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June 6, June 22, August 29, 2006 

New abstract, Nov. 1, 2006 

Minor rev. Dec 5 2006 

Ignorance and Indifference 
John D. Norton 

Center for Philosophy of Science and 

Department of History and Philosophy of Science 

University of Pittsburgh 

Pittsburgh PA 15260 

www.pitt.edu/~jdnorton 

 

 

The epistemic state of complete ignorance is not a probability distribution. In it, we 

assign the same, unique ignorance degree of belief to any contingent outcome and 

each of its contingent, disjunctive parts. That this is the appropriate way to represent 

complete ignorance is established by two instruments, each individually strong 

enough to identify this state. They are the principle of indifference (“PI”) and the 

notion that ignorance is invariant under certain redescriptions of the outcome space, 

here developed into the “principle of invariance of ignorance” (“PII”). Both 

instruments are so innocuous as almost to be platitudes. Yet the literature in 

probabilistic epistemology has misdiagnosed them as paradoxical or defective since 

they generate inconsistencies when conjoined with the assumption that an epistemic 

state must be a probability distribution. To underscore the need to drop this 

assumption, I express PII in its most defensible form as relating symmetric 

descriptions and show that paradoxes still arise if we assume the ignorance state to be 

a probability distribution. By separating out the different properties that characterize a 

probability measure, I show that the ignorance state is incompatible with each of the 

additivity and the dynamics of Bayesian conditionalization of the probability 

calculus. 

 



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1. Introduction 
 In one ideal, a logic of induction would provide us with a belief state representing total 

ignorance that would evolve towards different belief states as new evidence is learned. That the 

Bayesian system cannot be such a logic follows from well-known, elementary considerations. In 

familiar paradoxes to be discussed here, the notion that indifference over outcomes requires 

equality of probability rapidly leads to contradictions. If our initial ignorance is sufficiently 

great, there are so many ways to be indifferent that that the resulting equalities contradict the 

additivity of the probability calculus. We can properly assign equal probabilities in a prior 

probability distribution only if our ignorance is not complete and we know enough to be able to 

identify which is the right partition of the outcome space over which to exercise indifference. 

Interpreting zero probability as ignorance also fails multiply. Additivity precludes ignorance on 

all outcomes, since the sum of probabilities over a partition must be unity; and the dynamics of 

Bayesian conditionalization makes it impossible to recover from ignorance. Once an outcome is 

assigned a zero prior, its posterior is also always zero. Thus it is hard to see that any prior can 

properly be called an “ignorance prior,” to use the term favored by Jaynes (2003, Ch. 12), but is 

at best a “partial ignorance prior.” For these reasons the growing use of terms like 

“noninformative priors,” “reference priors” or, most clearly “priors constructed by some formal 

rule” (Kass and Wasserman, 1996) is a welcome development. 

 What of the hope that we may identify an ignorance belief state worthy of the name? 

Must we forgo it and limit our inductive logics to states of greater or lesser ignorance only? The 

central idea of this paper is that, if we forgo the idea that belief states must be probability 

distributions, then there is a unique, well-defined ignorance state and the project of this paper is 

to identify it. 

 The instruments more than sufficient to specify this state already exist in the literature 

and are described in Section 2. They are the familiar principle of indifference (“PI”) and also the 

notion that ignorance states can be specified by invariance conditions. I will argue, however, that 

common uses of invariance conditions do not employ them in their most secure form. The most 

defensible invariance requirements use perfect symmetries and these are governed by what I 

shall call the “principle of the invariance of ignorance” (PII). In Section 3, I will review the 

familiar paradoxes associated with PI, identifying the strongest form of the paradoxes as those 

associated with competing but otherwise perfectly symmetric descriptions. I will also argue that 



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invariance conditions are beset by paradoxes analogous to those troubling PI. They arise even in 

the most secure confines of PII since simple problems can exhibit multiple competing 

symmetries, each generating a different invariance condition. 

 In Section 4, I will argue that we have misdiagnosed these paradoxes as some kind of 

deficiency of the principle of indifference or an inapplicability of invariance conditions. Rather, 

they are such innocuous principles of evidence as to be near platitudes. They both derive from 

the notion that beliefs must be grounded in reasons and, in the absence of distinguishing reasons, 

there should be no difference of belief. How could we ever doubt the notion that, if we have no 

grounds at all to pick between two outcomes, then we should hold the same belief for each? The 

aura of paradox that surrounds the principles is an illusion created by our imposing the additional 

and incompatible assumption that an ignorance state must be a probability distribution. In the 

remaining sections, it will be shown that these instruments identify a unique, epistemic state of 

ignorance that is not a probability distribution. 

 Section 5 describes the weaker theoretical context in which this ignorance state can be 

defined. It is based on a notion of non-numerical degrees of confirmation that may be compared 

qualitatively; and it may be selectively enriched to bring it closer to the full probability calculus. 

In Section 6, we shall see that implicit in the paradoxes of indifference is the notion that the state 

of ignorance is unchanged under disjunctive coarsening or refinement of the outcome space; and 

that this same state is invariant under a transformation that exchanges propositions with their 

negations. These two conditions each pick out the same ignorance state, in which a unique 

ignorance degree is assigned to all contingent propositions. In particular, in that state, we assign 

the same ignorance state of belief to all contingent propositions and each of their contingent, 

disjunctive parts. We shall see in Section 7 that this state is incompatible with both the additivity 

of the degrees of belief and also with the property that allows for Bayesian conditionalization, 

when these properties are separated out by algebraic means, so that any logic of induction that 

employs this state can use neither property. Section 8 contains some concluding remarks. 

2. Instruments for Defining the State of Ignorance 
 The present literature in probabilistic epistemology has identified two principles that can 

govern the distribution of belief. They are both based on the simple notion that beliefs must be 

grounded in reasons, so that when there are no differences in reasons there should be no 



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differences in belief. Applying this notion to different outcomes gives us the Principle of 

Indifference (Section 2.1); and applying it to two perfectly symmetric descriptions of the same 

outcome space gives us what I call the Principle of Invariance of Ignorance (Section 2.2). 

2.1 Principle of Indifference 

 The “principle of indifference” was named by Keynes (1921, Ch.IV) to codify a notion 

long established in writings on probability theory. I will express it in a form independent of 

probability measures.1 

(PI) Principle of Indifference. If we are indifferent among several outcomes, that is, if 

we have no grounds for preferring one over any other, then we assign equal belief to 

each. 

Applications of the principle are familiar. In cases of finitely many outcomes, such as the 

throwing of a die, we assign equal probabilities of 1/6 to each of the 6 outcomes. If the outcomes 

form a continuum, such as the selection of a real magnitude between 1 and 2, we assign a 

uniform probability distribution. 

2.2 Principle of Invariance of Ignorance 

 A second, powerful notion has been developed and exploited by Jeffeys (1961, Ch.III) 

and Jaynes (2003, Ch. 12). The leading idea is that a state of ignorance can remain unchanged 

when we redescribe the outcomes; that is, there can be an invariance of ignorance under 

redescription. That invariance may powerfully constrain and even fix the belief distribution. 

Jaynes (2003, pp. 39-40) uses this idea to derive the principle of indifference as applied to 

probability measures over an outcome space with finitely many mutually exclusive and 

exhaustive outcomes A1, A2, … , An. If we are really ignorant over which outcome obtains, our 

                                                
1 For completeness, I mention that this principle is purely epistemic. It is to be contrasted with an 

ontic symmetry principle, according to which outcomes A, B, C, … are assigned equal weights 

if, for every fact that favors A, there are corresponding facts favoring B, C, …; and similarly for 

B, C,… . In the familiar cases of die throws and dart tosses, it is this physical symmetry that 

more reliably governs the assigning of probabilities. 



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distribution of belief would be unchanged if we were to permute the labels A1, A2, … , An in any 

arbitrary way: 

Aπ(i)’ = Ai                                                                       (1) 

where (π(1), π(2), … π(n)) is a permutation of (1, 2, … , n). A probability measure P that 

remains unchanged under all these permutations must satisfy2 P(A1) = P(A2) =  …= P(An). If the 

outcomes Ai are mutually exclusive and exhaust the outcome space, then the measure is unique: 

P(Ai) = 1/n, for i= 1, … , n. This is the equality of belief called for by PI.  

 This example illustrates a principle that I shall call: 

(PII) Principle of the Invariance of Ignorance. An epistemic state of ignorance is 

invariant under a transformation that relates symmetric descriptions.  

The new and essential restriction is the limitation to “symmetric descriptions,” which, loosely 

speaking are ones that cannot be distinguished other than through notational conventions. More 

precisely, symmetric descriptions are defined here as pairs of descriptions meeting two 

conditions: 

(S1) The two describe exactly the same physical possibilities; and each description 

can be generated from the other by a relabeling of terms, such as the additional or 

removal of primes, or the switching of words. 

An example is the permutation of labels of (1) above and a second is found below in (2a), (2b). 

(S2) The transformation that relates the two descriptions is “self-inverting.” That is, 

the same transformation takes us from the first description to the second, as from the 

second to the first. 

An example is the permutation that merely exchanges two labels; a second exchange of the same 

pair takes us back from the second description to the first. 

