Does our knowledge increase or decrease when we assign 0.5 probability to our belief, after being presented with new evidence?

Question

Suppose there was a girl who wanted to check if she was pregnant. She was relatively confident that she is NOT because contraceptives that she and her boyfriend used are very reliable. Also the pregnancy test that she had wasn't very trustworthy. She checked the test and it turned out positive. She then applied Bayes' theorem and calculated that after the test there is 0.5 probability that she is pregnant.

In other words, we can't be more or less confident that she is pregnant or that she is not, it can be either way. Did she learn something new? Can 50/50 probability of pregnancy be considered as knowledge? In mathematics when we quantify ignorance via probability we assign 0.5 probability to such event as pregnancy, so it looks like after the test we lost knowledge, we became ignorant due to presented evidence.

Answer 1

Well she learned something new in the sense that the state of her beliefs changed from holding the belief that she is not pregnant, to holding the belief that she might be pregnant.

I would also say that in light of Bertrand' paradox, the situation is a bit more complicated than saying P(e) = 0.5 should always mean maximum ignorance. For example, if we have a fair die, the probability of each side turning up is 1/6 so this should be maximum ignorance. But if we learn some new information and and conclude that the probability of the 3 side coming up is 1/2, we learn that the die is loaded; P(3) = 0.5 is not maximum ignorance here so I would say yes, a 50/50 probability can be considered knowledge here, especially after a large number of throws with the same die.

Another relevant quote from from the SEP article on Interpretations of Probability:

The following example (adapted from van Fraassen 1989) nicely illustrates how Bertrand-style paradoxes work. A factory produces cubes with side-length between 0 and 1 foot; what is the probability that a randomly chosen cube has side-length between 0 and 1/2 a foot? The tempting answer is 1/2, as we imagine a process of production that is uniformly distributed over side-length. But the question could have been given an equivalent restatement: A factory produces cubes with face-area between 0 and 1 square-feet; what is the probability that a randomly chosen cube has face-area between 0 and 1/4 square-feet? Now the tempting answer is 1/4, as we imagine a process of production that is uniformly distributed over face-area. This is already disastrous, as we cannot allow the same event to have two different probabilities (especially if this interpretation is to be admissible!). But there is worse to come, for the problem could have been restated equivalently again: A factory produces cubes with volume between 0 and 1 cubic feet; what is the probability that a randomly chosen cube has volume between 0 and 1/8 cubic-feet? Now the tempting answer is 1/8, as we imagine a process of production that is uniformly distributed over volume. And so on for all of the infinitely many equivalent reformulations of the problem (in terms of the fourth, fifth, … power of the length, and indeed in terms of every non-zero real-valued exponent of the length). What, then, is the probability of the event in question?

The paradox arises because the principle of indifference can be used in incompatible ways. We have no evidence that favors the side-length lying in the interval [0, 1/2] over its lying in [1/2, 1], or vice versa, so the principle requires us to give probability 1/2 to each. Unfortunately, we also have no evidence that favors the face-area lying in any of the four intervals [0, 1/4], [1/4, 1/2], [1/2, 3/4], and [3/4, 1] over any of the others, so we must give probability 1/4 to each. The event ‘the side-length lies in [0, 1/2]’, receives a different probability when merely redescribed. And so it goes, for all the other reformulations of the problem. We cannot meet any pair of these constraints simultaneously, let alone all of them.

Answer 2

It may be useful to keep track of what terms mean or at least state a definition. Wikipedia defines probability as follows:

Probability is a measure quantifying the likelihood that events will occur.

In the case of being pregnant or not, the current measure of the event of being pregnant is 0.5. That is all that number refers to. It is the current measure of the event of being pregnant.

Eric Schwitzgebel describes belief as a propositional attitude. In particular:

Contemporary Anglophone philosophers of mind generally use the term “belief” to refer to the attitude we have, roughly, whenever we take something to be the case or regard it as true. To believe something, in this sense, needn’t involve actively reflecting on it: Of the vast number of things ordinary adults believe, only a few can be at the fore of the mind at any single time. Nor does the term “belief”, in standard philosophical usage, imply any uncertainty or any extended reflection about the matter in question (as it sometimes does in ordinary English usage). Many of the things we believe, in the relevant sense, are quite mundane: that we have heads, that it’s the 21st century, that a coffee mug is on the desk.

He describes knowledge in terms of belief as follows:

Much of epistemology revolves around questions about when and how our beliefs are justified or qualify as knowledge.

Finally when something changes, that change need not be linear or a total ordering of events. Although an increase or decrease is a change, change itself may involve a more complicated ordering of events such as moving from one side of the room to the other or changing one's mind about the likelihood of an event.

---

Consider the question in the title:

Does our knowledge increase or decrease when we assign 0.5 probability to our belief, after being presented with new evidence?

That knowledge changes does not mean that such change is part of a linear ordering. Knowledge need not be increasing nor decreasing. Probability is a measure assigned to an event. Belief represents a propositional attitude about the probability of that event. Part of the justification for the new belief comes from the new evidence. Her current belief is that the probability is 0.5 that she is pregnant, not that she is or is not pregnant. She doesn't know now nor did she know previously whether she was pregnant.

---

Schwitzgebel, Eric, "Belief", The Stanford Encyclopedia of Philosophy (Summer 2019 Edition), Edward N. Zalta (ed.), forthcoming URL = https://plato.stanford.edu/archives/sum2019/entries/belief/.

Wikipedia contributors. (2019, May 30). Probability. In Wikipedia, The Free Encyclopedia. Retrieved 14:03, June 9, 2019, from https://en.wikipedia.org/w/index.php?title=Probability&oldid=899504762

Answer 3

I want to add two perspectives on the question.

1) The girl did learn something new because her distribution of beliefs now has more credence than before. From this point of view to increase knowledge means to increase credence of underlaying beliefs, no matter how it will change their distribution.

2)We can define ignorance as uniform discrete distribution of beliefs. In this case the girl lost knowledge, she became ignorant.