﻿ Note: Personal Identity - Probability (Theo Todman's Web Page)

# Theo Todman's Web Page - Notes Pages

## Personal Identity

### Probability

(Text as at 12/06/2020 10:48:00)

(For earlier versions of this Note, see the table at the end)

Plug Note1

• This note isn’t designed to provide a serious discussion of the mathematical theory of probability, but (eventually) to address the question of just what it is that makes the conclusion of a non-deductive argument more or less probable.
• Just what does probable mean2 in this context, given that this “probability” usually cannot be quantified – ie. given a number in the range [0,1]?
• In philosophical circles, the philosophy of rational belief closely follows Bayesian principles and conditional probabilities. So, while there is no objective probability for the truth or falsehood of our beliefs, we can supply subjective probabilities and revise these in the light of new evidence.
• I really don’t think this topic has much to do with my thesis on the topic of Personal Identity, though I will be on the look-out hereafter. It seems to be more relevant (as far as my own concerns go) in the philosophy of religion, as has already been noted, in regards to the probabilities of certain beliefs, miracles and the like. I’ve also written some brief Notes on Pascal’s Wager, repeated in a footnote below3.
• The above caveats not-withstanding, it is important to know something of the mathematical theory of probability – and also of statistics, to some degree. A representative selection of study material is therefore given below.
• One thing I have noticed is that the level of the mathematics in the philosophy of probability and related topics is very trivial, compared with that in the mathematics of probability itself. I’ve given a list of mathematical texts at the end of the reading lists.
• For a Page of Links4 to this Note, Click here. Far too many items for an updating run, but useful to cherry-pick from. However, despite the number, it might be worth doing an updating run so that the very many times I’ve used the term “probability” or “probable” are picked up.
• Works on this topic that I’ve actually read5, include the following:-
• As a footnote8, books on the purely technical and mathematical theory of probability and statistics that I actually own, and which I’d like to read include:-
• This is mostly a place-holder9.