 This principle is the most secure way of using invariance to fix belief distributions. What 

makes it so secure is the insistence on the perfect symmetry of the descriptions. That defeats any 

                                                
2 The simplest way to arrive at this result is to consider a transformation that merely exchanges 

two labels, Ai and Ak, say, for i and k unequal, If the probability measure is to remain unchanged 

under all such exchanges, then we must have P(Ai) = P(Ak) for each pair i, k, which entails the 

equality stated. 



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attempt to find reasons upon which to base a difference in the distribution of belief in the two 

cases; for any feature of one description will, under the symmetry, assuredly be found in the 

second. So any difference in the two epistemic states cannot be grounded in reasons, but must 

reflect an arbitrary stipulation. We shall see, however, that common invocations of invariance 

conditions in the literature do not adhere strictly to this symmetry in the transformations and are 

thus less secure. 

 If the outcomes form a continuum, the application of PII is identical in spirit to the 

deduction of the principle of indifference, though slightly more complicated. A clear illustration 

that gives the template for computing other cases is provided by applying PII to von Mises’ 

(1951, pp. 77-78) celebrated case of wine and water. We are given a glass with some unknown 

mixture of water and wine and know only that the mixtures lies somewhere between 1:2 parts 

water to wine and 2:1 parts water to wine. That is,  

the ratio of water to wine x lies in the interval 1/2 to 2;                        (2a) 

and the ratio of wine to water x’=1/x also lies in the interval 1/2 to 2.                       (2b) 

If we represent our uncertainty over x with the probability density p(x) and our uncertainty over 

x’ with the probability density p’(x’), the idea that our ignorance is unchanged by redescription 

turns out to fully specify both densities. The calculation that shows this has two parts. First we 

note that the transformation from x to x’ merely redescribes the same outcome, so the two should 

agree in assigning the same probabilities to the same outcomes. The outcome of x being in the 

small interval x to x+dx is the same outcome as x’ lying in x’ to x+dx’, where x’=1/x. Since the 

two outcomes must agree in probabilities we have:3 

p(x’)dx’ = -p(x)dx 

That is, more precisely, 

A. Agreement in probability      p’(x’) = -p(x) dx/dx’                                              (3a) 

In the second part, we note that there is a perfect symmetry between the two descriptions (2a) 

and (2b). Loosely speaking, that means that whatever our ignorance may be of the ratio of water 

to wine, it is just the same as our ignorance of the ratio of wine to water. Indeed had we 

mistakenly switched the labels in (2a) and (2b) it would make no difference to the problem 

posed. Formally that is expressed in the two descriptions (2a) and (2b) meeting the conditions 

                                                
3 The negative sign arises since the increments dx and dx’ increase in opposite directions. 



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(S1) and (S2) above. The first condition (S1) is met in that (2a) becomes (2b) if we switch the 

words “water” and “wine” and replace the variable x by x’; and (2a) and (2b) still describe 

exactly the same outcome space. 4 Condition (S2) is met since x relates to x’ in exactly the same 

way as x’ relates to x. That is, the function that transforms x to x’ is exactly the same as the 

function that transforms x’ to x; they are both the taking of the arithmetic inverse 

x’ = 1/x          x = 1/x’                                                     (4) 

In other words, they are self-inverting, since composing the transformation with itself yields the 

identity map. 

 We have complete symmetry of descriptions. So PII requires that the two probability 

distributions are the same: 

B. Symmetry          p’(.)  =  p(.)                                                    (3b) 

Since dx/dx’ = -x2, the system of equations (3a), (3b) and (4) entail that any p(x) must satisfy the 

functional equation 

p(1/x) (1/x)  =  p(x) x                                                                    (5) 

Notably, solutions of (5) do not include p(x) = constant. The most familiar solution is5 

p(x) = K/x                                                                            (5a) 

where the requirement that p(x) normalize to unity fixes K = 1/ln 4. 

 The example of wine and water cleanly embodies the symmetry of descriptions needed to 

trigger the requirements of PII. Other familiar cases of symmetry appear to be a little less 

symmetric in so far as the transformations between the descriptions are not self-inverting. Take 

for example the redescription of all the reals by unit translation: 

x’ = x – 1          x  =  x’ + 1                                               (6) 

The transformation is not self-inverting so the perfect symmetry of descriptions fails in that we 

proceed from the x description to x’ by adding unity; and from the x’ to x by subtracting unity. 
                                                
4 This symmetry can easily fails as it did in Von Mises’ original presentation. He took the ratio 

to lie in 1:1 to 2:1, so that permuting “wine” and “water” and replacing x with x’ does not lead to 

a description of the same outcome space. 
5 Briefly, arbitrarily many solutions can be constructed by stipulating p(x) for 1≤x<2 and using 

(5) to define p(x) for 1/2<x≤1, where the resulting function may need to be multiplied by a 

constant to ensure normalization to unity. 



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However, in this example, self-inverting transformations are readily found. The transformation 

(6) can be created by composing two transformations that are individually self-inverting: 

x’’ = -x          x  =  -x’’                                                     (6a) 

x’ = -1 – x’’        x’’  =  -1 – x’                                               (6b) 

So the invariance of epistemic state required by PII obtains for each transformation (6a) and (6b) 

individually and thus also for (6) as a whole. (This is analogous to the decomposition of an 

arbitrary permutation (1) into a sequence of many pair-wise exchanges, each of which is self-

inverting.) 

 Not all deductions of prior probabilities in the objective probability literature conform to 

the strict and most defensible conditions of PII, a perfect symmetry of descriptions. The most 

familiar use of invariance in the objective probability literature that lacks this symmetry is the 

deduction of the Jeffreys prior for a real-valued physical magnitude x that can take any value 

greater than zero. Jaynes’ method requires that our prior probability distribution p(x) be 

unaffected by changes in the units used. For example, Jaynes (2003, p. 382) requires that the 

prior probability distribution p(t) for a time constant t must be unchanged by a change in the 

units used to measure time. That is, if we measure the constant in different units t’=qt, for q a 

constant of unit conversion, we recover a new prior distribution p’(t’) that must be unchanged. 

This gives us two conditions analogous to those of the wine and water problem 

A. Agreement in probability     p’(t’)  =  p(t) dt/dt’                                              (7a) 

B. Symmetry                               p’(.) = p(.)                                                         (7b) 

Given that q can be any real, these two equations admit a unique solution6 

p(t)  =  constant/t                                                        (8) 

which is the Jeffreys prior. 

 The weakness of this deduction is that the condition B. Symmetry is not deduced from a 

perfect symmetry of the two descriptions. Rather it arises from something a little weaker and 

more nebulous. Jaynes writes of the two hypothetical experimenters using the different systems 

of units: “But Mr. X and Mr.X’ are both completely ignorant and they are in the same state of 

                                                
6 Since dt’/dt=q, the two equations entail p(qt) q = p(t). Holding t fixed and differentiating with 

respect to q we have p(qt) + qt dp(qt)/d(qt) = 0; that is, dp(t’)/dt’ = -p(t’)/t’. whose unique 

solution is the Jeffrey’s prior (8). 



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knowledge, and so [p] and [p’] must be the same function…” What Jaynes says here is wrong. 

Mr. X and Mr. X’ may know very little. But they do know how their units relate. If t is measured 

in minutes and t’ in seconds, so that q=60, then there is the following asymmetry, knowable to 

both: Mr. X’s measurement will be 1/60th that of Mr. X’. Switching t and t’ does not leave 

everything unchanged, as it did when we switched x and x’ in (2a) and (2b). So they are not “in 

the same state of knowledge.” Jaynes’ plausible presumption is that this is not enough 

asymmetry to overturn (7b) and perhaps he is right. However that reliance on plausibility falls 

short of the condition of perfect symmetry of descriptions needed by PII and routinely used in 

physics, Jaynes’ model science, to underwrite such invariances. 

 These two principles express platitudes of evidence whose acceptance seems irresistible. 

They follow directly from the simple idea that we must have reasons for our beliefs. So if no 

reasons distinguish among outcomes, we must assign equal belief to them; or if two descriptions 

of the outcomes are exactly the same in every non-cosmetic aspect, then we must distribute 

beliefs alike in each.7 Yet, as I will now review, both principles lead to paradoxes in the 

ordinary, probabilistic context. 

3. Their Failure if Belief States are Probability Distributions 

3.1 Paradoxes of Indifference 

 Since at least time of Keynes’ baptism (1921, Ch.IV) of the principle of indifference, it 

has been traditional to besiege the principle in paradoxes. Indeed they have become a fixture in 
                                                
7 One may think that subjective Bayesians can escape the platitudes since they allow prior 

probability distributions to be set arbitrarily in so far as the evidence fails to fix them. Since that 

is the only constraint on the priors, the resulting freedom in the range of admissible priors 

already conforms to both PI and PII. If no evidence picks outcomes A1 and A2 apart, then for 

any admissible prior that favors A1 in some way, there must be an admissible counterpart that 

favors A2 in the same way. Thus, if one’s beliefs are expressed by the range of admissible priors, 

they treat indifferent outcomes alike, as PI requires. Similarly this range must respect the perfect 

symmetry of descriptions of PII. In the wine-water example, for any admissible prior over the 

parameter x, there must be a corresponding admissible prior over the alternative parameter x’. 



