Alvin Seminar Handout

7/30/2019 Alvin Seminar Handout

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The Traditional Stories A New Approach to A, C & E Results & Open Questions Refs Proofs & Illustrations

Individual Coherence & Group Coherence

Branden Fitelson1

Department of Philosophy

&

Center for Cognitive Science (RuCCS)

Rutgers University

[email protected]

1This is joint work with Rachael Briggs (ANU), Fabrizio Cariani (NU), and

Kenny Easwaran (USC). We are working on a joint paper, to appear soon [1].

Branden Fitelson Individual Coherence & Group Coherence 1


Here are two traditional (individual) epistemic norms:

The Truth Norm for Belief (TB). Epistemically rational

agents should only believe propositions that are true.

The Consistency Norm for Belief (CB). Epistemically

rational agents should have logically consistent belief sets.

Fact. (TB) entails (CB). Suppose S violates (CB). Then, some

ofSs beliefs are false. Therefore, S violates (TB).

Here, entailment means satisfaction preservation. (TB)

is a local/narrow norm, and (CB) is a global/wide norm.

Problem. The norm of deductive consistency does not seemto line-up/jibe very well with evidential norms. Informally:

Evidential Norm for Belief (EB). Rational agents should onlybelieve propositions that are supported by their evidence.

In (sufficiently complex) preface cases, agents seem to

satisfy(EB) while violating(CB) [3, 13]. I take this as a datum.

Ill describe a new coherence norm that avoids this problem.



Suppose we have a panel of three judges (J1, J2, J3). Thispanel will vote on an agenda, which stems from:

Question. In the reunited Germany, should the German parliament

and the seat of government move to Berlin or stay in Bonn?

Suppose the panel votes on these two (atomic) premises:

P the parliament should move.

G

the seat of government should move.There is also the following conclusion whose truth-value isdetermined by the truth-values of the premises:

B both the parliament and the seat of government should move.

Suppose the judges render the following judgments (votes):

P? G? B?

J1 yes no no

J2 no yes no

J3 yes yes yes

For each judge, the conclusion column is determined bythe

premise columns (i.e., we assume each judge is cogent).



Example ofdoctrinal paradox/discursive dilemma ([14], [18]).

Doctrinal Paradox/Discursive Dilemma

P? G? B?

J1 yes no no

J2 no yes no

J3 yes yes yesMajority yes yes yes & no?

Naive majority rule for aggregating all judgments can lead

to inconsistent aggregations of premises+ conclusions.

Various alternative aggregation procedures have beenproposed, so as to ensure overall consistency. Example:

Premise-Based Procedure. Use majority rule on the

premises, and then just enforce deductive closure.

The premise-based procedure seems reasonable (esp. if the

premises make up the agenda that is explicitly voted on).



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Premise-Based Procedure

P? G? B?

J1 yes no no

J2 no yes noJ3 yes yes yes

Majority: yes yes (ignore)

Conclusion (from closure): yes

So, the premise-based procedure is a way of restoring

deductive consistency to a nave majority aggregation rule.

Another consistent approach is the conclusion-basedmajority procedure. In our example, this would return no

for the conclusion (B) and nothing for the premises (P/G).

Procedures silent on some members of some agendas are

called incomplete. These sometimes seem sensible [15].



There are other approaches to judgment aggregation that

are guaranteed to preserve consistency.

Among the most popular of these nowadays are the

so-called distance-based aggregation methods [17].

These methods are holistic they do not assume

independence. Idea: pick the closest (according to a global

distance measure) consistentprofile as the aggregate.

The upshot: traditional stories about individual (epistemic)

coherence presuppose that deductive consistency is a

requirement of individual epistemic rationality.

Traditional stories about judgment aggregation borrow thispresupposition, and (as a reult) require procedures to

render consistent outputs when given consistent inputs.

Next: a new story about (individual) coherence, and its

ramifications for the judgment aggregation literature.



Our new approach is inspired by work of de Finetti [4] and

Joyce [12, 11] on grounding coherence norms for credences.

Let b(p) = r Ss credence in proposition p is r. And,

let B(p) S believes that p. And, consider this analogy:

p is true

B(p)::

??

b(p) = r

Ramsey [19] gives reasons to be skeptical that this analogy

can be completed in an epistemologically useful way. His

main target is logical/a prioriprobability as truthmaker.

Hjek [8] argues that the analogy is useful, provided ??

gets filled-in with the objective chance of p equals r.

If we follow Hjeks advice, then provided that objective

chances must be probabilities (!) [10] we get probabilism

as the analogous coherence norm for credences.

