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

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.

• It is not the absolute value but a change in probabilities that corresponds to acquiring knowledge. Probability of an event and its complement always add up to 1, so any change increases the probability of something, in this case of pregnancy. See SEP on Bayesian models of learning. Commented Jun 8, 2019 at 23:33

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.

• "to holding the belief that she might be pregnant" It would be true only if she was 100 sure that she couldn't be pregnant. Otherwise she always believed that here is probability that she is pregnant. Also, if she was 100% sure that she wasn't pregnant, then it would be impossible to change her mind with any evidence. In other words, it doesn't seem like she acquired new belief. Commented Jun 8, 2019 at 14:59
• @user161005 All I meant was her belief "state" changed from P(pregnant) ≈ 0 to P(pregnant) ≈ 0.5. I don't hold many beliefs with probability exactly 1 or 0, most are in between. But the sense in which I would say the sun might not rise tomorrow (since the probability of it not rising is close to but not exactly 0), and the sense in which I say the result of the coin toss might be heads (since it's 50/50) are different. "Might" encompasses anything greater than 0 and less than 1, but I don't think anything that might be true has the same epistemic status as anything else that might be true. Commented Jun 8, 2019 at 16:06
• @user161005 Anyways, I think the above is just semantics. If we want to be consistently Bayesian in our approach, we should probably just dispense with informal notions of "maybe", "know", "justified", "uncertain", "probably true", etc. and stick to computing numbers. Whether or not she gained or lost knowledge is semantics; what definitely did happen is a change in probabilities in her beliefs. If we do want to define some of our informal notions, treating 50% as always maximum ignorance is problematic in light of Bertrand paradoxes. Commented Jun 8, 2019 at 16:11
• @user161005 But... I also see your point. Sometimes we go from "knowing" something to "not knowing" something in light of new evidence. In this particular case of yours, this might be what is happening, but some knowledge does seem to be gained. My answer was more of a caution about always seeing 50/50 as not knowing something, as in your title to the question. 50/50 can definitely be knowledge sometimes. (Being concise is not my strong point.) Commented Jun 8, 2019 at 16:17
• Your example show that uniform discrete distribution (UDD) between all outcomes would be maximum ignorance, and it's not always 0.5. On contrary, sometimes 0.5 can lead to non-UDD. I agree. But it doesn't disprove my example because in case with the girl 0.5 is result of UDD, so we can say that the girl is ignorant if we define ignorance as UDD of beliefs. Consequently we supposedly gain knowledge when we deviate from UDD. And lose knowledge when arrive at UDD of beliefs. Commented Jun 9, 2019 at 1:01

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

• +1 Really nice analysis Commented Jun 9, 2019 at 15:40
• @AdamSharpe Thank you. I up-voted and agree with your answer which inspired me to pay attention to this question. Commented Jun 9, 2019 at 16:39
• Interesting! You seem to perfectly separate between knowledge and probability of an event, which sounds fair since all knowledge is mainly based in the past, all probability is mainly based in the futur. However, it would be intresting to have your take on the borderline case where a probability reaches 99.99%, which, however big, remains different from a 100% probability of occurence. While for the second case the lady's belief equates a knowledge (similar to that of knowing that if you drop a ball it falls to the ground), the first case is trickier, since it cannot translate certainty. Commented Jun 10, 2019 at 6:43
• @Gloserio My own view is that only tacit knowledge (see Michael Polanyi) is real knowledge. For explicit knowledge, like knowing one is pregnant or not, something can get in the way and show what we we thought we knew was actually not true raising a Gettier problem. For this answer, I just wanted to distinguish terms, bringing attention to their differences, use definitions I could reference and then try to keep my answer in line with those definitions. Commented Jun 10, 2019 at 11:57

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.