

Some recent striking results

People who give P(A & B) as the answer to a question about the probability of truth of “if A then B”, or of winning a bet on it, are of relatively low cognitive ability.

People who give P(B|A) as the answer to a question about the probability of truth of “if A then B”, or of winning a bet on it, are of relatively high cognitive ability.

Cognitive ability(Evans, Handley, Neilens, & Over, 2007; Politzer et al.,

2010)

Given the de Finetti table, why do we not specify that P(A & B) is the correct answer to a question about the probability of truth of “if A then B” or the chances of winning the bet “if A then I bet B”?

A question the truth of a conditional presupposes that it makes an assertion that is true or false, and a question about the chances of winning a conditional bet presupposes that there is a bet. Thus in both cases the answer is P(A&B|A) = P(B|A).

A question about truth and winning

The expected value of a fair bet is 0. Our conditional bet would be fair if P(B|S) = 0.5, but in fact P(B|S) = 0.75. The expected value of Marie’s bet for her is:

P(S & B)(1) + P(S & not-B)(-1) + P(not-S)(0)

(.43)(1) + (.14)(-1) + (.43)(0) = 29 cents

For this bet to be fair, the odds should be 3 to 1, which corresponds to P(B|A) = 0.75.

Note that Marie’s bet is not fair

Further studies

• Should investigate conditional bets more fully, with a range of probabilities and expected values.

• Should give up assumption that we always know for sure whether A and B hold when we are judging the truth value of “if A then B”.

New study with Thomas Charreau

• Allowed us to investigate more fully probabilities and expected values.

• The participants were 19 Paris V students. They were asked to make choices between urns.

• The question did not refer to the probability of truth or of winning.

The new design: The urns

Urn B

● ● ● ●■ ■ ■ ■ ■

Two urns containing four kinds of chips: black circles (●), white circles (●), black squares (■) and white squares (■). A mechanical system draws randomly one chip from one of the two urns:

Urn A

● ● ● ●

■ ■ ■

The urn selection

Mary and Peter, two friends who are both honest, decide to gamble. Mary proposes to Peter the following bet, If the chip is a circle then I bet you that it is black. Peter accepts the bet and they each put one euro on the table. Which urn maximizes the chances that Mary will pocket the two euros? Circle one answer.

Urn A Urn B Urns A and B are equal

The truth table component

Now the automatic system draws a chip that is a white square. Here is that chip: ■What is the most likely outcome ? Circle one answer.

1) Mary will pocket the two euros.

2) Peter will pocket the two euros.

3) Mary will get her euro back and Peter will get his euro back.

The items (1st group)

Item 1 Item 2 Item 3 Item 4

Conj. eq A eq A

Cond. prob. B eq A eq

Exp. ut. B A A eq

Mat. imp. B B A B

Mat. bicond. A A eq A

Max. numb. eq eq eq eq

The items (2nd group)

Item 5 Item 6 Item 7 Item 8

Conj. eq A eq B

Cond. prob. A eq B eq

Exp. ut. A A A eq

Mat. imp. A B B A

Mat. bicond. eq A A A

Max. numb. eq eq eq eq

Item 2

Urn B ● ● ● ●

■ ■ ■ ■ ■

Mary proposes the bet to Peter, “If the chip is a circle

then I bet you it is black.” Which urn maximizes the

chances that Mary will pocket the two euros? P(b|c) =

.75 in both urns, but urn A has the higher expected

value for Mary.

Urn A

● ● ● ●

■ ■ ■

Analysis of data

We computed a compatibility score. Each answer is compatible with one, two, or three different interpretations. If the answer is compatible with just one interpretation, it gets a score of 1. If the answer is compatible with two interpretations, both get a score of 0.5. If the answer is compatible with three interpretations, each gets a score of 0.33.

Results: Urn selection

0

10

20

30

40

50

60

70

80

conj.

cond. prob.

exp. ut.

mat. imp.

bicond

max. numb.

Percentage of compatibility for each interpretation

The truth table task results

• The de Finetti table swamps all other interpretations.

• But note that we are still presenting the participants with a definite result. We say here is that chip: ■

• In many realistic cases, people cannot be sure about which truth table case holds: they are uncertain.

New study with Jean Baratgin, Guy Politzer, and Jean-Louis Stilgenbauer

• Allowed us to investigate tables for connectives where the “third” value is interpreted as uncertainty.

• There is a mechanism in which a chip is dropped and a photo is taken. Filters can be used that prevent the colour or the shape of the chip from being revealed.

• The participants see a “photo”, which can leave them in a state of uncertainty.

36

Conditional question: "If the chip is square then it is black"

If S then B BlackT U F

Square

T

U

F

37

Participants

• 192 French adults volunteered for the experiment. They already held a degree and so were mature students. They were in the second year of a BA program in Psychology at the University of Paris VIII Saint-Denis. They had various socio-professional origins (students, unemployed, employees, workers, executives). They had no specific background in logic or probability theory.

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Design One

• Participants were randomly allocated to two conditions (Standard and Bet). They were required to answer five blocks of nine random questions, each block corresponding to a specific connective (negation, conjunction, conditional, disjunction, and implication) . In each group, participants were randomly allocated to two different block orders:

1. negation, conjunction, conditional, disjunction, and material implication.

2. negation, implication, disjunction, conditional, and conjunction. In what follows, only the conditional will be considered.

