R+, R-, P+, P- (Plus, a little bit about desensitization)

Positive Reinforcement (+R): Increasing the probability of a behavior by delivering a stimulus immediately after a response or behavior is exhibited.
Negative Reinforcement (R-): when a certain stimulus occurs (an aversive stimulus) is removed after a particular behavior is exhibited. This particular behavior is likely to occur again in the future because of removing a negative consequence.
Positive Punishment (P+): it works by presenting a negative consequence after an undesired behavior is exhibited, making the behavior less likely to happen in the future.
Negative Punishment (: this happens when a certain desired stimulus is removed after a particular undesired behavior is exhibited, resulting in the behavior less likely to happen in the future.

Is it possible to only use R+?

Unfortunately, no. Here are some common mistakes that us R+ trainers make on occasion.
Scenario 1 (R-): You stand beside your horse and lift your leg, trying to convince him to mimic you and lift his leg as well. You try for a minute and he still doesn't do so. So you reach down and tap his leg with his hand. Once his leg comes up, you stop tapping, but you still give him a treat. You just used R- without intending to, because you introduced a negative stimulus (an annoying one, you tapping his leg repetitively) and then quit tapping once he lifted his leg up.
Scenario 2 (P+): You're trick training your horse and he keeps bringing his head in toward you for treats (if your horse has been trained correctly, this won't happen, but we'll use this in our scenario because a lot of people have trouble with this). As a result, you stick your elbow out and whenever he reaches his head in, he bumps into your elbow. He is now learning not to mob for treats because he received a negative consequence (bumping into your elbow) after trying to mob you.
Scenario 3 (: You're trick training your horse, and are about to give him a treat. Before you even open your hand (with the treat), he begins to try and grab the treat. Then you take the treat away and decide to give it to him later. Alternatively, you take the treat away and wait until he brings his head forward, or looks away, then you present the treat. Then you repeat this training (sticking your hand out and if he tries to grab the treat before you offer it, taking the treat away and waiting until he brings his head either forward in front of him or looks away) until he learns it. This is negative punishment.

There are many others situations that can happen where you will accidentally use R-, P+, and P-. But everyone, human and animal, goes through all of these. The below examples may seem off topic (since this is a horse forum, not a parenting forum!) but this is to explain that you don't have to feel bad, everyone has gone through it and it won't make a horse horribly depressed.
Scenario 1 (P+): A child gets an A+ on his report card, so his dad takes him out for ice cream.
Scenario 2 (R-): A child sweeps the kitchen, in order to avoid his mom nagging him about it.
Scenario 3 (P+): A child calls his little brother a bad word. As a consequence, his parents send him to his room for a time out.
Scenario 4 (: Two children fight over a toy. Then the parent takes the toy away.

Don't feel bad if you accidentally use R-, P+, or P-!

So if you find out that the other day you have accidentally used R-, P+, or P-, don't feel bad about it! It's not damaging, everyone does it accidentally. It's impossible to use R+ 100% of the time. Some traditional horse trainers (aka ones who primarily use R-) do accidentally use R+ on some occasions. Here are a few examples:
Scenario 1: A horse trainer is teaching his horse to ground tie. He stands away from his horse, and waits. His horse stands there for him, so he goes over to the horse and gives him a scratch on the neck.
Scenario 2: A horse trainer is teaching his horse to stand tied. He stands beside the horse and the horse stands still. The trainer then reaches over and scratches the horse on his forehead.
Scenario 3: A horse trainer is teaching his horse to enjoy arenas (many horses are scared of arenas, after being abused by them in the past). He unhalters his horse and allows his horse to walk around and roll. He's teaching the horse that the arena is a fun place to be, by providing a positive stimulus (exploration and scratching itches on his back by rolling).

There's no trainer that uses only R+ or only R-. I used to feel guilty if I ever accidentally used R-, P+, or P- on my horses. But then I realized all trainers will do it, whether on purpose or accidentally.

So does that mean there is no such thing as an R+ or R- trainer?

No, it does not. R+ trainers primarily use R+. R- trainers primarily use R-. You'll mainly see R+ trainers using targets, clickers, treats, and cordeos on their horses. You'll likely see R- trainers using rope halters, pressure and release, whips, and different tools on their horses. It just depends on what type of training (R+ or R-) the trainer himself/herself focuses on.

Is there any truly damaging way of training a horse?

Yes. That is called flooding, or shutting a horse down. Let me explain how R+ and R- trainers go about desensitizing horses.
R+ desensitizing: They tie a plastic bag to the end of a stick. They hold it out in front of the horse, and reward the horse with a treat or a pet if he looks at it, goes up to it, or touches it. They gradually work up to rubbing the horse with it, and reward him for standing, and ignore him moving. The horse's reward is the presenting of a treat, pet, or desired stimulus for standing still.
R- desensitizing: They tie a plastic bag to the end of a stick. They hold the horse on the halter, and begin to wave the stick around. Some trainers will start slowly, others start quickly. If the horse stands, they take the stick away and hide it behind their back. If the horse moves, they continue to follow the horse around with the plastic bag, shaking it, until the horse stops moving. Then they take it away. The horse's reward is the removal of the scary thing for standing still.

