The Logic of Decision
Symposium in honor of 
Richard Jeffrey
Philosophy of Science Association
October 24, 1998

Ethan Bolker
Department of Mathematics and Computer Science
UMass-Boston

Themes
•	Glimpses of history
•	Revisiting the existence theorem
–	two philosophical challenges:
 	noncommutative logic
 	doing without impartiality 
•	Everyday probability is hard
•	Decision theory with a human face

Revisiting the existence theorem
•	Find all binary relations pref that average:
 	A, B incompatible, 
 	A pref B 
 	
 	A pref (AB) pref B
•	Canonical example: the ordering on sets determined by computing the expected value of a function with respect to a probability measure.

The domain of pref 
•	In quantum mechanics, lattice of subspaces of Hilbert space replaces set of subsets of phase space - the expected value of an observable still averages.
•	196x goal: prove theorems in this noncommutative logic.
•	1998 challenge: is there a noncommutative logic of decision?
The Existence Question
•	Canonical example revisited
 	value(A) = integral f(s) dprob(s)
 	des(A) = value(A)/prob(A)
•	des induces pref
•	value(A) = des(A)  prob(A)
 	(value is discounted desirability)
•	When does pref come from value and prob?
The jeffrey function
•	For each x in (0,1) (the range of des) let
jeffrey(x,A) = x prob(A)  - value(A)
°	jeffrey(x,A) <= 0 just when x <= des(A)
°	jeffrey is a (signed) measure: when A and B are incompatible, 
jeffrey (x,AB) = jeffrey (x,A) + jeffrey (x,B) 
°	jeffrey(x,A) is an increasing function of  x
•	Challenge: What does jeffrey (x,A) mean?
 	(when we know we may rename it)
The Existence Theorem
•	Given a jeffrey such that
 	jeffrey(x,  · ) is a signed measure
 	jeffrey( ·, A) is an increasing function  
 	define des by
 	des(A) = the unique x such that jeffrey(x,A) = 0.
 	Then des averages, and every averaging des is of this form.
The role of impartiality
•	Impartiality is equivalent to the existence of value and prob such that
jeffrey(x,A) = x prob(A)  - value(A)
•	That’s all it does
–	like the parallel postulate
–	thin disguise for fair coins and arbitrary gambles
–	how much (what kind of) decision theory can you do with weaker (other) assumptions about jeffrey?
Into the third dimension
•	M   =  vector space of all signed measures
•	M+ =  cone of positive measures
•	V    =  span of jeffrey
•	V+  = M+  V
•	Impartiality  dim V = 2
dim V+  =  1  classic uniqueness
dim V+  =  2  weird (J/B) uniqueness
•	What does dim V  =  3  mean?
Everyday probability is hard

•	What is probability?
•	An elusive question in philosophy and psychology (only easy in mathematics)
•	Dick says
“The skill consists … in coming to have appropriate degrees of belief between 0 and 1 when conditions are less than ideal …”
•	A short experiment under ideal conditions 
Count runs of 4
… HTHHHHTTTTTH … 
expect (# runs) * (2/16) = 61/8 = 7.6 
0				 7
1				 8
2				 9
3				10
4				11
5				12
6				13

Observations
•	Human instinct’s weak on probability in general and Poisson processes in particular
•	Short memory approximates memoryless
•	You can win stone/paper/scissors …
•	You can cross a busy street
•	When will wide ties return?
•	We need the Logic of Decision
Radical probabilism
•	“It is a frequent complaint against academic moral philosophers …” (Braithwaite)
•	Dick argues that probability - a.k.a. uncertainty - is all there is
•	Life is a high risk enterprise
•	“… whenever there is a real issue between two of us, or whenever one of us is of two minds, both sides are ruled reasonable ...