Objective Chance → Moore’s Paradox

A Moore Statement (one that instantiates Moore’s Paradox) is a statement that could be true, but could not be believed.   For example, “It is raining but I don’t believe that it is raining”.

We find interesting new varieties of such statements when we replace the intuitive notion of belief with subjective probability.  Then there are two kinds of Moore Statements to be distinguished:

An Ordinary Moore Statement is one that could be true, but cannot have probability one.
A Strong Moore Statement is one that could have positive probability, but could not have probability one.

When we introduce statements about objective chance there are Moore Statements in our language.  Consider first the following (not a Moore statement) said when about to toss a die:

[1] The number six won’t come up, but the chance that six will come up is 1/6.

On this occasion both conjuncts can be true.  The die is fair, so the second conjunct is true, and when we have tossed the die we may verify that our prediction (the first conjunct) was true as well.

Moreover, [1] can be believed, perhaps by a gambler who bet that the outcome will be odd and is feeling lucky.  Or at least he could say, even with some warrant, that it seems likely (or at least a little likely) that [1] is the case.  The gambler could even say (and who could disagree, if the die is known to be fair?) that the probability that [1] is true is 5/6!

The way I will symbolize that is:            P(~Six & [ch(Six) = 1/6]) = 5/6.

In this sort of example we express two sorts of probability, one subjective and one objective.  Are there some criteria to be met?  Is there to be some harmony between the two?

Like so much else, there are some controversies about this.  I propose what I take to be an absolutely minimal constraint:

Minimal Harmony.  P(ch(A) > 0) = 1 implies P(A) > 0         If I am sure that there is some positive chance that A then it seems to me at least a little likely that A.

I really cannot imagine someone seriously, and rationally, saying anything like 

“I am certain that there is some chance that the six will come up, but I am also absolutely certain that it will not happen”.  

Except a truly deluded gambler, with a gambling strategy sure to lead to eventual ruin?    

To construct a Moore Statement we only need to modify [1] a little:

[2] The number six won’t come up, but the chance that six will come up is not zero

~Six & ~[ch(Six) = 0]

That [2] could be true we can argue just like we did for [1].  But [2] is a Moore Statement for it could not have subjective probability 1, by the following argument.

Assume that P([2]) = 1.  Then:

1. P(~Six) = 1
2. P(Six) = 0
3. P(~[ch(Six) = 0]) = 1
4. ~[ch(Six) = 0] is equivalent to [ch(Six) > 0]
5. P(ch(Six) > 0) = 1

contradiction between 2. and 5, violation of principle Minimal Harmony.

Here 1. and 3. follow from the assumption directly.  For 4. note that the situation being modeled here is the tossing of a die with chance defined for the six possible outcomes of that toss.

Not closed under conditionalization

This means also that [2] is a statement on which you cannot conditionalize your subjective probability, in the sense that if you do, your posterior opinion will violate Minimal Harmony.

So we have here another case where the space of admissible probability functions is not closed under conditionalization. 

I will make all this precise in the Appendix.

REFERENCE

My previous post called ‘Stalnaker’s Thesis → Moore’s Paradox’

APPENDIX.  Semantic analysis: language of subjective probability and assessment of chance

As an intuitive guiding example we can think of a model of a tossed die.  There is a set of possible worlds, and in each there is a die (fair or loaded in some fashion) that is tossed and a number that is the outcome of the toss.  To represent the die we need only the corresponding chance function, e.g. the function that assigns 1/6 to the set of worlds in which the outcome is x (for x = 1, 2, 3, 4, 5, 6).  Then, a special feature of this sort of model, there is the set of probability functions on these worlds, representing the different subjective probabilities one might have for (a) what the outcome is, and (b) in what fashion the die is loaded.

Definition.  A probability space M is a triple <K, F, PP> where K is a non-empty set, F is a Borel field of subsets of K, and PP is a family of probability measures with domain F.  

The members of K we call “worlds” and the members of F, the ‘measurable sets’, we call propositions.

Definition.   A subset PP* of PP in probability space M = <K, F, PP> is closed under conditionalization iff for all P in PP* and all elements A of F, P( -|A) is in PP* if p(A) > 0

Definition.  A probability space with chance M is a quadruple <K, ch, F, PP> where <K, F, PP> is a probability space and ch is a function that assigns to each world w in K a probability function ch(w) defined on F. 

Definition.  For world w in K, GoodProb( w) = {P in PP: for all A in F,   if P(ch(w)(A) > 0) = 1 then P(A) > 0}.

Theorem.  GoodProb( w) is not closed under conditionalization.

Proved informally the Moore Paradox way, in the body of this post.     

The relevant language has as vocabulary a set of atomic sentences, connectives & and ~, propositional operators (subnectors, in Curry’s terminology) P and ch, relational symbols = and >, and a set of numerals including 0. 

There is no iteration or nesting of P or ch, which form terms from sentences.

Simultaneous inductive definition of the set of terms and sentences:

1. An atomic sentence is a sentence
2. If A is a sentence then ~A is a sentence
3. If A, B are sentences then (A & B) is a sentence
4. If A is a sentence and no terms occur in A then ch(A) is a term 
5. If A is a sentence and P does not occur in A then P(A) is a term
6. If t is a term and n is a numeral then (t = n) and (t > n) are sentences.

Truth conditions for sentences:

For M =  <K, ch, F, PP> a probability space with chance, and P a member of PP, a P-admissible interpretation ||…. || of the language in M is a function that maps the sentences to propositions, and numerals to numbers (with 0  mapped to 0), subject to the conditions:

1. ||A & B|| = ||A|| ∩ ||B||
2. ||~A|| = K – ||A||
3. ||ch(A) = n|| = {w in K: ch(w)(||A||) = ||n||}
4. ||ch(A) > n|| = {w in K: ch(w)(||A||) > ||n||}
5. ||P(A) = n|| = {w in K: P (||A||) = ||n||}
6. ||P(A) > n|| = {w in K: P (||A||) > ||n||}

Note that ||P(A) = n|| is in each case either K or empty, and similarly for ||P(A) > n||.

We call a sentence A true in world w exactly if w is a member of ||A||.

For example, if A is an atomic sentence then there is no constraint on ||A|| except that it is a proposition.  And then sentence P(A & ch(A) > 0) = n is true under this interpretation (in all worlds) exactly if P assigns probability ||n|| to the intersection of set ||A|| and the set of worlds w such that ch(w)(||A||) is greater than zero.  And otherwise that sentence is not true in any world.