I realise there is already a question on this matter, but I wasn't entirely satisfied with the answer given so that's why I am opening a new question.
So the problem is to find the probability that $\{(0,0) \leftrightarrow (1,0)$} for $p = 1/2$. If we denote the edge between the two points by e, and the two points by $0$ and $1$, what I have so far is:
\begin{align*} P\left(0 \leftrightarrow 1\right) &= P\left(0 \leftrightarrow 1 : e=1\right)P(e=1) + P\left(0 \leftrightarrow 1: e=0\right)P(e=0) \\ &= 1/2 + 1/2P(0 \leftrightarrow 1:e=0) \end{align*}
Now how I've seen the argument go, is consider the complement of the last probability, so $P(0 \nleftrightarrow 1: e=0)$. If this is the case, then there exists a closed circuit in the dual lattice around either of the points $0$ and $1$, since $p=1/2$ clusters are a.s. finite. What is then argued is that if we remove the edge $e'$ (dual of $e$), then we have a loop connected the end points of $e'$ together, which is dual to having $e=1$ in the original lattice. Thus the probability of the two must be the same.
What I don't understand about this argument, is for duality don't we need both situations to imply each other? We have $\{\text{endpoints of e' connected in } \mathbb{L}^2_d\}\subset\{ e=1$ in $\mathbb{L}^2\}$ but not the other way around. Or am I missing something? And how would this help us find the above probability?
And if this doesn't work, how can the probability be calculated using duality?
