Bet 1 Bet 2 Arun is French Not Arun is French Arun is French Not Arun is French 1 060 060 050 1 050 Total Payout 010 Cr Arun is French 06 ID: 931833
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Slide1
Dutch Books, Coherence and Logical Consistency
Slide2Bet 1
Bet
2Arun is FrenchNot (Arun is French)Arun is FrenchNot (Arun is French)£(1 - 0.60)£(-0.60)£(-0.50)£(1 - 0.50)
Total Payout = £(-0.10)
Cr(
Arun
is French) = 0.6
Cr(Not (
Arun
is French)) = 0.5
Slide3Intuitively Incoherent?
Can
be Dutch Booked?Agents who violate the probability axioms
Slide4Bet 1
Bet
2Arun is FrenchNot (Arun is French)Arun is FrenchNot (Arun is French)£(1 - 0.60)£(-0.60)£(-0.50)£(1 - 0.50)
Total Payout = £(-0.10)
Cr(
Arun
is French) = 0.6
Cr(Not (
Arun
is French)) = 0.5
Slide5Bet 1
Arun
is FrenchNot (Arun is French)£(-0.50 + 1)£(-0.50)Total Payout = £(-0.50)Cr(Arun is French) = 0.5Cr(Not (Arun is French)) = 0.5
Slide6When we say that the agent is
guaranteed
to lose money, what do we mean?
Slide7An agent who has that credence function, and so accepts those bets, will lose money – no matter what the rest of the world is like.
When we say that the agent is
guaranteed to lose money, what do we mean?
Slide8Bet 1
Bet
2Arun is FrenchNot (Arun is French)Arun is FrenchNot (Arun is French)£(1 - 0.60)£(-0.60)£(-0.50)£(1 - 0.50)
Total Payout = £(-0.10)
Cr(
Arun
is French) = 0.6
Cr(Not (
Arun
is French)) = 0.5
Slide9
Claim
Cr(.)Arun is French0.6Not (Arun is French)0.5...... Just from this information alone, about the agent’s credence function, the bookie can design a set of bets that she knows will lose the agent money.
Slide10Cr(
Arun
is French) = 0.6Cr(Cr(Arun is French) =0.6) = 0.9
Slide11
Claim
Cr(.)Arun is French0.6Cr(Arun is French)=0.60.9...... Just from this information alone, about the agent’s credence function, the bookie can design a set of bets that she knows will lose the agent money.
Slide12
Claim
Cr(.)Arun is French0.6Cr(Arun is French)=0.60.9...... Bet 1Cr(Arun is French) = 0.6Not (Cr(Arun is French) =0.6))
£(0.90-1)
£(0.90)
Total Payout = £(-0.10)
Slide13Intuitively Irrational?
Can
be Dutch Booked?Agents who violate the probability axiomsAgents who are uncertain of their own credence function
Slide14Incoherence in Outright Beliefs
An agent has incoherent outright beliefs, iff the set of contents of her beliefs is
logically inconsistent. A set of sentences is logically inconsistent iff there is no interpretation under which those sentences are all true. All fish are pencilsAll whales are mammalsx(PxQx)
Slide15Arun
is French
Barack Obama is AmericanAgent not classed as incoherent.
Slide16Arun
is French
It’s not the case that Arun is FrenchPaPaAgent classed as incoherent.
Slide17Arun
is French
I don’t believe that Arun is FrenchIt’s not the case that Rover believes that Barack Obama is AmericanBarack Obama is AmericanIt’s not the case that I disbelieve that Barack Obama is AmericanAgent not classed as incoherent.
Slide18In every possible world where the agent has that credence function (and so accepts those bets), she will be lose money.
When we say that the agent is
guaranteed to lose money, what do we mean?Under any interpretation of the relevant claims, the agent will lose money.