Apart from [10], there are serious epistemic problems with

this analogy. Hjek himself discusses a crucial example.



the coin that I am about to toss is either two-headed or two-tailed,

but you do not know which. What is the probability that it lands

heads? . . . reasonably, you assign a probability of 12

, even though

you knowthat the chance of heads is either 1 or 0. So it is rational

to assign a credence that you knowdoes notmatch the . . . chance.

This is disanalogous to rational belief, since it is never

rational to believe something that you knowis not true.

We think this is a counterexample to any useful (narrow)

truth norm analogy between full belief and partial belief.

Ramsey has a different strategy for trying to ground

probabilism as a coherence norm for credences.

He offers a pragmatic argument for probabilism, which hasbecome known as the Dutch Book Argument (DBA).

I wont discuss the (DBA) today (idea: if Ss credences are

non-probabilistic, S is susceptible to sure monetary loss).

de Finetti [4] and Joyce [12, 11] have offered epistemic

arguments for probabilism via considerations ofaccuracy.Branden Fitelson Individual Coherence & Group Coherence 8


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Joyce [12, 11] thinks of Ss credence b(p) as Ss estimate of

the (numerical) truth-value ofp. He gives an argument for

probabilism that is based on the accuracy of Ss estimates.

In order to understand the accuracy norm(s) appropriate tocredences (in Joyces sense), we first need to say something

about how to measure the (in)accuracy of credences.

Following Joyce, well associate the truth-value T (at each

world w) with the number 1 and the truth-value F with 0.

The inaccuracy of b(p) at w is bs distance (d) from ps

numerical truth-value at w (this is gradational inaccuracy).

Norm of Gradational Accuracy. A rational agent must evaluate

partial beliefs on the basis of their gradational accuracy, and she

must strive to hold a system of partial beliefs that has an overall

level of gradational accuracy not guaranteed to be lower than

that of any alternative system she might adopt.



Example. Suppose S has just two (contingent) propositions{P ,P} in their doxastic space. Then, there are two salientpossible worlds (w1 in which P is T, and w2 in which P is F).And, the overall inaccuracyof b at w [I(b,w)] is given by:

I(b,w1) = d(b(P), 1)+ d(b(P ), 0).I(b,w2) = d(b(P), 0)+ d(b(P ), 1).

Various measures (d) of distance from 0/1-truth-value

have been proposed/defended in the historical literature.

de Finetti [4] endorsed the following measure of distancefrom truth-value (in one of his arguments for probabilism):

s(x,y) = (x y)2.

The distance measure s gives rise to a measure of overallinaccuracy (Is), which is known as the Brier Score. In our toyexample, the Brier Scores of b in worlds w1 and w2 are:

Is(b,w1) = s(b(P), 1)+ s(b(P ), 0) = (b(P) 1)2 +b(P )2.

Is(b,w2) = s(b(P), 0)+ s(b(P ), 1) = b(P)2 + (b(P ) 1)2.



If one adopts the Brier Score as ones measure of bs

inaccuracy, then one can give an accuracy-dominance

argument for a probabilistic coherence norm for credences.

de Finetti [4] proved the crucial Brier-dominance theorem:

Theorem (de Finetti). b is non-probabilistic if and only if

there exists a probabilisticcredence function b such that

b

has a strictly lower Brier Score than b at every world.It helps to visualize what happens to a non-probabilistic b:

0.0 0.2 0.4 0.6 0.8 1.0

0.0

0.2

0.4

0.6

0.8

1.0



There are various controversial assumptions in Joyces

accuracy-dominance argument for probabilism [9]. We have

two forthcoming papers criticizing Joyces argument [6, 5].

We can, by analogy, apply the Joycean strategy to full belief.

When we do, we get a compelling coherence norm for full

belief. And, we avoid the problems faced by Joyces

argument for probabilism (i.e., it works better for full belief!).

The basic idea is to move away from the old accuracy

analogy toward a new accuracy-dominance analogy.

Let B be the set of qualitative judgments (beliefs/disbeliefs)

of an agent S (at a time). A new (global/wide) analogy:

B is accuracy-dominated

B is incoherent::

b is Brier-dominated

b is not a Pr-function

Filling-in this analogy requires answering two questions:

How should we gauge the (in)accuracyof B at a world w?

How should we explicate B accuracy-dominatesB?



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For simplicity, we restrict our discussion to finite, logicallyomniscient, opinionatedagents who make definite judgmentsregarding each proposition in some Boolean algebra B.