### In-Page Footnotes:

Footnote 2:
• I made a stab at this question in my youth, prior to philosophical training. The results (such as they are) are here.
• To save contaminating the “printable” version of this Note with all the links to this old work, I’ve extracted the relevant items here:-
1. A requirement of great importance is the ability to assign a probability to any statement about the world (or within a model), in accord with the likelihood of it being a true statement.
2. Ideally, we would like to assign a mathematical probability to any statement, ie. a real number in the range 0 to 1, with 0 representing impossibility & 1 representing certainty. As in the frequency theory of probability, the assigned number should represent the proportion of situations in which the statement is expected to turn out to be true.
1. However, in most cases this approach has a spurious aura of precision, because most situations are not regular or repeatable events.
2. It must be noted that where the possible outcomes of an experiment form a continuum or other infinite set, the a priori probability of any particular outcome is 0. Nonetheless, we would not describe any particular outcome, a posteriori, as unlikely. Rather, we would divide the domain of possible outcomes into equivalence classes (whose total probability may be obtained by summation or integration) and judge the outcome of the experiment by the probability of its class.
3. In practical life, where it is unreasonable to assign a numerical probability to an event, we do assign non-mathematical probabilities to statements and base our actions on them.
1. For instance, a jury may decide that a defendant is guilty "beyond reasonable doubt". That is, using the mathematical model, the defendant is probably guilty with a probability that approaches 1. However, the ascription of a number (say 0.95) to this probability does not have as precise a meaning as the accuracy of the number suggests. A possibility is to apply a tolerance factor (eg. 0.95 +/- 0.05), which would indicate the degree of uncertainty.
4. It also makes sense to say that certain statements are more probable than others, even when they do not refer to the same domain of experience.
1. For instance, one could take odds on the contention that a colony will be founded on the moon before 100m is run in under 9.0s.
2. By taking odds one might be able to assign a probability to any statement, though, because of the marginal utility of goods, definition would be lost at the ends of the spectrum (unless comparative probabilities such as those in the previous example are adopted).
3. However, because the set of all possible statements about possible experience is not closed, we cannot easily assign mathematical probabilities to them without slipping into non-linearity. That is, the accumulation of new statements might require a renormalisation of the whole scale of probabilities to smooth out anomalies brought about by bunching.
5. It would seem to be possible to assign a priori probabilities to statements about the world, the probability being assigned a priori to that particular potential experience, by reference to other actual experiences, though not a priori to all experience.
1. This assumes that there is regularity in physical & even human behaviour that may be assumed to apply unless there is strong evidence to the contrary.
6. A statement with a low a priori probability may yet have a higher a posteriori probability because of the strength of actual testimony or experimental evidence.
Footnote 3:
• I considered the question of Pascal’s Wager in my youth, prior to philosophical training. The results (such as they are) are here.
• To save contaminating the “printable” version of this Note with all the links to this old work, I’ve extracted the relevant items here:-
• Pascal's Wager is not to be accepted.
1. There is an argument, known as Pascal's Wager, to the effect that although we are unsure whether there is a God or not, we still ought to believe in him on the basis of expected reward (using "expected" in the probabilistic sense of V * p, where V = "value of outcome" & p = "probability of occurrence of outcome").
2. The argument proceeds as follows. Since the gain obtained if our belief in the Christian God turns out to be well placed is infinite (eternal bliss) whereas the loss incurred by believing in vain is finite (loss of some pleasure during a finite life), we are being rational to believe however low the probability of our belief being correct is taken to be, because our expected gain is infinite, whereas our expected loss is finite.
3. This argument is fallacious for several reasons
1. It does not work in the very case for which Pascal is arguing - ie. there is no evidence in the Bible or from Christian tradition to suggest that God would award eternal life to anyone whose faith was of this "insurance policy" type.
2. It is not possible to "turn on" belief at will. The most that can be done is to make a pretence, which would not deceive an omniscient God.
3. The argument is the ultimate abuse of the laws of probability, which are aimed at events that occur, or may be induced to occur, many times.
4. Admittedly, we have on occasion used probabilities in this irregular sense ourselves, but we have not multiplied these numbers by infinities.
5. As is well known, the product of two variables, one of which tends asymptotically to infinity while the other tends to zero, may be zero, infinity or any value in between depending on the relationship between the variables. Consequently, we have no assurance that the "expected reward" of our pseudo-faith is non-zero.
6. Because it is difficult to justify assigning a zero probability to any non self-contradictory statement, Pascal's wager could be applied equally validly to any belief with potentially infinite rewards. If the argument is correct, we would be as rational to become Muslims as Christians. Indeed, we would be more rational to become Muslims, since we would be more likely to reap our reward, if there is one, under the terms of Islam than under those of Christianity.
4. The reason I have stressed this point is because religion is for many an insurance policy, and many seem to accept the Wager on this basis. It would be easy to slide from belief to this position. To do so would be to evade all the issues.
Footnote 6:
• Despite the title, this is mostly about probabilistic – and especially Bayesian – reasoning.
Footnote 7:
• I have not itemised the papers in this book
Footnote 8:

### Table of the Previous 5 Versions of this Note:

 Date Length Title 14/07/2019 18:05:46 808 Probability 11/03/2018 20:19:41 2108 Probability 18/12/2010 19:58:05 582 Probability 21/05/2010 10:35:02 570 Probability 21/05/2010 10:27:50 514 Probability

Note last updated Reading List for this Topic Parent Topic
12/06/2020 10:48:00 Probability Logic of Identity

To access information, click on one of the links in the table above.

 Aeon Papers, 2, 3, 4

To access information, click on one of the links in the table above.

### Authors, Books & Papers Citing this Note

 Author Title Medium Extra Links Read? Hains (Brigid) & Hains (Paul) Aeon: 2019+ Paper 2, 3, 4 Yes Shoemaker (David) Personal Identity and Immortality Paper Yes