Cheryl's birthday and Dynamic Epistemic logic

Compared to the majority of my posts that focus on political philosophy and ethics, this post will be much more lighthearted. I'm writting my dissertation on dynamic epistemic logic and thought it would be fun to discuss some applications. In particular, one can very easily solve the famous Cheryl's birthday problem.

First a quick comment about dynamic epistemic logic. The purpose of the discipline is to model the knowledge and information states of different agents. That's standard epistemic logic. By making it dynamic essentially means that we are actively changing the models in our logic as new information comes into play. The focus is typically based on communication, the fact that a person comes to know information directly impacts the entire model. 

The main idea is this: We have different states of the world, or rather what the agent thinks could be the case. We also have relations between states, which basically says that the agent is unable to distinguish between which world it will be. Now the actual state is in fact one of the states. We say that an agent knows that $p$ in state $s$ if in all worlds related (including itself), $p$ is the case. Otherwise they do not know that $p$. The idea is simple, we're in a certain state, but we intuitively it could be another state. Yet if in all states we deem it could be instead, $p$ is still true, then we know for sure that $p$ is true. We might not know perse which state we're in, but we can say the proposition is correct.

Now Chery's birthday problem is as follows (https://www.theguardian.com/science/alexs-adventures-in-numberland/2015/apr/13/how-to-solve-albert-bernard-and-cheryls-birthday-maths-problem):

Albert and Bernard just met Cheryl. “When’s your birthday?” Albert asked Cheryl.

Cheryl thought a second and said, “I’m not going to tell you, but I’ll give you some clues”. She wrote down a list of 10 dates:

• May 15, May 16, May 19
• June 17, June 18
• July 14, July 16
• August 14, August 15, August 17

“My birthday is one of these”, she said.

Then Cheryl whispered in Albert’s ear the month—and only the month—of her birthday. To Bernard, she whispered the day, and only the day.

“Can you figure it out now?” she asked Albert.

• Albert: I don’t know when your birthday is, but I know Bernard doesn’t know either.
• Bernard: I didn’t know originally, but now I do.
• Albert: Well, now I know too!

When is Cheryl’s birthday?

We can model this using the following diagram:

Here we have the different days, as states of the world. For example M15 is the state where Cheryl's birthday is May 15th. The relations between states concern distinguishability. If Albert is unable to tell two dates apart (e.g. M15, M16 then we have a linking arrow $a$. The same for Bernard. I have connected the related arrows for us. Clearly, Albert can't tell the months apart and Bernard can't tell the dates apart. This is because Albert is told which month her birthday, but not which of the days of the month. Thus all states of the same month are linked. Bernard is told the date, and so all dates are linked.

Now we process the information we're given.

Albert says that he doesn't know which day it is, but that he knows that Bertrand also doesn't know which day it is. Now, albert not knowing which it is means that the real state is $a$ connected. In fact, just looking at the graph makes this obvious since each state is $a$ connected to another. However the fact that he knows bertrand doesn't know means that the real state lies in an $a$ connected set that are each $b$ connected. The reason for this is that if the actual state had no $b$ connections, then Bertrand would know. For albert to know (i.e be certain of) Bertrand's ignorance means that every state connected to the actual state must have $b$ connections.

This gives us the far simpler diagram. Here every state was once b connected to another state. Now we can also remove all the arrows that connect to states that we have already ruled out.

Now we are told that Bertrand now knows Cheryl's birthday. This obviously means that the real state is one in which there are no b connections. And so we get the even simpler diagram:

Finally we're told that now, Albert suddenly knows. Which means that the actual state has no $a$ connections. And so we're done, Chery's birthday is July 16th.

The impressive thing about dynamic epistemic logic is that the statement of additional epistemic propositions massively changes the model. In this sense, we get what quite surprisingly the fact that agents repeatedly say what they do or do not know, actually resolves our model and gets us these solutions.

If one is further interested, they can feel free to try the same methodology onto: https://www.theguardian.com/science/2015/may/25/can-you-solve-it-cheryls-birthday-logic-puzzle-part-2-denises-revenge.

Once one is familiar with these strategies and the theory behind dynamic epistemic logic, this is very straightforward (if rather tedious to be honest). One thing to note, with more basic puzzles like these you can easily do them without having a particularly formal method. However by formalising, it not only simplifies things for you in the long run, but also makes it possible for computers to do this, by following a very exactly algorithm.

MRA's and Feminists

In this post I'm going to further discuss some previous symmetry arguments, using the example of MRA's, or men's rights activists. In particular I am going to discuss the conditions in which one can argue for lack of symmetry if they wish to avoid double standard arguments.

Let's start with the following premise: it is thought in current society that feminism is in general a good thing. Now, this isn't universally accepted perse, and there is a huge amount of disagreement about what feminism is. Yet I do think there is some kind of prevailing opinion that one should in general be favour of feminism, that one is almost doing something morally wrong otherwise. For now I suggest three kinds of definitions for feminism. The first is a movement that aims for gender equality (let us leave gender as an atomic concept), the second is a movement that aims for the improvement of women's rights, on the grounds of gender equality. The third is a movement that tackles structural gender imbalances, the focus typically being to remove what is perceived to be a patriarchal society. Whilst each definition is slightly different, it seems there is this thought that gender equality at least is something that is good and should be strives for. Someone who denies this tenet is viewed as bigoted and doing something wrong. In this sense, few would deny the principles of the first definition.

