I have just begun reading about Epistemic Logic from Reasoning About Knowledge, Fagin et. al. $\mathcal K_i$ is defined as the possibility relation between two worlds, and the author says that for the most part - one would want $\mathcal K_i$ to be an equivalence relation. I do not understand the...

Imagine $n$ children playing together. The mother of these children has told them that if they get dirty there will be severe consequences. So, of course, each child wants to keep clean, but each would love to see the others get dirty. Now it happens during their play that some of the children, ...

I've been reading Epistemic Logic from Fagin et. Al's Reasoning about Knowledge, and I came across the following discussion: $\varphi \implies K_i\varphi$ is not valid. $\varphi \implies K_i\varphi$ says that if $\varphi$ is true then agent $i$ knows $\varphi$. An agent does not necessarily know...

Why is this identity true? $$\sum_{k=1}^{\infty} \frac{\sin(k\pi/2)}{k^2} = G$$ where $G$ is Catalan's constant.

The problem I have been trying for a while now to show that this monster $$\begin{align} &\int_0^{\pi/4}\tan(x)\sum_{n=1}^{\infty}(-1)^{n-1}\left(\psi\left(\frac{n}{2}\right)-\psi\left(\frac{n+1}{2}\right)+\frac{1}{n}\right)\sin(2nx)\,\mathrm{d}x \\ +&\int_0^{\pi/4}\cot(x)\sum_{n=1}^{\infty}\left...

I have the conjecture for the integral $$ \int_{\frac{5\pi}{36}}^{\frac{7\pi}{36}} \ln (\cot t )\>dt +\int_{\frac{\pi}{36}}^{\frac{3\pi}{36}} \ln (\cot t )\>dt = \frac49G $$ where $G$ is the Catalan constant, following some heuristic effort. But, I am unable to derive it formally despite having t...

I am aware that Catalan's constant appears in the evaluation of many definite integrals, as well as in the evaluation of certain infinite series, and is a special value of a function closely related to the Riemann zeta function, and so on. But is there a way in which we might think of this const...

Some years ago, someone had shown me the formula (1). I have searched for its origin and for a proof. I wasn't able to get true origin of this formula but I was able to find out an elementary proof for it. Since then, I'm interested in different approaches to find more formulae as (1). What oth...