Wednesday, 25 February 2009

We believe in probabilities, we do not believe in terms of probabilities

In Warrant: the Current Debate1, Alvin Plantinga criticises Bayesian Coherentism. Bayesian Coherentism is a new fangled way of dealing with an old idea: that we should be consistent in the probabilities we apply to propositions and should act on those propositions in accordance with their probability.

As Plantinga points out Bayesian Coherentism does at least acknowledge that we have degrees of belief in propositions. Plantinga, for example, believes “that Banff is in Scotland, that there was such a thing as the American Civil War, that I am more than ten years old, and that 7 + 5 = 12; and I believe these in ascending order of firmness.”2 Whilst many epistemologies lack the means to deal with these distinctions, Bayesianism simply attaches probability statements to them and introduces a precise method coping with diverse probabilities: Bayes Theorem (for an explanantion of the theorem even I can understand, see Eliezer Yudkowsky's)

All this is to the good. To the bad are the difficulties in applying probability to everyday reasoning and the applicability of Bayesian arguments for coherence, usually in the form of gambling theory, to people who may not want to bet. Almost as a coupe de grace Plantinga pulls out the Paradox of the Preface to argue that, despite the Bayesian's protestations, it is rational to believe propositions with contradictory probabilities.

Plantinga goes further than is necessary, he need not argue that we can believe propositions with probabilities that are incoherent because...well...applying probabilities to propositions is bollocks.

Propositional and Empirical Probability

I can say that the probability of the next throw of a fair die coming up six is 1/6. Whilst it can be argued that this expresses a degree of belief that the next throw of a fair die will come up six, it doesn't express the degree of belief I hold in the proposition “the next throw of a fair die coming up six is 1/6”. To make it a little clearer and to remove the need for inverted commas, let's call “the next throw of a fair die coming up six is 1/6” by a name: “Ichabod”. Let's name a non-probabilistic statement, say “there is a pint of beer in front of me”, “Ahitub”. I am pretty confident that there is a pint of beer in front of me, but not entirely sure. There is a degree of belief I have in Ahitub, despite the proposition itself containing no probability statement. Using the language of probability it can be said that I believe Ahitub to X%. Applying the language of probability to Ichabod, it can be said that I believe Ichabod to Y%. The probabilistic elements of proposition, the 1/6 probability predicted of the die, are the “empirical probability”. The probabilistic terminology applied to the degree of belief in the propositions themselves is the “propositional probability”. Now, how big is Y%?

It appears that I can vary a propositional probability by varying the probabilistic element within a treatment. Whilst I only believe “there is a pint of beer in front of me” is probable, I believe “there is probably a pint of beer in front of me” completely. Actual numbers, given a few auxiliary assumptions, can be given in the case of dice. Let us take it that a die is a cube, it has six equal sides. A “fair” die is so constructed that no one side is favoured over another when thrown and that its edges are so constructed that, when thrown, it must show one side and one side only.

If we cast the propositions in terms of empirical certainty there are six possible propositions:

1.The die, on the next throw, will come down “one”
2.The die, on the next throw, will come down “two”

one of which must be true. Thus the combined probability of one of the six propositions being true is 1. Given that the die is a “fair” die the distribution of that probability over the propositions will be even: each empirically certain proposition is believed 1/6.

Given our auxiliary assumptions, the empirical probabilities of the six possible outcomes are:

1.The probability of the next throw coming up “one” is 1/6
2.The probability of the next throw coming up “two” is 1/6

the combination of which must come to 1. Something will happen and, as our auxiliary assumptions limit that something to one of the six I must believe “the summation of the probabilities is 1” with certainty. Our assumptions also determine that no individual outcome is preferred: I must also believe the distribution of that probability with certainty.

