Conditional probability of events not arising from a well-defined function on S

Given a probability space $$(S,\mathcal{F},P)$$ where $$S$$ is countably infinite, $$\mathcal{F}$$ is the full $$\sigma$$-algebra, and $$\forall a \in S, P(a) > 0$$, is it reasonable to speak of $$P(Y|X)$$ when $$Y$$ and $$X$$ are constructed with respect to $$(y,x)$$ with $$y,x\in S$$ and not uniquely from $$x$$ or $$y$$ for $$X$$ or $$Y$$ respectively?

For more context, I'm struggling to work through a conversation with my advisor about conditional probability in a probability space defined on something like a computation tree for a nondeterministic Turing machine. I'm interested in discussing the probability of being in state $$y$$ at time $$t+k$$ given the machine was in state $$x$$ at some previous time $$t$$. The actual probability space and compound events are more involved in their description, but I hope my misunderstanding can be resolved without going into the actual construction. Below is a picture of events $$X$$ and $$Y$$ constructed from $$x$$ and $$y$$ after inferring $$k$$. $$X$$ contains every state visited in $$k$$ or fewer time steps having visited state $$x$$ at time $$t$$, and $$Y$$ contains every state that could have been visited at time $$t$$ or later in order to visit state $$y$$ at exactly time $$t+k$$.

I would like to speak of $$P(Y|X)$$, but my advisor has stated that the function $$f\colon S \to \mathcal{P}(S)$$ that produces some compound event $$A$$ from $$a\in S$$ cannot be ambiguous, $$\it{i.e.}$$ $$f(x)$$ cannot vary with $$y$$ (which implies the "hitting time" $$k$$) or with $$k$$ directly. If the function is ambiguous, then the claim is that it's fine to use $$\frac{P(X\cap Y)}{P(X)}$$, but I cannot call it a conditional probability and cannot use any resulting work in probability theory about conditional probabilities without reconfirming the appropriate theorems myself.