10 

the routine, now nearly ritualized dismissals of the classical interpretations of probability.8 

Laplace (1825, p.4) famously defined probability as the ratio of favorable to all cases among 

“equally possible cases,” which he defined as “cases whose existence we are equally uncertain 

of.” While Laplace seems to have taken the notion of an equally possible case as a primitive term 

not in need of further definition within his system, the later literature insists that on pain of 

circularity, he must provide an independent specification of which are the equally possible 

cases.9 The principle of indifference is pressed into service and the outcome is shown to be 

paradoxical.10 

 The paradoxes reveal that the principle of indifference only partially captures the notion 

of indifference presumed in the literature. All the paradoxes have the same structure. We are 

given some outcomes over which we are indifferent and thus assign equal probability. The 

outcomes are redescribed; indifference once again dictates that the outcomes are of equal 

probability; and it turns out the two assignments of probability contradict. Thus, it is also 

routinely assumed that the indifference described by the principle can persist when we redescribe 

the outcome space in certain ways, to be specified below. 

 The paradoxes divide into two classes. The first employs a finite outcome space and often 

a very small one of mutually exclusive outcomes. In one of Keynes’ (1921, Ch. IV) examples, 

the unknown country of a man is one of 

France,    Ireland,    Great Britain 

so, by the principle of indifference, we assign a probability 1/3 to each. We can coarsen the same 

space by forming a disjunctive outcome “British Isles,” to arrive at 

France,    British Isles (=Great Britain v Ireland) 
                                                
8 For surveys of these paradoxes both in the context of the classical interpretation and otherwise, 

see Galavotti (2005, § 3.2), Gillies (2000, pp. 37-49), Howson and Urbach (1996, pp. 59-62) and 

Van Fraassen (1989, Ch. 12) 
9 This insistence has seems unfair to me. On pain of circularity or infinite regress, we cannot 

demand explicit definitions for all our terms. Some may be introduced by implicit definitions, 

that is by assertions in which they figure in some essential way. 
10 It has also been suggested that the logical interpretation of probability is troubled by these 

paradoxes. (Gillies, 2000, p. 37) 



11 

By the principle of indifference we now assign the probability of 1/2 to each. We have now 

assigned both probability 1/3 and 1/2 to France. In another example, we do not know whether the 

color of the cover of a book is 

red,   blue,   black 

and assign probability 1/3 to each. However we also have the coarsened space 

red,     –red = (blue or black) 

to whose outcomes we assign the incompatible probability of 1/2. The additivity of the 

probability calculus clearly drives the paradoxes. Since it requires that probabilities sum to unity, 

it precludes leaving the probabilities of the two outcomes red and -red of the coarsened space at 

1/3, as the first space requires. Alternatively, additivity forces the probability of  -red to be the 

sum of the probabilities of blue and black, so that it cannot equal the probability assigned to red 

in the first space. 

 This analysis can be continued with Keynes’ other examples, as well as those offered 

elsewhere. Cramer (1966, pp. 15-16) describes a game in which two coins are tossed and player 

A wins if there is at least one head.11 On one analysis, we are indifferent over the outcomes 

HH,   HT,   TH,   TT, 

                                                
11 This example is equivalent to Keynes’ example of two cards, each drawn from different 

packs. One card proves to be black so by an application of the principle of indifference we can 

infer that the probability that the other is black is 1/2. The inference requires that we count the 

disjunctive outcome of (black-red or red-black) as a single outcome and assign it a probability 

equal to the only remaining outcome of black-black. A similar treatment of disjunctive outcomes 

lies at the heart of the mathematically much more complicated urn sampling problem discussed 

by Keynes (1921, pp. 49-50) arising in discussions of Laplace’s rule of succession, with Howson 

and Urbach (1996, pp. 55-59) laying out the detailed computations. An urn contains n balls that 

may each be black or white. We may indifferent over each of the possible numbers of white balls 

in the urn, in which case each possible number has probability 1/(n+1). Or we may be indifferent 

over each constitution, that is each of the 2n assignments of black and white to each ball in 

succession. That there are exactly n/2 white balls (for n even), is a single outcome in the first 

case. In the second, it is a disjunction of n!/((n/2)!(n/2)!) constitutions, each of which is a distinct 

outcome. 



12 

where “HT” represent the outcome of a head and then a tail. In another analysis, the first two 

outcomes are replaced by a single H = (HH v HT) since if a head is tossed on the first throw, 

player A has won and the second coin will not be tossed. So now we are to be indifferent over 

the three outcomes of the coarsened space 

H = (HH v HT),   TH,   TT, 

which, by analogous reasoning, yields incompatible probability assignments. 

 This second class of paradox employs a continuous outcome space, indexed by a 

continuous parameter, and the redescription arises through a manipulation of this parameter. We 

shall see, however, that all the paradoxes still employ the same notion that indifference can 

persist through a coarsening of the outcome space. This second class of paradox is often 

associated with so-called “geometrical probabilities” (Borel, 1950, Ch.7) since these cases 

commonly arise in geometry. The locus classicus is Bertrand’s12 (1907, Ch. 1) discussion of a 

series of examples. In the simplest, we are to pick a real number at random between 0 and 100. 

We are indifferent to the number being in either half of this interval, 0 to 50 or 50 to 100, so we 

assign probability 1/2 to each. Our outcome space is 

0 to 50,      50 to 100 

and we assign probability 1/2 to each. If however we consider the squares of these numbers, then 

we can divide the corresponding interval of squares of the chosen number, 0 to 10,000 into four 

equal parts, to create a refined outcome space (the inverse of coarsening) in which the original 

outcome of 50 to 100 is divided into three parts 

0 to 2500 (= 02 to 502),      2500 to 5000,      5000 to 7500,      7500 to 10,000. 

 By indifference, we assign a probability 1/4 to each. The contradiction is immediate. The 

interval 0 to 50 was first assigned a probability 1/2. That same interval corresponds to the 

interval of squares 0 to 2500, to which we now assign the incompatible probability 1/4. 

 The basic idea of this construction can be replicated in many ways. Van Fraassen  (1989, 

pp. 302-303) describes a conversion of it into a perfect cube factory that produces iron cubes 

                                                
12 Bertrand seemed to make no connection with something like the not yet named principle of 

indifference. Rather he took his constructions merely to illustrate the danger of analyses that 

involve the infinities associated with continuous magnitudes. The existence of incompatible 

outcomes was for him simply evidence that the original problem was badly posed. 



13 

with sides between 0 and 2 cm. If we are indifferent to the length of the side, then we assign 

probability 1/2 to the outcome that one cube has a side with a length between 0 and 1 cm. For the 

1cm sided cube lies midway in the interval of side lengths of 0 to 2 cm. If we are indifferent to 

the surface area of the cubes, we end up assigning a probability 1/4 to this outcome. For side of 

area 1 cm2 lies 1/4th into the interval of side areas of 0 to 22=4 cm2. If we are indifferent to the 

volume of the cube, we end up assigning a probability 1/8 to this outcome. For the cube of 

volume 1 cm3 lies 1/8th into the interval of volumes of 0 to 23=8 cm3. 

 Bertrand himself gave a series of more complicated examples. The most discussed was 

the problem of deciding whether a randomly chosen chord of a circle is greater in length than the 

side of an inscribed equilateral triangle. Bertrand gave three constructions that yielded 

probabilities of 1/3, 1/2 and 1/4. In another example, he asked after the distance between two 

points chosen at random on a sphere. The probabilities of various distances differ according to 

whether the choosing is uniformly distributed on a great circle or uniformly over the area of the 

sphere. 

 These many multiplications of the one construction share a defect. They all depend on the 

assumption that it is appropriate to distribute beliefs indifferently in both the original description 

and the new coarsened or refined description. Perhaps that is just a mistake. Perhaps, as Gillies 

(2000, p. 46) notes, it is appropriate to exercise the principle of indifference in just one but not 

the other description, because of some difference intrinsic to the first description. The side, the 

area and the volume of a cube are all mathematically different in their properties and perhaps 

something about those differences dictate that we should be indifferent only over the length of 

the side. Sentiments such that these presumably lay behind Borel’s (1950, pp. 81-83) response to 

Bertrand’s problem of selecting two points on a sphere. Borel essentially insisted that selecting a 

point at random on a sphere must mean that the selection is uniformly distributed over the 

sphere’s surface area. 

 It seems far-fetched to me to imagine that we may find some property of one description 

that warrants us exercising indifference only over it. And if there is some reason that privileges 

one description over another, do the paradoxes not return if we are so ignorant that we do not 

know it? Nonetheless, the threat remains: we may imagine that we can find some distinctive 

property of one description that licenses the use of an accordingly embellished principle of 

indifference, where that distinctive property would not be present in the other description. 