We either believe the empirically certain proposition “the die, on the next throw, will come down “six”” with a propositional probability of 1/6 or believe the empirical probability statement “the probability of the next throw coming up “six” is 1/6” with propositional certainty. I leave it for someone who has the mathematical ability to show that this holds for all values of empirical and propositional probability but, at the limits, it is clear that:

Claim 1: Propositional probability of X with Empirical probability of Y
is equal to
Empirical probability of X with Propositional probability of Y

Claim 2: my propositional belief in Ichabod, described in terms of probability, is 100%

Unavoidable Propositional Uncertainty

So far, so good. However, let us consider another die. This die is supplied by 'Doc' Plantinga, a philosopher of international renown who has built up a lucrative sideline in scamming games of chance. This die is also a cube, it has six equal sides and is so constructed that when thrown, it must show one side and one side only. This die, however, is anything but a “fair” die. 'Doc' Plantinga has almost certainly loaded the die so that one side is favoured over the others when thrown. In predicting the likelihood of a particular number being thrown I must factor in the likelihood of that side being weighted for and the extent of the distortion (whether it biases the die to that number or guarantees that number).

Unfortunately, although I am pretty confident that 'Doc' has fixed the die I do not know in which direction. The same factor must be applied to the same original probability for each side and the result for all six sides must still add to 1. Thus the 'new' empirical probability is the same as the old: 1/6. From the two claims above it follows that my “propositional probability” for “the probability of 'Doc' Plantinga's die coming up six on the next throw” (call this proposition “Solomon”) is 100%.

Claim 3: Whatever 'degree of belief' I attach to a statement of empirical probability, the propositional probability of that statement is 100%

Whoever thought otherwise?

On reflection it is quite ridiculous to apply probabilities to our attitudes to propositions. Whilst we may estimate probabilities, our actions (both practical and intellectual) are binary: we either act as if we believe or we do not. We do not “probably” drink a pint of beer, nor do we press down really hard on the pencil if we particularly believe the proposition we are writing. All logics reduce to two value logic: that 1/6 chance either is, or isn't, 1/6 and it is either true or false to say of a variable in n-value logic that it stands at value n. Even in games of chance, deliberate exercises in allowing probabilistic judgements, we are rarely in a position to align our actions with our estimates of probabilities. We either bet on 'Dog's Dinner' in the 3:30 or do not, I am given one hand in poker and must either call, raise or fold.

So what is going on?

Our resort to probabilities results from their success elsewhere, the necessity of something to describe degrees of belief and the idea that probability is the only game in town. I do not believe Solomon to anywhere near the degree I believe Ichabod. What other concept can be used to distinguish my attitude to Ichabod from my attitude to the Solomon? What is different? It is my attitude to revising my beliefs in the light of future information. If the fair die came up six on each of the first three throws I would put this down to an unusual, but not outrageously unlikely, event and continue to believe that the odds of the next throw coming up six to be 1/6. If 'Doc' Plantinga's die came up six on each of the first three throws I would start to question whether this was the side he had weighted for and start to revise my estimate of the odds of the next throw with this die coming up six. Right now, if I believe Solomon, I believe it with the same probability as Ichabod but I believe it a darn sight less tenaciously.

This solves the “Paradox of the Preface”

As the Paradox of the Preface, which Plantinga uses to justify inconsistency, arises from the probability calculus removing propositional probability can be expected to solve the paradox. And so it does. I shall use Plantinga's formulation of the “Paradox of the Preface” as he casts it neatly in terms of belief, nicely isolating that part of the traditional view of knowledge:

“I write a book named I Believe! reporting therein only what I now fully believe. Being decently modest, I confess in the preface that I also believe that at least one proposition in I Believe! is false (although I have no idea which ones[s]). Then my beliefs are inconsistent, in the sense that there is no possible world in which they are all true; but might they not nevertheless be perfectly rational?”3

He is being perfectly rational and not at all inconsistent. Let us expand the statement “I believe X” to encompass two statements:

1.I have adopted X, for the present, and will act as if it is true
2.I am happy to revise that belief in the light of future experience

And let us imagine a strangely sceptical Plantinga who believes just five propositions, P1, P2, P3, P4 and P5. The text I Believe! can be summarised:

1.I have adopted P1 and am prepared to revise it in the light of future experience
2.I have adopted P2 and am prepared to revise it in the light of future experience
3.I have adopted P3 and am prepared to revise it in the light of future experience
4.I have adopted P4 and am prepared to revise it in the light of future experience
5.(The Preface) I have adopted P1, P2, P3 and P4 and am prepared to revise P1, P2, P3 and P4 it in the light of future experience

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