*snip*
Yeah, this is exactly where I struggle with it. The statement of the fictional action already includes the conclusion of the deduction; the player has already made the important logical leap ("there could be a secret door here because there's water coming from somewhere there shouldn't be") - and all the character can now do is test that hypothesis against the physical information available. I think this is clear when you ask the question: 'What exactly is the character _doing_ in the fiction while they "deduce"'? I don't see what they can actually be reasoning about. Really, all that can move the situation on are things like:
- Tapping the walls
- Examining the flow of water more carefully
- Looking for seams/hinges/opening mechanisms
... and to me those all sound like "Searching" - intelligently directed examination of the environment to gather evidence - rather than "deduction"; and I don't really see how they represent a significantly different skill from any other type of searching in the game. The real "deduction" came much earlier - when the presence of water implied a space from which it might have flowed - and "rolling to see what your character thinks" is a form of play that has never worked well for me.
It might sound like semantic nitpicking, but I think the failure of these skills to map well onto our everyday model of how we mentally interact with the world goes a long way towards explaining why they generate so much debate.
And I should expand beyond my earlier explanation that the passive/explicit Perception checks I mentioned above only get the PC to the point you are describing here where something is out of place and needs to be investigated more fully. Even with a failed Perception check, a PC could feel that the wall was so suspicious that they still investigate it anyway. A successful Intelligence (Investigation) check reveals the presence of the secret door and (usually) how to open it.