14 

 This threat to the cogency of the paradox can be defeated fully by means of ideas 

introduced in the discussion of PII. Von Mises’ wine-water problem (above) provides a 

traditional illustration of the paradoxes of indifference. When we are indifferent to x, the ratio of 

water to wine, we distribute probability uniformly over x. So we are indifferent to the three 

outcomes with x in the ranges: 

x = 1/2 to 1,     x = 1 to 3/2,      x = 3/2 to 2 

Under the redescription in terms of x’, the ratio of wine to water, we are indifferent to the 

outcomes 

x’ = 2 to 3/2,     x’ = 3/2 to 1,     x’ = 1 to 1/2 

The paradox follows. We assign probability 1/3 to outcome x’=1 to 1/2, although it corresponds 

to the disjunctive outcome (x= 1 to 3/2) v (x = 3/2 to 2), each of whose disjuncts is also assigned 

probability 1/3. 

 What is distinctive in this formulation of the wine-water problem is that, as described 

above, there is a perfect symmetry between the two descriptions that includes the transformations 

used to relate them, which are self-inverting. So there can be no intrinsic difference between the 

two for an embellished version of the principle of indifference to exploit. Any distinctive feature 

that the principle may call up from one will, by the symmetry, assuredly be found in the other. 

 Indeed the example is even more troublesome for the principle of indifference. Exactly 

because of the perfect symmetry of the two descriptions, we should assign the same distributions 

of belief in each. The class of probability densities that respect the symmetry was specified by 

(5). It turns out not to contain the uniform distribution licensed by the principle of indifference. 

That is, the question is not to decide to which of the two descriptions the principle of indifference 

may be applied. Rather it turns out that the principle is applicable to neither. This last example is 

the strongest form of the paradoxes of indifference. 

 What we learn from this little survey of the paradoxes of indifference is a fact about 

routine expectations in the literature: the indifference expressed by the principle of indifference 

persists through coarsenings and refinements of the outcome space. While this expectation of 



15 

persistence may be challenged in many of the familiar paradoxes as a way of defeating the 

paradoxes, 13 we can find symmetric cases in which it cannot be challenged. 

3.2 Paradoxes of Invariance 

 While it is not generally recognized, it turns out that the use of invariance conditions as a 

way of specifying probability distributions is beset by paradoxes akin to those that trouble the 

principle of indifference. The paradoxes of indifference arose because greater ignorance gave us 

more partitions of the outcome space over which to exercise indifference. Each new partition 

corresponds to a new mathematical constraint on our probability measure. They quickly combine 

to produce a contradiction. The same thing happens with invariance conditions. Each ignorance 

is associated with an invariance. Thus the greater our ignorance, the more invariances we expect 

and the greater threat that these compounding invariances impose contradictory requirements on 

our probability distribution. 

 It takes only a little exploration to realize this threat. It is easily seen in a simple example 

that sufficient ignorance forces contradictory invariance requirements. Consider some magnitude 

x whose values lies in the open interval (0,1); and that is all we know about it. What is our prior 

probability density p(x) for x? The problem remains unchanged if we reparameterize the 

magnitude, retaining the range of parameter values in (0,1). In order to have the most defensible 

restrictions, let us consider only self-inverting transformations as reparameterizations, so that we 

have a perfect symmetry between the two descriptions and PII applies. The simplest of these is 

just 

x’ = 1 - x                                                       (9a) 

One may imagine that such self-inverting transformations are rare. They are not and very many 

can be found. Loosely speaking, as will be seen from the construction below, they are about as 

                                                
13 Keynes’ (1921, pp. 52-64) efforts to eliminate the paradoxes of indifference depend upon 

blocking the application of the principle to systems connected by disjunctive coarsening or 

refinement. Similarly Kass and Wasserman (1996, §3.1) propose that the paradoxes be avoided 

“practically” in that we use “scientific judgment to choose a particular level of refinement that is 

meaningful for the problem at hand.” 



16 

common in the world of functions on reals as are symmetric functions.14 The self-inverting (9a) 

is the λ=∞ member of the λ-indexed family of self-inverting transformations15 

! 

x'= f"(x) = " # "
2

+(" #1)
2
#(" # x)

2                                  (9) 

displayed graphically in Figure 1. Another simple member is λ=1, which is 

! 

x'= f1(x) =1" 2x" x
2                                                       (9b) 

                                                
14 The device used to generate the paradoxes of indifference, the rescaling of variables, is not 

sufficient to generate competing invariances, such as are needed to generate the paradoxes of 

invariance. That is, let p(x) and p’(x’) satisfy the conditions (10a) and (10b) and rescale the 

variables to X=f(x) and X’=f(x’), where both rescalings are effected by the same monotonic 

function f(.). Two new probability distributions are induced by 

P(X) = p(x) dx/dX      P’(X’) = p’(x’) dx’/dX’ 

It follows immediately that P(.) and P’(.) are the same functions since p(.) and p’(.) are the same 

functions; and dx/dX is the same function of X as dx’/dX’ is of X’. That is, the induced 

distributions P and P’ will satisfy B. Symmetry, and, by their construction, A. Agreement in 

probability as well. 
15 One can affirm that (9) is self-inverting by directly expanding fλ(fλ(x)) to recover x or by 

noting that x’= fλ(x) is equivalent to 

(λ-x’)2 + (λ-x)2 = λ2 + (λ-1)2                                                       (a) 

The expression for x’= fλ(x) is found by solving (a) for x’ in terms of x; and the expression for 

x= fλ-1(x’) by solving (a) for x in terms of x’. The two functions recovered must be the same 

since x and x’ enter symmetrically into (a). The properties of (9) are more easily comprehended 

geometrically. As equation (a) indicates, each curve in the graph is simply the arc of a circle with 

a center at (x’,x) = (λ, λ) and radius R satisfying R2 =  λ2 + (λ-1)2. 



17 

 
Figure 1. A Family of Self-Inverting Transformations 

That these functions are self-inverting manifests as a symmetry of the graph about the x’=x 

diagonal, represented as a dashed line in Figure 1. Indeed any function with this symmetry 

property will be self-inverting. It ensures that if x=a is transformed to x’= fλ(a), then x’=a will 

also be transformed to the same x= fλ(a) by the inverse transformation, as shown in the figure.16 

                                                
16 It is more cumbersome but possible to generate self-inverting functions by algebraic 

manipulation. To generate such a function with the domain [L,H], where L<H and L may be -∞ 

and H may be ∞, select some K such that L<K<H. Now pick any strictly decreasing function 

x’=gh(x), defined for K≤x≤H, the “h” high part of the domain, such that gh(K)=K and gh(H)=L. 

Since gh is strictly decreasing, it is invertible. Define gl(x) for L≤x≤K, the “l” low portion of the 

domain, by stipulating for x in this domain that gl(x) = gh-1(x). The self-inverting function f(x) is 

defined by f(x)= gl(x), for L≤x≤K, and f(x)= gh(x), for K≤x≤H. Even if gh and gl are arbitrarily 

differentiable, the combined f may fail to be differentiable at the join x=K. Arbitrary 

differentiability may be assured for f, such as in (9), by suitable selection of the derivatives of gh 

at x=K and x=H. 



18 

 The probability distribution p(x) must remain invariant under each transformation fλ, 

according to PII, for each yields a perfectly symmetric redescription of the problem. We can see, 

however, that no p(x) can exhibit this degree of invariance. To see this, we do not need all the 

properties of the family of self-inverting transformations fλ. We need only assume that it has two 

members, h1 and h2, say, such that h1(x)< h2(x) for all 0<x<1. That is trivially achieved using 

any pair of members of fλ, since, for fixed x in (0,1), fλ(x) is strictly increasing in λ. As before, 

PII requires that p(x) and its transform p’(x’) under a self-inverting transformation x’= fλ(x) 

must satisfy 

A. Agreement in probability     p’(x’)  =  -p(x) dx/dx’                                              (10a) 

B. Symmetry                               p’(.) = p(.)                                                         (10b) 

Now select any X, such that 0<X<1. For x1’(x)= h1(x), we have from (10a) and (10b) that 

! 

p(x)dx = " p(x
1
')
dx

1
'

dx
dx = p(x

1
')dx

1
'

x1 '=x1 '(X)

1

#
x=0

X

#
0

X

#  

Similarly for x2’(x)= h2(x), we have 

! 

p(x)dx = " p(x
2
')
dx

2
'

dx
dx = p(x

2
')dx

2
'

x2 '=x2 '(X)

1

#
x=0

X

#
0

X

#  

Subtracting and relabeling the variable of integration, we have 

! 

0 = p(y)dy
x1(X)

x2(X)

"                                                                   (11a) 

Since p(x)≥0 for all x, it follows that17 p(x)=0 for all x1’(X)<x< x2’(X). Since X is chosen 

arbitrarily, we may select X so that any nominated value of x lies in the interval (x1’(X), 

x2’(X)).18 Hence it follows that p(x) = 0 for all 0<x<1, which contradicts the requirement that a 

probability distribution normalize to unity. 