I.e., for each p B, S either believes p [B(p)] or disbelieves

p [D(p)], and not both (and no suspension of judgment).First step: say what it means for beliefs/disbeliefs to be(in)accurate. This is easy and uncontroversial. Of course:

B(p) is (in)accurate at w iffp is true (false) at w.

D(p) is (in)accurate at w iffp is false (true) at w.

Second step: say what it means for one judgment set B

(over an algebra B) to be accuracy-dominated by another B.

The analogous way to think about accuracy-dominance forsets of qualitative judgments involves comparing numbersof inaccurate judgments in each of the judgment sets, i.e.,

One set of judgments B (strictly) accuracy-dominates

another B (over a full algebra B) iff B contains strictly fewer

inaccurate judgments than B at everypossible world.



On this approach, we obtain the following coherence norm:

(CB) S should not have a qualitative judgment set B that is

(strictly) accuracy-dominatedby some alternative set B.

What is the precise content of (CB

)? Is it non-trivial? And,how does it relate to our evidential norm (EB)?

It turns out that (CB) is non-trivial, and it is very closely

aligned (in lock-step with) with our evidential norm (EB).

First, consider (CB) from the point of view of its violation.

(I) Theorem. S violates(CB) iffB contains a subset such that,

at every possible world, most members of are inaccurate.

(CB) is strictly weaker than (CB).

(CB) (CB). IfS satisfies (CB), then Ss judgment B set will

be perfectlyaccurate in some w ( B is non-dominated).

(CB) (CB). Notably, minimal inconsistencies (lotteries,

prefaces) need notviolate (CB). [See, also, Extras #1.]



Next, consider (CB) from the point of view of satisfactionof the norm. Heres a sufficientcondition for satisfaction:

(II) Theorem. S satisfies(CB) iftheir B can be probabilistically

representedin the following strict12 -threshold fashion:

(LB) There exists a probability function Pr such that, p B:

B(p) iffPr(p) >12

, and D(p) iffPr(p)

1

2

Pr(p|

E) 12 B(p) B and

Pr(p) < 12 D(p) B. This implies Theorem (II).



Consider a language w/16 state descriptions s1,...,s16. Let:

p s1 s2 s3 s4 q s1 s5 s6 s7r s2 s5 s8 s9 s s3 s6 s8 s10t s4 s7 s9 s10 {p , q , r , s , t}

Here are four key facts about the set .

(i) Any two sentences in are logically consistent.

because any pair shares a state description.

(ii) Any three sentences in are logically inconsistent.

because every state description occurs exactly twice.

(iii) Any four sentences in are coherent (if jointly believed).

Theorem (I) + the fact that it is notguaranteed that such a

judgment set will contain a subset such that, at every

world, a majority ofs members are inaccurate.

(iv) is not coherent (if jointly believed).

Theorem (I) + the fact that, at every world, a majority of

members of any such judgment set mustbe inaccurate.



p q r s t

J1 B B B B D

J2 B B B D B

J3 B B D B B

J4 B D B B B

J5 D B B B B

Majority B B B B B

Each judge can be coherentbecause judgment sets with 4/5

beliefs (and 1/5 disbeliefs) over can be non-dominated.

This is because there will be worlds in which a majority of

such judgments are accurate. (For example: in worlds that

make state description s1 true, p, q and t are all true.)

However the (80%!) majority believes all members of. And,

any judgment set containing these judgments must be

dominated. So, majority rule doesnt preserve coherence.

On the next slide, well sketch a proof of our positive

Theorem (III). The key will be to use Theorem (II).Branden Fitelson Individual Coherence & Group Coherence 23


Recall, Theorem (II) ensures that that if a judgement set B is

representable by some probability function via a strict12 -threshold, then that judgment set B must be coherent.

For majority acceptance on individually consistent and

complete inputs this is clearly true. The probability

function in question is just the pattern of individual votes:

For all p, Pr(p) # of judges for p# of total judges .

To verify this, note that Pr() satisfies the Pr-axioms.Additivity is the only axiom that deserves comment.

Suppose p, q are m.e. If p is accepted by ry

of the judges

and q is accepted bysy

of the judges, then (by consistency +

completeness) p q will be accepted byr+s

yof the judges.

By Theorem (II) and the existence of Pr(), it follows that

majority rule on consistent and complete profiles always

yields coherentaggregations. That is, if judges satisfy (CB),

then their majority aggregate must satisfy (CB). Branden Fitelson Individual Coherence & Group Coherence 24

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