That's a tough question... because all training/learning occurs through a form of motivation. The animal has to be motivated to do xyz by something... and so without using positive reinforcement... what are we left with to motivate the horse to learn something new? Something that we want them to do or learn?
Horses do not have an inherent desire to please us, respond to us in a positive way, or learn many of the things we want them to do (like stand tied to a wall.. SUPER fun for a horse). There is always an underlying motivation to everything they do with us/for us. Usually, it's a form of operant conditioning, but there's potential for an intrinsic motivator (like play) to factor in.
In theory, a potentially intrinsic motivator like play could be the motivator through which learning can be achieved, but it's very hard to use true "play" to teach a horse to tie to a trailer, to stand for shots, to pick up their hooves, to jump a jump course... Play is fantastic for exploring range of motion, for building a relationship, for rehabilitation, and perhaps other areas.. but it's not a practical way to approach all horse training, even "at liberty" training.
True play is also not something that's easily achieved with a horse, especially an adult horse.. and we often grossly misinterpret frustration and stress behaviors for playful behaviors. There is potential for other intrinsic motivators, which I won't dive into deeply in this post ( I will however link resources to learn more at the end), but again... it's going to be nearly impossible to teach a horse to walk a circle around us, to not kick at us, to jump over an obstacle, or change directions via intrinsic motivation alone, though we can definitely try integrating self reinforcing and self motivating factors into the training sessions to improve the quality of the training and the relationship the horse has with the training and us.
There are also other learning theories that can explain how a horse learns, how information is built upon, and explain different kind of motivation... but, there is very little information on if/how this can be applied to animal training at this time.
SO, at the end of the day, it's clear we need operant conditioning to teach specific behaviors we would like, and if we aren't using positive reinforcement we are left with negative reinforcement, positive punishment, and negative punishment. And none of these allow the horse full autonomy/choice from my personal experience and from what science tells us, so can the horse every truly be "at liberty" without choice?
For negative reinforcement to work, we have to apply pressure until we get a response from the horse that we want. If the horse tells us "no", by not responding or responding incorrectly, and we stop applying pressure... that would be ineffective training and the desired progress will never be achieved. All we will achieve is teaching the horse that if they don't respond the pressure will stop, they will quite literally learn to do nothing in order to get relief from the pressure, and we will teach them to say "no" every single time. OR.. we will teach them to respond "incorrectly", such as turning left when we ask for right or backing up when we ask for forward.... All those beautiful liberty circles and fun things we picture doing with our horses at liberty will never happen.
The other two are forms of operant conditioning are punishments, which are used to STOP or decrease a behavior, not create it. So, that wont work... we need the horse to "do" something, not "stop doing" something. Punishers also don't give the horse a choice because the point of punishment is to correct a behavior we do not like. That's our choice, not theirs.
So this leaves us with.. positive reinforcement. Now, we don't have to use food.. we could use something else of value to the horse, like scratches in a sweet spot....and we don't have to use a clicker (which is just a bridge signal), but in order to give the horse control over the outcome of the training, choice in whether they respond or not, and choice on how they perform.. TRUE "at liberty" training... I do believe we have to use positive reinforcement at least to a certain degree (if not completely).
photo credit ; danielle vargo photography
* Disclaimer: I'm fully aware that there is much more research to be done on motivation, intrinsic motivation, play, learning theories etc. This article is not meant to replace self education on the subject or to be taken as "fact", but instead it's intended to inspire questions/reflection and to encourage honest analysis of mainstream "liberty training".