                                                
17 Or, if p(x) is discontinuous, it may differ from zero at most at isolated points, so that these 

non-zero values do not contribute to the integral. Therefore they are discounted, since they 

cannot contribute to a non-zero probability of an outcome. 
18 Proof: For a given 0<x<1, by we have from the initial supposition on h1 and h2 that x1’(x) < 

x2’(x). A suitable X is any value x1’(x) < X < x2’(x). For, from x1’(x) < X, we have x = 

x1’(x1’(x)) > x1’(X), where we use the fact that x1’ is strictly decreasing in x. Similarly from X < 



19 

4. What Should We Learn from the Paradoxes? 
 The moral usually drawn from the paradoxes of indifference is a correct by short-sighted 

one: indifference cannot be used as a means of specifying probabilities in cases of extensive 

ignorance. That there are analogous paradoxes for invariance conditions is less widely 

recognized. They are indicated obliquely in Jaynes’ work. He described (1973, §7) how he 

turned to the method of transformational invariance as a response to the paradoxes of 

indifference, exemplified in Bertrand’s paradoxes. They allowed him to single out just one 

partition over which to invoke indifference, so that (to use the language of Bertrand’s original 

writing) the problem becomes “well-posed.” The core notion of the method was (§7): 

Every circumstance left unspecified in the statement of a problem defines an 

invariance which the solution must have if there is to be any definite solution at all. 

We saw above that this notion leads directly to new paradoxes if our ignorance is sufficiently 

great to yield excessive invariance. Jaynes reported (§8) that this problem arises in the case of 

von Mises’ wine-water problem: 

On the usual viewpoint, the problem is underdetermined; nothing tells us which 

quantity should be regarded as uniformly distributed. However, from the standpoint 

of the invariance group, it may be more useful to regard such problems as 

overdetermined; so many things are left unspecified that the invariance group is too 

large, and no solution can conform to it. 

     It thus appears that the “higher-level” problem of how to formulate statistical 

problems in such a way that they are neither underdetermined nor overdetermined 

may itself be capable of mathematical analysis. In the writer’s opinion it is one of the 

major weaknesses of present statistical practice that we do not seem to know how to 

formulate statistical problems in this way, or even how to judge whether a given 

problem is well posed. 

When the essential content of this 1973 paper was incorporated into Chapter 12 of Jaynes’ 

(2003) final and definitive work, this frank admission of the difficulty no longer appeared, even 

                                                                                                                                                       

x2’(x), we have x2’(X) > x. The function x1’ is strictly decreasing since it is invertible, x1’(0)=1 

and x1’(1)=0; and similarly for x2’. 



20 

though no solution had been found. Instead Jaynes  (2003, pp. 381-82) sought to dismiss cases of 

great ignorance as too vague for analysis on the manifestly circular grounds that his methods 

were unable to provide a cogent analysis: 

If we merely specify ‘complete initial ignorance’, we cannot hope to obtain any 

definite prior distribution, because such a statement is too vague to define any 

mathematically well-posed problem. We are defining this state of knowledge far more 

precisely if we can specify a set of operations which we recognize as transforming the 

problem into an equivalent one. Having found such set of operations, the basic 

desideratum of consistency then places nontrivial restrictions on the form of the prior. 

My diagnosis, to be developed in the sections below, is that Jaynes was essentially correct in 

noting that invariance conditions may overdetermine an ignorance belief state. Indeed the 

principle of indifference also overdetermines such a state. In this regard, we shall see that both 

instruments are very effective at distinguishing a unique state of ignorance. The catch is that this 

state is not a probability distribution. Paradoxes only arise if we assume in addition that it must 

be. We thereby fail to see that PI and PII actually work exactly as they should. 

5. A Weaker Structure 
 In order to establish that PI and PII do pick out a unique state of ignorance, we need a 

structure hospitable to non-probabilistic belief states. Elsewhere, drawing on an extensive 

literature in axiom systems for the probability calculus, I have described such a structure 

(Norton, forthcoming). For a precise synopsis of its content, see Appendix. Informally, its basic 

entity is [A|B] is introduced through the properties of F. Framework. It represents the degree to 

which proposition B inductively supports proposition A, where these propositions are drawn 

from a (usually) finite set of propositions closed under the Boolean operations of  ∼ (negation), ∨ 

(disjunction) and & (conjunction). The degrees are not assumed to be real valued. Rather it is 

only assumed that they form a partial order so that we can write [A|B] ≤ [C|D] and [A|B] < 

[C|D]. These comparison relations will be restricted by one further notion. Whatever else may 

happen, we do not expect that some proposition B can have less support that one of its 

disjunctive parts A on the same evidence. That is, we require monotonicity: if A⇒B⇒C, then 

[A|C]≤[B|C]. This much of the structure will provide the background for the analysis to follow. 



21 

 The structure has further notions that will prove incompatible with the ignorance state to 

be defined. The first, introduced through the property A. Addition, is an addition operator ⊕, 

which allows the combining of degrees of support of mutually contradictory propositions to yield 

the degree of support of their disjunction. It is the surrogate of the additivity of the probability 

calculus. The second, introduced through the property B. Bayes Property, is a multiplication 

operator ⊗, which allows for the distinctive dynamics of updating associated with Bayes’ 

theorem. Both of these properties are compatible with the probability calculus and express 

essential elements of it. 

 The existence of these two operators is logically independent of one another; we can have 

systems that have either one without the other. We shall see, however, that the unique state of 

ignorance will be incompatible with each of them individually. 

 This structure is formulated in terms of “degrees of support.” On the supposition that we 

believe what we are warranted to believe, I will presume that our degrees of beliefs agree with 

these degrees of support. 

6. Characterizing Ignorance 
 Once we dispense with the idea that a state of ignorance must be represented by a 

probability distribution, we can return to the ideas developed in the context of PI, PII and their 

paradoxes and deploy them without arriving at contradictions. We can discern two properties of 

a state of ignorance: invariance under disjunctive coarsenings and refinements; and invariance 

under negation. As we shall see below, each is sufficient to specify the state of ignorance fully 

and it turns out to be the same state: a single ignorance degree of belief “I” assigned to all 

contingent propositions in the outcome space. 

6.1 Invariance under Disjunctive Coarsenings and Refinements 

 We saw in Section 3.1 above that the paradoxes of indifferences all depended upon a 

single idea: if our ignorance is sufficient, we may assign equal beliefs to all members of some 

partition of the outcome space and that equality persists through disjunctive coarsenings and 

refinements. This idea is explored here largely because of its wide acceptance in that literature. I 

have already indicated above in Section 3.1 that the idea is less defensible in cases in which there 

is not a complete symmetry between the two descriptions. We shall see in Section 6.2 below that 



22 

the same results about ignorance as derived here in Section 6.1 can be derived from PII using 

descriptions that are fully symmetric, related by self-inverting transformations. 

 Let us develop the idea of invariance of ignorance under disjunctive coarsenings and 

refinements. If we have an outcome space Ω partitioned into mutually contradictory propositions 

Ω = A1 v A2 v … v An, an example of a disjunctive coarsening is the formation of the new 

partition of mutually contradictory propositions Ω = B1 v B2 v … v Bn–1, where 

B1= A1, B2= A2, … , Bn–1= An–1 v An                                               (12) 

All disjunctive coarsenings of A1, A2, …, An are produced by finitely many applications of this 

coarsening operation along with arbitrary permutations of the propositions, as defined by (1) 

above. A partition and its coarsening are each non-trivial if none of their propositions is Ω or ∅. 

The inverse of a coarsening is a refinement. 

 Assume that we have no grounds for preferring any of the members of the non-trivial 

partition of Ω = A1 v A2 v … v An, then by PI we assign equal belief to each, and, by 

supposition, this ignorance degree of belief [∅|Ω]<I<[Ω|Ω] is neither certainty nor complete 

disbelief: 

[A1|Ω]  = [A2|Ω] = ... = [An|Ω] = I                                                   (13a) 

Now consider the coarsening (12) and assume that we have no grounds for preferring any of the 

members. Once again there exists a possibly distinct ignorance degree of belief I’, neither 

certainty nor complete disbelief, such that 

[B1|Ω]  = [B2|Ω] = ... = [Bn–1|Ω] = I’                                               (13b) 

Since from (12) B1= A1, we have [A1|Ω] = [B1|Ω] so that 

I’=I 

Since all coarsenings are produced by successive applications of (12) along with permutations, it 

follows that the one ignorance degree of belief I is unique. Finally, if C is a non-trivial 

disjunction of some proper subset of the {A1, A2, …, An}—written  C = Aav…vAb—some 

sequence of coarsening and permuations allows us to infer that its degree of confirmation is I. 

That is, we infer the distinctive property of ignorance presumed in the paradoxes of indifference 

[Aa|Ω]  =  ...  = [Ab|Ω] =  [ Aav…vAb |Ω]  = Ι                                                   (14) 



23 

6.2 Invariance of Ignorance Under Negation 

 We saw in Section 2.2 above that one application of PII was Jaynes’ deduction of the 

principle of indifference by requiring that a state of ignorance over a finite outcome space 

remains invariant under a permutation of the propositions. This same idea can also be applied to 

a transformation that switches propositions with their negations. If we are really ignorant over 

some outcome A, then our degree of belief in A would be unchanged if A had been somehow 

switched with its negation ∼A. A short parable may help clarify the transformation. 

 Let us imagine that we are to receive a message pertaining to the two compatible 

outcomes “land” and “sea” by means of a secret code that assigns numbers to each outcome. The 

code was devised by a very dedicated logician, so there are numbers for all possible 16 logical 

combinations of the outcomes. For greater security, we have two code books to choose from. 