People who flood use a different technique than either of the two. These trainers were very, very common back in the old days. I don't even like to call them trainers, because to me, this method falls under abuse. Here are some common techniques:
-Tying a horse's legs up, pushing him to the ground, and throwing a tarp on him and leaving him like that for a few hours.
-Tying a horse's legs up, pushing him to the ground, and spraying a hose on him.
-Tying a horse's legs up, pushing him to the ground, and beating him with a whip until he doesn't react (this apparently gets a horse used to pain, so when the rider gets on, he can spur/whip the horse as much as he wants without the worry of the horse freaking out or bucking).

Flooding is different than other types of desensitization. With flooding, the scary thing is not removed at all (unlike in R- desensitizing, where they remove the scary thing once the horse stands) when the horse displays positive signs like staying still. The horse is also normally tied up and restricted in such a fashion to where there is no escape. They'll commonly tie his head either to a high pole so that he cannot break free, hobble his front legs and beat him with things, or completely tie him down and push the horse to his side. R+ trainers don't restrict their horses like this. Some choose to use halters in their desensitizing. R- trainers do use halters in their training, but they usually keep the lead rope long, and allow the horse to move around if he needs to. The trainers who flood don't allow the horse any way of escape, they make it so he cannot even move. They also do not reward the horse with a positive stimulus (treats or pet) once the horse stops moving. Nor do they remove the negative stimulus (like R- trainers do) once the horse stops moving. They usually keep the horse restrained, with the scary object on him or by him, for several hours at a time.

This causes a horse to shut down. Their eyes will gloss over, they will have a permanent "dull" look on their face, show very little expression, and they will usually be deemed as "bombproof". This is not to say all bombproof horses were shut down. This is certainly not the case. But the horses you see that hang their head there, have a dead look in glossy eyes, their ears are always stiffly back, he has no curiosity to speak of, etc. then you're probably looking at a shut down horse, who was flooded. The shut down horses are actually not bombproof at all. When presented with something that scares them, they freeze up and accept death. These horses have been taught that when something scary happens, there is nothing they can do except freeze up, stay still, and wait to die. This is also a form of learned-helplessness. This is the cruelest way of training a horse. It destroys his spirit and his natural curiosity. People think of desensitizing as "getting a horse used to that object in case you encounter it on the trail". That is far from the truth. Desensitizing is teaching a horse how to deal with fear. It teaches them how to wind down if they see something stressful. If you desensitize your horse enough, you'll find that within a month, a few months, a year, or even a few years, they see something scary, get a little started, then drop their head, lick and chew, and cock their leg as if nothing happened. This is from all the desensitizing. The desensitizing puts horses into the dealing with their fear and winding down. This isn't to say these horses will never have a big spook, but they will usually quickly calm down as if nothing happened, because they have been conditioned to do so. Flooded horses however, are in constant fear. They see a scary object and instead of becoming afraid and then winding down and becoming calm, he sees it and thinks he's about to die, so he freezes up and waits (since he has been taught that response through flooding). These horses are usually the ones that people say "he literally never spooks!" (I'm not saying that horses who rarely spook are shut down, just saying it's a common comment that people make if their own horse is shut down). Then a few years later, their horse will actually spook, and they'll say, "it just came out of nowhere! He bolted and ran for 10 minutes, and he didn't calm down for hours! I never saw that coming!" This is because the horses that were flooded were never taught to deal with fear. If they do get that scared, it usually takes a very long time to calm them down again. Whereas if you desensitized a horse instead of flooded them, they will usually wind down quickly. They may be afraid and a nervous wreck as long as the scary object is present (a tractor in this example). But as soon as the tractor goes away, they either calm down right away, or it only takes a few minutes. Horses that were properly desensitized (and not flooded) rarely, if ever, stay wound up after passing the scary thing.

This is because the desensitized horses were taught that standing still, lowering their head, blinking, licking and chewing, cocking a hind leg, or any kind of sign of relaxation either got them a treat or praise (R+) or it caused the scary thing to go away (R-). Flooding damages a horse for life (it's hard to earn the trust of a shut down horse) and makes him into a zombie. His only sign of life is a beating heart. He doesn't have a twinkle in his eyes, or any curiosity. That has all been killed the second he was flooded or tortured. Desensitized horses become braver, more confident, and learn to deal with fear and stress.

So to answer your question... Yes, there is a damaging way of training a horse. It's done through flooding, shutting a horse down, or through straight out abuse (such as beating, bike chain bits, twitching, shocking, tasering, soring, gingering, or any method that causes emotional or physical harm to a horse.

Your views on R+, R-, P+, and P-?

I use all of these techniques. Everyone does. I use a mix between R+ and R-, depending on what I'm teaching or unteaching. It's impossible to only do R+, or only do R-. You may see a R- trainer say, "I hate R+ training, it makes horses disrespectful!" But it is very likely that they have used it at one point without realizing it.

Here are some pictures.

R+


R-


Flooding