# Augustus de Morgan and Probability

3/3/20

In PROBABILITY DOES NOT EXIST (Part II) by Bayesian Spectacles, they argue that Augustus de Morgan emphasized the individualistic and subjectivist notion of probability. I disagree.

In their examples, from de Morgan's An Essay on Probabilities and on Their Application to Life Contingencies and Insurance Offices, they focus on the following quotes

"And thus we see that the real probabilities may be different to different persons."

...

"Probability is the feeling of the mind, not the inherent property of a set of circumstances."

Now if that were the case that de Morgan emphasized the individualistic notion of probability, I'm not sure why then there would be page after page of probability calculations which are not different for different persons in that book. The individualistic and subjectivist notion of probability certaintly is not reflected in this book in my opinion.

In any case, I believe it is more likely that de Morgan is talking about "uncertainty" and not "probability". Probability, in the von Mises sense, is converging relative frequency with irrelevancy of place selection ("randomness"). Asking someone about when they feel their ship will come into port is not "probability" but "uncertainty" or something else that involves subjective judgments.

When read in context, I believe de Morgan is not stating that "probability" is different for different people, just that their estimate or calculation or judgment of probability can be different given different assumptions (different models or different reference sets). If person A believes all the balls in a box of red and green balls are red, then what A believes the probability of drawing a ball is is simply mistaken however, and not correct just because they believe it to be so (no matter how strongly they believe).

I believe de Morgan knew the difference between "probability" and "uncertainty" fairly well, and he somewhat explains the distinctions on page 87 by discussing the words "probability", "chance", "presumption", "possibility", "facility", and "expectation". For "probability", de Morgan writes (underlines are mine)

"...while in the word probability we feel disposed rather to think of the external arrangements on the knowledge of which the strength of our presumption ought to depend..."

Clearly he is saying a presumption is not the same as a probability. This is demonstrated easily by seeing relative frequency converging no matter your beliefs, when likelihoods swamp priors no matter your prior, and asymptotic results holding well like the Central Limit Theorem as n gets large, and learning more about a population as n/N approaches 1. Elsewhere, de Morgan writes that P(E|H0) is different from P(H0|E), that the first is objective and the second is subjective. For the latter, I'd opine that using the probability notation P(event) is somewhat misleading, and it is more correctly written as Belief(H0|E).

I'd say de Morgan was overall more on the "logical" and subjective probability side, but that he knew that subjective belief is different than probability. His views were later directly challenged by John Venn in Venn's The Logic of Chance. In a footnote to the 6th chapter, "The subjective side of probability. Measurement of belief", Venn writes that the chapter was:

"Originally written in somewhat of a spirit of protest against what seemed to me the prevalent disposition to follow De Morgan in taking too subjective a view of the science."