Their values are (shh—don’t tell!): 

 



24 

Code in Book A Secret Message Code in Book B 

1 ∅ = ∼Ω 16 

2 land & sea = ∼(∼land v ∼sea) 15 

3 ∼land & sea = ∼(land v ∼sea) 14 

4 land & ∼sea = ∼(∼land v sea) 13 

5 ∼land & ∼sea = ∼(land v sea) 12 

6 sea = ∼(∼sea) 11 

7 land = ∼(∼land) 10 

8 (land & sea) v (∼land & ∼sea) = 

∼((land & ∼sea) v (∼land & sea)) 

9 

9 (land & ∼sea) v (∼land & sea) = 

∼((land & sea) v (∼land & ∼sea)) 

8 

10 ∼land = ∼(land) 7 

11 ∼sea = ∼(sea) 6 

12 land v sea = ∼(∼land & ∼sea) 5 

13 ∼land v sea = ∼(∼land & sea) 4 

14 land v ∼sea = ∼(∼land & sea) 3 

15 ∼land v ∼sea = ∼(land & sea) 2 

16 Ω = ∼∅ 1 

Table 1. Code Books Illustrating the Negation Map 

 

Prior to its receipt we are in complete ignorance over which message may come and begin to 

contemplate how credible the content of each message may be. On the presumption that Book A 

is in use, we assign beliefs not over which message may come, but over the truth of the 16 

possible messages. We must assign maximum belief to the content of code 16, since we know 

that Ω is necessarily true; we must assign minimum belief to the contents of code 1, since the 

contradiction ∅ is always false; and we assign intermediate, ignorance degrees of belief to 

everything in between. In convenient symbols 

[1|Ω] = [∅|Ω]  [2|Ω] = I2  [3|Ω] = I3 … [14|Ω] = I14  [15|Ω] = I15  [16|Ω] = [Ω|Ω] 



25 

 We now find that it is not Book A, but Book B that will be used. How will that affect our 

distribution of belief? We must exchange our degrees of belief for the message with codes 1 and 

16, since code 1 now designate the necessarily true Ω and code 16 the necessarily false ∅. What 

of the remaining messages? We had assigned degree of belief I6 to what we thought was the 

message “sea”. It now turns out to be the message “∼sea”. If our ignorance is sufficient, that will 

have no effect on our degree of belief. “sea”? “∼sea”? We just do not know! That is, we must 

assign equal degree of belief to each, so that I6 = I11. This analysis can be repeated for all the 

remaining outcomes 2 to 15. In each case, the switching of the codebooks has simply switched 

one message with its negation, as the table reveals. For example, under the Book A, code 14 

designates “land v ∼sea.” Under Book B, code 14 designates its negation “∼land & sea” and the 

original “land v ∼sea” is designated by code 3. So, by analogous reasoning, I3 = I14. We can also 

infer that all four values are the same by using the property of monotonicity mentioned in Section 

5 above.19 Continuing in this way, we can conclude that all intermediate degrees of ignorance 

have the same value I = I2 = … = I15. Rather than displaying the argument in all detail, it is 

sufficient to proceed to the general case, whose proof covers this special case. 

 An outcome space  Ω consists of propositions C1, C2, …, Cm generated by closing a set 

of atomic propositions A1, A2, …, An, under the usual Boolean operators ∼, v and &, taking note 

of the usual logical equivalences. The remapping of codebooks corresponds to the negation map 

N between set of proposition labels C1, C2, …, Cm and a duplicate set of proposition labels, C1’, 

C2’, …, Cm’, in which  

∼Ci’ = N(Ci)                                                                           (15) 

What is important about this negation map (15) is that it is self-inverting—the negation of a 

negation returns the original proposition (or, in this case, its label clone). Thus the sets Ci and C’i 

are symmetric descriptions20 and, if we are in ignorance over the outcomes, PII may be applied. 

                                                
19 Since ∼land & sea ⇒ sea, we have I3≤I6; and since ∼sea ⇒ land v ∼sea, we have I11≤I14. 

Recalling I6 = I11 and I3 = I14, we must have I6 = I11 = I3 = I14. 

20 This symmetry may not be evident immediately since the negation map (15) can take an 

atomic propositions (such as A1) and map it to a disjunctive propositions (here A2v…vAn) and 



26 

The analysis proceeds with the two-component calculation already shown in Section 2.2. Since 

the map simply relabels the same outcomes, we have 

A. Agreement in degrees of belief          [∼Ci’|Ω’] = [Ci| Ω] 

But since we have a perfect symmetry in the two descriptions of the outcomes, we also have for 

the contingent propositions (i.e. those that are not always true or always false)21 

B. Symmetry          [Ci’|Ω’] = [Ci| Ω] 

It follows immediately from these two conditions A. and B. that [∼Ci’|Ω’] = [Ci’|Ω’]; or, re-

expressed in the original proposition labels 

[∼Ci|Ω] = [Ci|Ω]  =  Ii                                                           (16) 

where this condition holds for every contingent proposition in the outcome space Ω. So far, we 

cannot preclude that the ignorance degree Ii defined is unique for each distinct pair of contingent 

propositions Ci, ∼Ci. As before, monotonicity allows us to infer that all Ii are equal to a common 

value I. To see this, take any two contingent propositions C and D. There are two cases: 

(I) C ⇒ D or D ⇒ C or ∼C ⇒ D or C ⇒ ∼D. Since they are the same under relabeling, 

assume C ⇒ D, so that ∼D ⇒ ∼C. We have from monotonicity that [C|Ω] ≤ [D|Ω] and 

[∼D|Ω] ≤ [∼C|Ω]. But we have from (16) that [∼C|Ω] = [C|Ω] and [∼D|Ω] = [D|Ω]. 

Combining, it now follows that [∼C|Ω] = [C|Ω] = [∼D|Ω] = [D|Ω]. 

(II) Neither C ⇒ D nor D ⇒ C nor ∼C ⇒ D nor C ⇒ ∼D. This can only happen when C&D 

is not ∅. Since C&D ⇒ D, we can repeat the analysis of (I) to infer that [∼(C&D)|Ω] = 
                                                                                                                                                       

conversely; whereas our earlier examples of self-inverting maps (such as an exchange of two 

labels Ai and Ak) mapped atomic propositions to atomic propositions. This greater complexity 

does not compromise the facts that Ci and C’i label the same set of propositions and that the map 

between them is self-inverting, which is all that is needed for the symmetry. 
21 Restricting B. Symmetry to contingent propositions only is really a stipulation on the type of 

ignorance being characterized. We are assuming that the ignorance does not extend to logical 

truths, such as Ω, and logical falsities, such as ∅. Without the restriction, we would recover a 

more extensive ignorance state in which we would be uncertain even over logical truths and 

falsities. There is no contradiction in such a state, but it is of lesser interest, since knowledge of 

logical truths can at least in principle be had without calling upon external evidence. 



27 

[C&D|Ω] = [∼D|Ω] = [D|Ω]. Similarly, we have C&D ⇒ C, so that [∼(C&D)|Ω] = [C&D|Ω] 

= [∼C|Ω] = [C|Ω]. Combining we have the result sought: [∼C|Ω] = [C|Ω] = [∼D|Ω] = [D|Ω]. 

We have now used PII to infer that, in cases of ignorance, every contingent proposition in the 

outcome space Ω must be assigned the same ignorance value I. This is the same result as arrived 

at in Section 6.1 above by means of the idea that the ignorance state must be invariant under 

disjunctive coarsenings and refinements. Thus we affirm that both approaches lead us to a unique 

state of ignorance, in which all contingent propositions are assigned the same ignorance value I. 

  There is a more formal way of understanding the generation of this unique ignorance 

state from the condition of invariance under negation. Elsewhere (manuscript), I have 

investigated how the familiar duality of truth and falsity in a Boolean algebra may be extended to 

real-valued measures defined on the algebra. To each additive measure m, there is a dual additive 

measure M, defined by the dual map M(A)=m(∼A), for each proposition A in the algebra. The 

dual of an additive measure is itself additive but not in the familiar way. The familiar rule that 

allows us to add the measures of propositions (whose conjunction is ∅) when they are disjoined 

is replaced by an unfamiliar rule that allows us to add the dual measures of propositions (whose 

disjunction is Ω) when they are conjoined. Because additive measures and their duals obey 

different calculi, additive measures are not self-dual. 

 We arrive at the ignorance state for the case of real valued measures by the simple 

condition that the measure be self-dual in its contingent propositions. That is, it is a monotonic, 

real-valued measure22 m that remains invariant under the dual map M(A)=m(∼A), for contingent 

propositions A, so that it satisfies the defining condition mI(A)=mI(∼A). If we set by convention 

that the extreme values of this ignorance map are mI(∅) = 0 and mI(∅) = 1, then it follows23 that 

all contingent propositions must be assigned the same real value mI(A)=I, where I is some fixed 

real, 0<I<1. Clearly mI is not additive. 

