Your question has two main facets. The first is that you did not grasp the way logic does not fall to the liar paradoxes. The second is that there are deeper reasons as to why we have such apparently innocuous sentences in natural language that seem to defy assimilation into formal logic systems....
But if you're just looking for common fallacies that people have when they don't use logical reasoning, Wikipedia has a long list hahaha..
@MoreAnonymous I'll get back to you later. I need to go off soon.
Hi, may I ask why $\frac{W_{ik}}{W_{ik}H_{kj}}$ (the denominator is not $ \sum_{k}{W_{ik}H_{kj}}$) is not equal to $\frac{1}{H_{kj}}$? Since it is not sum over $k$ yet?
$W$ and $H$ are matrices with dimension $ik$ and $kj$, respectively.
@Jan That is not the point. If W is a real matrix, then W_{i,k} is an entry in W, which is a real number, so what you wrote has nothing to do with matrix multiplication. Just answer my question first: Is it true that x/(xy) = 1/y if x,y are reals?
@Jan Yes and that's all about it. What you wrote is also true only if W[i,k] and H[k,j] are both nonzero. Otherwise it can be considered meaningless (division by zero).
Whether you later sum over all k later or not is not relevant.
I discovered the following relation for arbitrary $d_r$:
$$ \lim_{k \to \infty} \lim_{n \to \infty}\ \sum_{r=1}^n d_r \left( f(\frac{k}{n}r)\frac{k}{n} \right) = \lim_{s \to 1} \! \underbrace{\frac{1}{\zeta(s)} \sum_{r=1}^\infty \frac{d_r}{r^s}}_{\text{removable singularity}} \int_0^\infty...
