Hi, I've trained a client with a similar problem - she had a 6 yr old rescue mix - always lifting his leg around the house. She gave up on trying to train him, until she found us.
In terms of positive reinforcement, you 'can' positively 'reinforce' peeing outside - but the problem is, (AND A BIG PROB) is that you MAY reinforce 'marking' e.g. you say 'good boy for marking that tree!!'
This is what I advised, and so far, the dog has hardly lifted his leg in the house again.
1. Correction: you have to be able to correct him 'hard enough'. Something that will even make him 'yelp' a little.
I am STRICTLY against using unnecessary force with a dog, but when it comes to toilet training, if he can hold it for 10 hours in the crate, he can hold it for 5 hours in the house.
2. Catch him by surprise - the moment he sneaks off to another room, slowly follow him, or walk by him as if you are ignoring him and when you see him do his bad thing, voice and physically correct him. Keep your voice calm but noticeable in correction, but the force has to be strong e.g. grab him by the scruff of the neck or whatever you normally do - but double/quadruble the strength. The idea is to be clear to him that toilet in house is- nono.
3. After the correction, quickly let him go, so the correction is barely more than 2-3 seconds. He MAY (my clients dog did this...) go and try to mark a second object or third in succession - then you must v. firmly correct him. You would probably have to be REALLY QUICK!.
But remember, stay calm and pretend that everything is fine, but once his leg goes up, you become his worst nightmare.
4. If you have corrected him good enough, and he understands, he may go to a corner to lie down or under a chair etc. At this point, you should slowly go over and 'see if he is ok'. You may notice, he will be shy of you, and possibly lick you. But just resume life as if nothing happened.
5. Very important to catch him everytime. For everytime he pees, the more he learns that it is ok.
6. If you need to use a leash, then do so, but the problem will be that the dog can learn it is not ok to pee on the leash, and half ok if he is off=leash in the house.
7. I would confine him to one room, supervised, if a few days go with no mistakes, open another room.
8. If he goes into other rooms, you want to 'spy' on him - so if he DOES do something, you can correct him, but in the dog's mind, he doesn't know you are watching him.
POSTIVE:
IF you only want to stick with positive - additionally, you can give him wet food for meals (nice food) when the house is clean with additional treats etc. as bonus.
But when he pees in the house, all nice food is taken away.. he is fed dry food and there are no treats for the rest of the day.
HOME:
When you return from a walk, to re-enter the house, get him to wait by the door, and when he puts a paw in, (without being asked) you correct him and usher him back behind the door line.
IMPORTANT: no voice commands are used. e.g. NO sit/stay.
You simply walk in and if he walks in too, you correct him.
Then after a few minutes , you call his name to enter.
I like to give freedom for my house dogs, and to accept the mentality of: you can do anything you want, but you cant do... 1.2.and 3. That way, we can live together nicely and they know what they can and cannot do.
From my training experience, if the owner can correct the dog effectively enough, (effectively meaning clearly) the dog will think twice about doing certain things.
Oh, and my client's dog literally changed in 3 days.