                                                
22 A monotonic measure is defined analogously to the property of monotonicity (Appendix). If 

A⇒B⇒C, then m(A|C)≤ m(B|C). 
23 Proof: set mI(Ci) = [Ci|Ω] = Ii. Then the uniqueness of Ii follows from the demonstration of 

the uniqueness of Ii of (16) and requires the monotonicity of the measure. 



28 

6.3 Comparing Ignorance Across Different Outcome Spaces 

 The arguments of Sections 6.2 and 6.3 establish that, for each outcome space, there exists 

a unique ignorance degree of belief for all contingent propositions in it. That leaves the 

possibility that this ignorance degree of belief is different for each distinct outcome space. It is a 

natural expectation that the same ignorance degree of belief can be found in all outcome spaces. 

Naturalness, however, is no substitute for demonstration. The difficulty in mounting a 

demonstration is that, if B. Bayes Property is not assumed, the framework set up in Section 5 is 

too impoverished to enable comparison of degrees of belief between different outcome spaces. 

Some further assumption is needed to enable the comparison. In this section, I will show that 

introducing a very weak notion of independence and assuming that it is occasionally instantiated 

is sufficient to allow us to infer that the same ignorance degree of belief arises in all finite 

outcome spaces. 

 Consider an outcome space “A” defined by the logical closure under Boolean operations 

of m mutually exclusive and exhaustive, contingent propositions A1, A2, … Am, so that  Ω = A1 

v A2 v … v Am. We shall assume that we are in a state of complete ignorance over A so that 

IA  =  [Ai|Ω]A 

for i = 1, …, m. The subscripts A allow the possibility that the ignorance degree of belief and 

other degrees of belief are peculiar to the outcome space A. A second outcome space “B” is 

defined analogously with n propositions B1, B2, … Bn, for which  

IB  =  [Bk|Ω]B 

where k = 1, …, n. Finally we define a product outcome space “AB” as generated in the same 

way by the mn mutually exclusive and exhaustive propositions, (A1&B1), (A1&B2), …, 

(Am&Bn). Degrees of belief relative to this product outcome space are designated by [.|Ω]AB. 

 A very weak notion of independence of the two spaces is: 

Weak independence of outcome spaces A and B. The degree of belief in some 

proposition of A is unaffected by the mere knowledge that outcomes in B are 

possible; and conversely. That is expressed by the condition 

[Ai | Ω] A = [Ai | Ω] AB and     [Bk | Ω] B = [Bk | Ω] AB                           (17) 

for all admissible i, k. 



29 

This condition is much weaker than the usual condition of probabilistic independence. In the 

latter, the probability assigned to an outcome of one space is unaffected when the outcome is 

conditionalized on the supposition that some outcome of the other space obtains. In (17) we 

conditionalize only on the knowledge that the other outcome space exists, not that one of its 

outcomes obtains. 

 An example illustrates the differing strengths. Nothing in the discussion above precludes 

the propositions B1, B2, … Bn of the second outcome space being merely a permutation of the 

propositions A1, A2, …, Am of the first outcome space. In that case, ordinary probabilistic 

independence between the two spaces would fail. However the weaker sense of (17) would still 

hold. To see that this weaker sense is not vacuous, imagine a second example in which the 

propositions of the second outcome space are B1=A1, B2=A2, …, Bn=Am-1; that is, the second 

space allows everything in first but denies Am. (In effect this is the space generated from the A 

outcome space by conditioning on A1v…vAm-1.) Since the B but not A outcome space presumes 

Am is false, we would not expect (17) to hold; for learning the range of possibility admitted by B 

supplies new information that can alter judgments of degrees of belief. Indeed relation (17) must 

fail in case i=m, for [Am|Ω]A=IA but24 [Am|Ω]AB =[∅|Ω]AB. 

 If outcome spaces A and B are independent in the sense of (17), then we can show that 

their ignorance degrees of belief are the same. First note that for this case, we must also have an 

ignorance distribution in the product space AB, with an ignorance degree IAB 

[Ai | Ω] AB =  [Bk | Ω] AB =  IAB 

for admissible i, k. From the weak notion of independence (17), we also have 

IA = [Ai | Ω] A = [Ai | Ω] AB     and     IB =  [Bk | Ω] B = [Bk | Ω] AB 

Combining we have 

IAB   =   IA   =   IB                                                                   (18) 

so that the ignorance degree of belief in two independent outcome spaces is the same. 

                                                
24 In the outcome space AB, Am is represented by the disjunction Am = (Am&B1)v…v(Am&Bn) 

= (Am&A1)v…v(Am&Am-1) = ∅ v … v ∅  =  ∅.  



30 

 This last conclusion is enough to enable us to conclude the uniqueness of the ignorance 

degree of belief for all outcome spaces, even ones that are not independent in the sense of (17). 

To see this, imagine that the outcome spaces A and B are not independent and that their 

ignorance degrees of belief are IA and IB. We need only assume that there exists a third outcome 

space, C, that is independent from each of A and B in the sense of (17), with ignorance degree of 

belief IC. It now follows from (18) that IC = IA and IC = IB. Combining them, we have IA = IB, 

which establishes the equality of the degrees of ignorance for any two outcome spaces A and B. 

7. Incompatibility with Degrees of Belief as Probability 

Measures 
 It is evident that the assigning of the same ignorance degree of belief to all contingent 

propositions in the outcome space is incompatible with the distribution of belief being a 

probability measure. What we shall see in this section is that this incompatibility runs quite deep. 

It contradicts two, important, independent aspects of the probability calculus.  

7.1 Incompatibility with A. Addition 

 The surrogate here for the additivity of the probability calculus is the existence of an 

addition operator ⊕ with properties A. Addition defined in the Appendix. It is incompatible with 

the ignorance state defined in Section 6. To see this, assume that we have such a state on an 

outcome space Ω that includes two mutually contradictory propositions A and B. For them we 

have 

[A v B|Ω] = [A|Ω] = [B|Ω] = I 

where I is distinct from [Ω|Ω] and [∅|Ω]. If, for purposes of contradiction, we presume that we 

also have an addition operator defined on the space, that operator relates these degrees as 

[A v B|Ω] = [A|Ω] ⊕ [B|Ω] 

where [A|Ω] ⊕ [B|Ω] is strictly increasing, and thus invertible, in both arguments. Combining, 

we have 

[A|Ω] ⊕ [B|Ω]  = [A|Ω] = [B|Ω] = I 

That is, the ignorance degree I acts as a zero of the operator in the restricted sense that 

composing it with itself leaves it unchanged 



31 

[A|Ω] ⊕ I = I ⊕ [A|Ω] = [A|Ω] 

However, since A v ∅ = ∅ v A = A, the minimal value [∅|Ω] also is a zero of ⊕ in the more 

general sense that composing it with anything leaves the latter unchanged: 

[A|Ω] ⊕ [∅|Ω] = [∅|Ω] ⊕ [A|Ω] = [Av∅|Ω]  = [A|Ω] 

Comparing the last two expressions and recalling that ⊕ is invertible in both arguments, we 

conclude that 

I = [∅|Ω] 

That is an unacceptable outcome, for [∅|Ω] is the minimum degree of belief assigned to the 

contradiction ∅ that we are certain is false. The incompatibility is thereby established. 

 That the ignorance state is incompatible with A. Addition was to be expected on quite 

general grounds. The presence of this operator amounts to a particular interpretation for the 

degrees in the system: as they range from high to low, they span justification of complete belief 

to complete disbelief. As is described in greater detail in Norton (forthcoming, Section 4.1), this 

follows from a reciprocal relationship between belief and disbelief: high belief in A entails high 

disbelief in ∼A, and conversely. (More or less ignorance concerning A is not reciprocally related 

to less or more ignorance in  ∼A.) If one translates this reciprocal relationship into functional 

terms, the existence of the ⊕ operator follows. 

7.2. Incompatibility with B. Bayes Property 

 B. Bayes Property, with its multiplication operator ⊗, gives the system the characteristic 

dynamics of conditionalization associated with Bayes’ theorem. From it, for hypothesis H and 

evidence E, we infer a version of Bayes’ theorem 

[H&E|Ω] = [H|E] ⊗ [E|Ω] = [E|H] ⊗ [H|Ω] 

Since the operator ⊗ is almost25 always strictly increasing and invertible in both arguments, the 

usual dependencies follow. The posterior [H|E] increases when, other terms fixed, the 

expectedness [E|Ω] decreases; or the likelihood [E|H] increases; or the prior [H|Ω] increases. 

 An ignorance state on some outcome space Ω is incompatible with B. Bayes Property, as 

long as we also require that it be possible for there to remain some uncertainty after we 
                                                
25 For example, when [E|H] = [∅|H], [E|H] ⊗ [H|Ω] is not strictly increasing in [H|Ω]. [H|E] ⊗ 

[E|Ω] is undefined when [E|Ω] = [∅|Ω]; and similarly for [E|H] ⊗ [H|Ω]. 