Now of course certain groups that call themselves feminist, might be viewed as very much extreme. It seems there are phenomena of groups who may be very up misandrist (man / men hating). This of course seems to be morally impermissible on the same grounds of misogyny, at least intuitively speaking. There may also be an intuition that on some level, misandary is morally better than misogyny, but  I will discuss this argument in due course. Anyway, under the first and second, there seems to be something rather questionable about this practice. A number of people who call themselves feminists would decry these misandrist groups. Now something strange I notice is that people may dislike these misandrists, not particularly because they hate men, but because they provide ammunition for supporters of men's rights activists. It seems useful now to discuss what a men's right activist is. As with feminism, there is a challenge in defining this. Some groups like to perceive them as misogynist, a group that was generated as a backlash to feminism. Now, I think the latter arguably is true, but one needs to flesh out what backlash means.

Does backlash mean opposing feminism? If so, then it must oppose at least one of our following definitions. If it opposes definition 1, then it means that they must deny gender equality. Opposing definition 2 suggests either that they deny gender equality, or that they think there is something problematic with only improving women's rights. If they deny definition 3 then they may deny structural oppression or deny the patriarchy. To me, only the denial of 1st definition is obviously problematic. Yet, it doesn't seem obvious to me that this is what MRA's are doing. In principle at least, MRA's are meant to be attaining certain types of rights for men, for example in social care, or domestic violence. It is not necessarily case that they are denying gender equality, in fact many cases seem to be promoting gender equality. At least for the first definition of feminism, MRA's can very much integrate into it. With the second definition, the two groups could in theory be parsiminous. We might for example define MRA's or Menism as the feminist equivalent: improving men's rights for the purpose of gender equality. Whilst many would think that women are the oppressed gender, I think one would have to be deliberately blind to deny that there are any instances  where men are the suffering gender.

Now, one may criticise me by claiming that MRA's whilst in principle can be consistent with feminism, in practice are just mysognist and backwards. This may be true of certain groups, yet one should consider now whether a double standards argument should be invoked. We in principle don't decry feminism just because certain 'feminists' are decreed misandrist. (Also note the following argument: some people who claim to be feminists would deny that misandrists are indeed feminists, since they will violate certain definitions of feminism). Why should we decry mens rights movements as a whole just because certain men have anti-women views? Even if most men's rights movements were indeed mysogynist, it seems wrong to decry any given group that aims to champion men's rights in a negative manner. Presumably anyone who is a true definition 1 feminist, should in fact be very much supportive of such MRA's. And vice versa of course. What I'm trying to say is that one really should not have immediately negative reactions to these groups. It should not be inherently negative to care about men's rights. In fact anyone who truly cares about gender equality should be necessarily supportive of them, at least the 'good ones'.

For a final challenge, let's consider the following asymmetry argument. Perhaps we think a la definition 3 that we live in a hard patriarchy with wide spread oppression of women. We think that perhaps there are certain instances where men suffer comparatively, but any general movement in favour of them is just morally wrong. Why should we be helping these men out, when women are suffering so much already. Surely for gender equality, we should be improving women's rights, not catering for the small cases of men's rights inequality. Whilst this argument is interesting (if extremely depressing actually if one considers individuality arguments), there are a number of issues that need to be addressed. The first thing is to justify the premise, the idea that we indeed live in a patriarchy with widespread oppression of women. Whilst many people assume this, one really does need a justification for this. This interestingly leads to an epistemic problem. Since there are extremely few cases of explicit gender discrimination against women (the vast majority seem to be against men), one must rely on implicit cases to justify the perceived oppression or patriarchy. Whilst this can certainly be done, it does face difficulties with subjectivity and intentionality. For example suppose we have such thing as the gender pay gap. That the average women gets paid less than the average man. Now if we wish to use this as an example of our opression we need to show that there is discrimination. Because it is illegal to do this explicitly, one would need to show that there is something particularly wrong with why women are less likely to go into certain industries. The simple fact of statistics is not enough: one requires an explanation to use this as fuel for our oppression argument. This is important, because we would apply the same standards towards men. Suppose someone said men were being oppressed, and gave a number of examples (e.g. high suicide rates, worse academic attainment), one would need an argument to explain why these examples fail to attain. 

To conclude, for us to really justify decrying MRA's, one has to set up quite a sophisticasted argument that is protected from double standards. You need to convincingly argue that we can vilify 

MRA's by their worst movements, but not feminism. One then needs to argue somehow that we don't need to care about men's rights because there is this pervading patriarchy. Instead, I suggest we just look at individual groups, both feminist and MRA groups, and see whether they have justification. In principal at least, MRA and Feminists should work in sync, for they are promoting the same thing: attaining gender equality.