Apparently there was more than a little bit of formality to those matches.
I wasn't really referring to any formality surrounding the match itself. The OP was talking about something he termed "formal and informal combat." So we're talking about the action of combat itself. The OP uses as his example "MMA" and "TMA," and equates TMA's solely with sport / point fighting tournaments, as if those things define the arts. By TMA tournaments being more "formal," I assume that he means a stricter rule set, or resetting after a point is made, etc. He may also be referring to participants practicing their style more "by the book," and being somewhat more "rigid" in their approach to fighting, though I'm only speculating.
In any case, all of the above are modern phenomenon for most traditional martial arts - excluding, of course, western arts, many of which did derive from sport, and particular wrestling sports such as shuai jiao or sumo. I would also include here particular modern arts reinvented from older arts for a sportive emphasis, such as judo or kendo for example. Now, having hopefully covered my butt sufficiently, most older, traditional martial arts do not derive from sport / tournament matches, contrary to what the OP's impression seems to be.
Now, describe to me how combat that takes place in MMA is any less "formal" (as in rigid, controlled, or rule-bound) than a roof top fight, or some such duel, and you might have a relevant point.
Do you understand what "Professional" means? If you get paid to do martial arts then you're a professional martial artist. This ranges from teaching martial arts to performing them. The guy who owns the TKD Dojang down on the corner is every bit as much a "Professional Martial Artist" as is Anderson Silva.
World English Dictionary
[TABLE="width: 100%"]
[TR="class: tr1"]
[TD="class: td1, colspan: 2"]
professional (prəˈfɛʃən ə l) [/TD]
[/TR]
[TR]
[TD="colspan: 2"][/TD]
[/TR]
[TR="class: tr2"]
[TD="class: td2, colspan: 2"]
— adj[/TD]
[/TR]
[TR="class: tr3"]
[TD="class: td3n1, width: 1%, align: right"]
1.[/TD]
[TD="class: td3n2"]
of, relating to, suitable for, or engaged in as a profession[/TD]
[/TR]
[TR="class: tr3"]
[TD="class: td3n1, width: 1%, align: right"]
2.[/TD]
[TD="class: td3n2"]
engaging in an activity for gain or as a means of livelihood[/TD]
[/TR]
[TR="class: tr3"]
[TD="class: td3n1, width: 1%, align: right"]
3.[/TD]
[TD="class: td3n2"]
a. extremely competent in a job, etc[/TD]
[/TR]
[TR="class: tr3"]
[TD="class: td3n1, width: 1%, align: right"]
[/TD]
[TD="class: td3n2"]
b. (of a piece of work or anything performed) produced with competence or skill[/TD]
[/TR]
[TR="class: tr3"]
[TD="class: td3n1, width: 1%, align: right"]
4.[/TD]
[TD="class: td3n2"]
undertaken or performed for gain or by people who are paid[/TD]
[/TR]
[TR]
[TD="colspan: 2"][/TD]
[/TR]
[TR="class: tr2"]
[TD="class: td2, colspan: 2"]
— n[/TD]
[/TR]
[TR="class: tr3"]
[TD="class: td3n1, width: 1%, align: right"]
5.[/TD]
[TD="class: td3n2"]
a person who belongs to or engages in one of the professions[/TD]
[/TR]
[TR="class: tr3"]
[TD="class: td3n1, width: 1%, align: right"]
6.[/TD]
[TD="class: td3n2"]
a person who engages for his livelihood in some activity also pursued by amateurs[/TD]
[/TR]
[TR="class: tr3"]
[TD="class: td3n1, width: 1%, align: right"]
7.[/TD]
[TD="class: td3n2"]
a person who engages in an activity with great competence[/TD]
[/TR]
[TR="class: tr3"]
[TD="class: td3n1, width: 1%, align: right"]
8.[/TD]
[TD="class: td3n2"]
an expert player of a game who gives instruction, esp to members of a club by whom he is hired[/TD]
[/TR]
[/TABLE]
Language is dependent on context. Words carry multiple meanings, depending on their use, and the intent of the speaker. Now, I kind of expect people to try to understand my intended meaning, or overall point, as opposed to just nit-picking in order to criticize, or completely circumventing my intent and pointing at irrelevant examples.
However, regardless of your definition, as I believe I also used professional in the sense of someone paid for a profession, your point is still irrelevant. A Professional Mixed Martial Artist is paid to fight in Mixed Martial Arts matches. A Professional Martial Arts instructor is paid to instruct Martial Arts, not to fight in Mixed Martial Arts matches. Moreover, it should at least be recognized that Martial Arts has not always been as commercialized as it is today. While there certainly were those whose main occupation was teaching or practicing martial arts throughout history, there were also many for whom practicing and teaching martial arts was not a profession, and this remains the case even to this day.
I think your idea of "formal" and "informal" may need some tweaking. Particularly in antiquity, there were many instances where joining armies on the battlefield was much more "formal" than you seem to give them credit for.
This is a bit subjective, but you may make a valid point here. Military training and practice is by definition what you might call "formal." At least in practice, if not application, doing things in a disciplined, "by the book" manner is important. And I am aware that in antiquity, battles were sometimes a very formal occasion. But, if we are talking about individual combat itself, that is somewhat outside the relm of formality surrounding a particular engagement.
Not as a "measurement" as a description or definition.
The people asking the question aren't "measuring" MMA vs. TMA, they're asking "is MMA a 'real' martial art" (as can be seen by following posts in this and its sister thread).
Actually the OP was trying to equate MMA to martial arts as a whole. He asserted that it was no different than Traditional Martial Arts, except that it is more "informal," and was contrasting MMA competitions to tournament competitions.
You specifically wrote, "They ask why there aren't any traditional martial artists in professional MMA matches." I was pointing out that this assumption is flat out wrong. There are many "professional MMA" competitors who have a TMA background, many of whom continue their TMA training.
It is irrelevant if a professional MMA competitor is from a TMA background or not, or whether they continue their TMA training. They may even use some of their TMA training in MMA, yes. But their training and fighting methodology is unique to MMA. In essence, what they do in the ring is usually better described as "Mixed Martial Arts." And yes, I do realize there are exceptions to that rule, such as the Gracies, but many have pointed out already that, as MMA evolves, it seems to be developing more and more its own distinct methodology and repertoire which best suit it. I tend to agree with those who say that it might one day evolve into a distinct martial art in its own right.