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Design

Bet condition Standard condition

Five blocks=five connectives

Negation

I bet that… The chip is square

Conjunction

I bet that... The chip is square and black

Disjunction

I bet that... The chip is square or black or both

Indicative conditional

I bet that... If the chip is square, then it is black

Material Implication

I bet that... There is no falling square that is not black

40

Design

• Each block=nine questions

• Each of the nine questions corresponds to a cell of the truth table of the connective * under review; each block thus yields a complete truth table.

P*Q QT U F

P

T Question 1 Question 2 Question 3

U Question 4 Question 5 Question 6

F Question 7 Question 8 Question 9

For each of the following photographs, indicate whether Pierre's statement is true or false by using the mouse to select the number that coincides with your response.

Marie chooses one chip at random and drops it.

Pierre says:

"If the chip is square then it is black"

Pierre's statement may be true or false. For you to decide, you will have at your disposal photographs taken as just explained.

Photograph 1

"If the chip is square then it is black"

True FalseNeither true nor

false

For each of the following photographs, indicate whether Pierre's statement is true or false by using the mouse to select the number that coincides with your response.

Marie chooses one chip at random and drops it.

Pierre says:

“The chip is square and it is black"

Pierre's statement may be true or false. For you to decide, you will have at your disposal photographs taken as just explained.

Photograph 1

“The chip is square and it is black"

True FalseNeither true nor

false

For each of the following photographs, indicate whether Pierre wins or loses his bet by using the mouse to select the number that coincides with your response.

Marie chooses one chip at random and drops it.

Pierre, who has not seen the chip drop, says:

"I bet that if the chip is square then it is black"

Photograph 1

“I bet that if the chip is square then it is black"

Certainly won

Certainly lost Neither won nor lost

For each of the following photographs, indicate whether Pierre wins or loses his bet by using the mouse to select the number that coincides with your response.

Marie chooses one chip at random and drops it.

Pierre, who has not seen the chip drop, says:

"I bet that the chip is square and it is black"

Photograph 1

“I bet that the chip is square and it is black"

Certainly won

Certainly lost Neither won nor lost

Macro and micro analyses

• For each answer category, we have detailed the specific table that corresponds to the three valued-logic literature.

• The proximity with the specific table is determined with the number of differences.

Example:

Suppose the participant gives the following answer

Macro analysis

Further categorized as the de Finetti conditional with one difference

Micro analysis

Categorized as conditioning

∣ T U F

T T F

U

F U U

Result (macro) for the conditional question: "If the chip is square then it is black"

0,0%

10,0%

20,0%

30,0%

40,0%

50,0%

60,0%

70,0%

Standard 1

Standard 2Bet 1

Bet 2

ConditioningImplicationConjunction EquivalenceOther Undetermined

Order effect: In order 1, the conditional question arrives just after the conjunction question. That seems to increase the conjunction answer to the detriment of the conditioning and implication answers under standard condition and only the implication answer in the bet condition.

In agreement with results of Politzer et al. (2010)- The same three main categories

The significant difference for conditioning answers (when the two orders are pooled together) stems from this order effect.

Micro analysis of conditioning answers

De Finetti (1936)Geach (1949)Reichenbach (1949)Rescher (1962)Morgan (1979)Farrel (1986)Blamey (1989)Mcdermot (1996)

Jeffrey (1963)Farrell (1949)(GNW) Goodman, Nguyen, Walker (1991)

Jeffrey (1963)Adams (1966)Schay (1968)Calabrese (1988)Cantwell (2008)

References ?

0 = 0 difference1 = 1 difference2 = 2 differences

1 2

differences with de Finetti

QuickTime™ et undécompresseur

sont requis pour visionner cette image.

Some reflections on this study

• Will be a challenge comparing the full results, for negation, conjunction, disjunction, and the material conditional, to all the tables that have been given for these connectives.

• Will be able to generalize the technique by having the chip randomly selected from a given frequency distribution and then dropped into the machine.

• Would then be able to ask questions like, “Does seeing the photo make it more or less likely that …?”

Two definitions of validity

• Milne (2010) discussed two types of semantics with two definitions of validity

• There is probability semantics and its probability validity – an inference is valid if and only if the uncertainty of its conclusion cannot be coherently greater than the sum of the uncertainties of its premises.

• There is alternatively the pay-off, or truth value, validity of the (restricted) de Finetti table. An inference is valid if and only if its conclusion is true when all its premises are true and at least one of its premises is false when its conclusion is false .

The definitions produce different results

Milne (2010) pointed to cases in which the two definitions differ. One was hypothetical syllogism / transitivity:

A → B, B → C ├ A → C

Pay-off, truth value validity, defined on the restricted de Finetti table, makes this valid.

But probability validity makes it invalid.

Milne (2010) called for more experiments to see whether people endorse this scheme and which validity definition they conform to in general.

An example

A → B, B → C ├ A → C

If you strike the match it will light. If it lights it is not wet. So if you strike the match it is not wet.

People might sometimes appear to accept transitivity, but that may well be because they represent it as Cut (Pfeifer & Kleiter):

A → B, A & B → C ├ A → C

However, the above example can be modified to appear to put Cut in doubt.

Conclusion

• Milne (2010) was certainly right. More experiments are needed.

• These experiments should be on the general tables that correspond to people’s judgments about uncertainty, the probability judgments they make about premises and conclusions, and which inferences they endorse as valid.