32 

conditionalize in a state of ignorance. To see this, assume that we have two contingent 

propositions A and B in Ω where A⇒B and B. Bayes Property holds. We have 

[A|Ω] = [A|B] ⊗ [B|Ω] 

Since [A|Ω] = [B|Ω] = I, we have 

I = [A|B] ⊗ I                                                          (19) 

Indeed, if we set A=B, this becomes 

I = [A|A] ⊗ I                                                        (19a) 

Since ⊗ always invertible in the first argument, it follows from (19) that [A|B] will be the same 

for all contingent A and B, where A⇒B. Worse, from (19a) we see that this unique value is 

certainty 

[A|B] = [A|A] = [Ω|Ω] 

That is an extraordinarily unintuitive result. It tells us that if we are completely ignorant over 

1000 mutually exclusive outcomes A1, …., A1000 but condition A1 on the first 999, we become 

certain of A1 since [A1| A1v ….vA999] = [Ω|Ω]. 

 We may seek to avoid these outcomes by portraying the ignorance degree I as a zero of 

the operator ⊗ so that (19) would not be invertible. Indeed that would have the very desirable 

outcome of enabling non-trivial conditionalization. Since (19) would now no longer be 

invertible, we could maintain ignorance degrees for both [A|Ω] and [B|Ω] and then be free to let 

other considerations fix the value of [A|B].26 Attractive as it is, this possibility contradicts B. 

Bayes Property, which does not admit a zero in the second position of the operator ⊗. The 

natural zero in [A|B] ⊗ [B|C] would be [B|C] = [∅|C]. It is precluded since B. Bayes Property 

requires B to be “admissible,” that is, not of minimum measure. The analogous prohibition in 

probability theory is that if A⇒B and P(A)=P(B)=0, then P(A|B) = P(A)/P(B) = “0/0” is not 

defined. 

                                                
26 This possibility is realized in the “theory of random propositions,” described in Norton (1994, 

§7.3). 



33 

8. Conclusion 
 How should an epistemic state of ignorance be represented? My contention here is that 

we have long had the instruments that uniquely characterize it in the principle of indifference and 

the principle of invariance of ignorance. However our added assumption that epistemic states 

must also be probability distributions has led to contradictions that we have misdiagnosed as 

arising from some deficiency in the two principles. 

 There are other proposals for representing states of ignorance. In the Shafer Dempster 

theory of belief functions (Shafer, 1976, pp. 23-34), ignorance is represented by a belief function 

that assigns zero belief both to a proposition A and its negation, Bel(A) = Bel(∼A) = 0, but unit 

belief to their certain disjunction, Bel(Av∼A) = 1. A weakness of this proposal is that it is what I 

shall call “contextual.” That is, our ignorance concerning some outcome A is not expressed 

simply by the value assigned directly to A. Bel(A)=0 can mean ignorance if Bel(∼A) = 0, or it 

can mean disbelief if Bel(∼A) = 1. Its meaning varies with the context. One may also represent 

ignorance through convex sets of probability measures; complete ignorance consists of the set of 

all possible measures on some outcome space. I have elsewhere (Norton, forthcoming, §4.2) 

explained my dissatisfaction with this last proposal. Briefly my concern is the indirectness of 

using probability measures, which do have properties A. Addition and B. Bayes Property, to 

simulate the behavior of distributions of belief that do not. One difficulty suggests that the 

simulation is not complete. We expect an epistemic state of ignorance to be invariant under the 

negation map (15). Convex sets of probability measures are not invariant under that map, for, 

under that map, an additive probability measures is transformed to a dual additive measure, 

which obeys a distinct calculus. 

 Finally, one may well wonder about the utility of the epistemic state of ignorance defined 

here. It invokes a single degree of belief that is neither complete belief nor disbelief, assigned 

equally to all contingent propositions in the outcome space, and is resistant to both addition and 

Bayesian updating. Might such a state really arise in some non-trivial problem? My contention 

elsewhere (forthcoming, §8.3) is that it already has. There is an inductive logic naturally adapted 

to inferences over the behavior of indeterministic physical systems. Its basic belief state 

coincides with the ignorance state described here. 



34 

Appendix 
The following is a synopsis of the system described in Norton (forthcoming). 

F. Framework 

 A (usually) finite set of propositions (sometimes assumed mutually exclusive and 

exhaustive) A1, A2, … is closed under the familiar Boolean operations  ∼ (negation), ∨ 

(disjunction) and & (conjunction) and, occasionally, countable disjunction. The formula A ⇒ B 

(“A implies B”) means that the propositions are so related that ∼A∨B must always be true. The 

universal proposition, Ω, is implied by every proposition in the algebra and is always true. The 

proposition, ∅, implies every proposition and is always false. 

 The symbol [A|B] represents the degree to which proposition B confirms proposition A. 

It is undefined when B is of minimum degree, which means that B=∅, or there is a C such that 

B⇒C and [B|C]= [∅|C]. The sentences [A|B] ≤ [C|D] and [C|D] ≥ [A|B] means ‘D confirms C at 

least as strongly as B confirms A.’ The relation ≤ is a partial order; that is, it is reflexive, 

antisymmetry and transitive. The sentences [A|B] < [C|D] and [C|D] > [A|B] hold just in case  

[A|B] ≤ [C|D] but not [A|B] = [C|D]. For all admissible27 propositions A, B, C and D: 

[∅|Ω] ≤ [A|B] ≤ [Ω|Ω] 

 [∅|Ω] < [Ω|Ω] 

 [A|A] = [Ω|Ω] and  [∅|A] = [∅|Ω] 

 [A|B] ≤ [C|D] or [A|B] ≥ [C|D] (universal comparability) 

if A⇒B⇒C, then [A|C]≤[B|C] (monotonicity) 

A. Addition. 

 For any admissible proposition Z and mutually contradictory propositions X and Y, there 

exists an addition operator ⊕ such that [X∨Y|Z] = [X|Z] ⊕ [Y|Z] where ⊕ is strictly increasing in 

both [X|Z] and [Y|Z]. 

B. Bayes Property is the conjunction of N. and M.: 

 N. Narrowness. 

 For any proposition A and any admissible B, [A|B] = [A&B|B] 

                                                
27 Here and elsewhere, “admissible” precludes formation of the undefined [.|B], where B is of 

minimum degree. 



35 

M. Multiplication. 

 For any proposition A and admissible propositions B and C such that A ⇒ B ⇒ C, there 

exists a multiplication operator ⊗ such that [A|C] = [A|B] ⊗ [B|C] where ⊗ is strictly increasing 

and thus invertible in both arguments (excepting [B|C], when [A|B]=[∅|B]). 

R. Real Values. 

 For any admissible propositions A, A’, B and B’, the set of values possible for degrees of 

confirmation [A|B] can be mapped one-one onto a closed set of reals such that the mapped real 

values f([A|B]) > f([A’|B’]) just in case [A|B] > [A’|B’]. 

All these properties combined are sufficient to entail the existence of real valued degrees of 

support that can be rescaled to yield a conditional probability measure. 

References 
Bertrand, Joseph (1907) Calcul des Probabilités. 2nd Ed. Gauthier-Villars: Paris; repr. New 

York: Chelsea. 

Borel, Émile (1950) Elements of the Theory of Probability. Trans. John E. Freund. Englewood 

Cliffs, NJ: Prentice-Hall, Inc., 1965. 

Cramer, Harald (1966) The Elements of Probability Theory and Some of its Applications. 

Huntington, New York: Robert E. Kreiger Publishing Co., 2nd Ed., 1966; reprinted 1973. 

Galavotti, Maria Carla (2005) Philosophical Introduction to Probability. Stanford: CSLI 

Publications. 

Gillies, Donald (2000) Philosophical Theories of Probability. London: Routledge. 

Howson, Colin and Urbach, Peter (1996) Scientific Reasoning: The Bayesian Approach. 2nd ed. 

Chicago and La Salle, IL: Open Court. 

Jaynes, E. T. (1973) “The Well-Posed Problem,” Foundations of Physics, 3, pp. 477-493. 

Jaynes, E. T. (2003) Probability Theory: The Logic of Science. Cambridge: Cambridge 

University Press. 

Jeffreys, Harold (1961) Theory of Probability. 3rd Edition. Oxford: Oxford University Press. 

Kass, Robert E. and Wasserman, Larry (1996) “The Selection of Prior Distributions by Formal 

Rules,” Journal of the American Statistical Association, 91, pp. 1343-70. 

Keynes, John Maynard (1921) A Treatise of Probability. London: MacMillan; reprinted New 

York: AMS Press, 1979. 



36 

Laplace, Pierre-Simon (1825) Philosophical Essay on Probabilities/ 5th ed. Trans. Andrew I. 

Dale. New York: Springer-Verlag, 1995. 

Norton, John D. (1994) "The Theory of Random Propositions," Erkenntnis, 41, pp. 325-352. 

Norton, John D. (forthcoming) “Probability Disassembled,” British Journal for the Philosophy of 

Science. 

Norton, John D. (manuscript) “Disbelief as the Dual of Belief.” 

Shafer, Glen (1976) A Mathematical Theory of Evidence. Princeton: Princeton University Press. 

Van Fraassen, Bas (1989) Laws and Symmetries. Oxford: Clarendon. 

Von Mises, Richard (1951) Probability, Truth and Statistics. 3rd German ed.; 2nd revised 

English translation. London: George Allen and Unwin, Ltd., 1957; repr. New York: 

Dover, 1981.