Slide19
Arun
is French0.5Anna is short0.6Anna is not short0.4 Barack Obama is American
Slide20Bet 1
Bet
2Arun is FrenchNot (Arun is French)Arun is FrenchNot (Arun is French)£(1 - 0.60)£(-0.60)£(-0.50)£(1 - 0.50)
Total Payout = £(-0.10)
Cr(
Arun
is French) = 0.6
Cr(Not (
Arun
is French)) = 0.5
Barack Obama is American
Barack Obama is American
Not (Barack Obama is American)
Not (Barack Obama is American)
Slide21Cr(
Arun
is French) = 0.6Cr(Cr(Arun is French) =0.6) = 0.9Bet 1Cr(Arun is French) = 0.6Not (Cr(Arun is French) =0.6))£(0.90-1)£(0.90)Total Payout = £(-0.10)Total Payout = £0.90HCr(Arun is French) = 0.6Not (HCr(Arun is French) = 0.6)
Slide22Intuitively Irrational?
Can
be Dutch Booked?Agents who violate the probability axiomsAgents who are uncertain of their own credence function
Slide23Intuitively Irrational?
Can
be Dutch Booked?Agents who violate the probability axiomsAgents who are uncertain of their own credence functionAgents who violate the Reflection PrincipleAgents who violate Conditionalization
(diachronic)
(diachronic)
Slide24t0
t1
Cr0(D) = 0.7Cr1(D) = 0.9Cr0(Cr1(D) = 0.9) = 0.2Cr0(D/Cr1(D) = 0.9) = 0.7Cr1 (D) = 0.9 Not (Cr1 (D) = 0.9)£(-0.04+0.20)£ -0.04Cr1 (D) = 0.9 & D
(Cr
1 (D) = 0.9) & not D
Not
(Cr
1
(D) = 0.9)
£ (0.70 -1)
£ (0.70)
£0
D
Not(D)
£ (-0.90
+ 1)
£ -0.90
£-0.04
To be offered at t
1
iff
Cr
1
(D) = 0.9
Slide25t0
t1
Cr0(D) = 0.7Cr0(Cr1(D) = 0.9) = 0.2Cr0(D/Cr1(D) = 0.9) = 0.7Cr1 (D) = 0.9 Not (Cr1 (D) = 0.9)£(-0.04+0.20)£ -0.04Cr1 (D) = 0.9 & D (Cr1 (D) = 0.9) & not DNot (Cr1 (D) = 0.9)
£ (0.70 -1)
£(0.70)
£0
D
Not(D)
£ (-0.90
+ 1)
£-0.90
£-0.04
To be offered at t
1
iff
Cr
1
(D) = 0.9
HCr
1
(D) = 0.9
Not (HCr
1
(D) = 0.9)
(HCr
1
(D) = 0.9) & D
(HCr
1
(D) = 0.9) & not-D
Not (HCr
1
(D) = 0.9
Slide26Intuitively Irrational?
Can
be Dutch Booked?Agents who violate the probability axiomsAgents who are uncertain of their own credence functionAgents who violate the Reflection PrincipleAgents who violate Conditionalization
(diachronic)
(diachronic)
Slide27t0
t1
Cr(H) = 0.7CrE(H) = 0.9Cr(E) = 0.2Cr(H/E) = 0.7ECrE (H) = 0.9ENot (E)£(-0.04+0.20)£ -0.04
E & H
E & not (H)
Not E
£ (0.70 -1)
£(0.70)
£0
H
Not(H)
£ (-0.90
+ 1)
£-0.90
£-0.04
To be offered at t
1
iff
E
Slide28Intuitively Irrational?
Can
be Dutch Booked?Agents who violate the probability axiomsAgents who are uncertain of their own credence functionAgents who violate the Reflection PrincipleAgents who violate Conditionalization
(diachronic)
(diachronic)
Slide29What does it mean to say that an agent is
guaranteed to lose money?
It means that the agent would accept a set of bets as fair, and at any possible world where she would accept those bets as fair, the bets lose her money.It means that the agent would accept a set of bets as fair, and under any interpretation of the claims involved, the bets lose her money. My new understanding is motivated by a classic account of logical consistency.On my new understanding, just the (intuitively) right Dutch Book Arguments go through.