LONG POST...
The moment I can relate a partial action, standard action, and a move equivalent action in any two formulas, then I can PROVE, by eliminating SA from the equation, that PA = mea.
First off, let me tell you that I *do* see your mathematical argument, however, I disagree with some of your logical steps are flawed (in that some of the "formulas" you use are not valid) and I will try to use concrete examples from the SRD to explain my perceived flaws in your reasoning. I will state my first assumption here... "you can always choose to do less than your maximum potential but you can never choose to do more than your maximum potential." This seems reasonable to me. I will reference only those SRD definitions relevant to my argument. Let us look some of the action types as found in the SRD:
From the SRD
Move-Equivalent Action: Move-equivalent actions take the place of movement in a standard action or take the place of an entire partial action. The combatant can normally also take a 5 foot step.
I read "take the place of an entire partial action," as "you can make a partial action into a MEA, but you cannot make an MEA into a partial action." IOW, it's a one-way conversion. You can do some actions in the space/time/effort it takes to do an PA that you don't have enough space/time/effort to do as an MEA, hence the need for a class of actions designated as MEA (that can be performed as either a PA or an MEA) and PAs (that can be performed as PAs but not MEAs).
The relevant inequality would be MEA < PA.
From the SRD
Standard Action: A standard action allows a combatant to do something and move a combatant's indicated speed during a combat round. A combatant can move before or after performing the activity of the action.
It seems to me that this definition is fairly straight-forward... it tells me I can do something ("do something" = partial action, hereafter, PA) and move (=Move-Equivalent Action, hereafter MEA). Nowhere in this definition is it explicitly stated (though it IS implied) that a Standard Action (hereafter StdA) = PA plus MEA, but I grant that the formula just stated is the correct reading of this definition.
Notably, it does NOT say I can do something and do something else (IOW, 2 PAs). Movement and actions *are* considered separately in the 3e system... and when combined with the premise (above) that "you can make a partial action into a MEA, but cannot make an MEA into a partial action," you then have a need for the term "Move-Equivalent Action" to describe those actions that
The Full-Round action is not an interesting case, though as a sidebar, I would suggest that three "tiers" of actions are established in the rules...
1.) Full-Round Action / Standard Action
2.) Partial Action
3.) Move-Equivalent Action
Obviously, the main difference between Full-Round and Standard actions is that a full-round action is one really long thing that takes the whole round, while a SA is basically choosing to do two shorter things that each take part of the round. I don't believe that the previous sentence is in dispute.
<snip>Initiative is termed iniiative, and not turn order for a very simple reason. When it is not "your turn" - you cannot INITIATE an action.
<snipped meat of Initiative discussion>
Thus - it is possible to take three possible "categories" of action when you don't have the initiative.
1. a readied action - if you have met the conditions needed for it
2. a free action in "response to" the action being initiated
3. a "non-action" - such as falling, taking damage ect... that represents the consequence of that action occuring at that moment.
Not useful to my discussion, but as an aside, I like it very much... it is very neat and concise explanation of Initiative - probably best I've seen in a long time. But back to the discussion at hand...
Standard A = MEA + Partial A - given definition standard action
Standard A = MEA + MEA - given definition Double Move
Again, if you must use formulae, your first is correct. However, you run into trouble with the second statement... the general case of all Standard Actions and the specific case of the Double Move are not necessarily "equal." More importantly, this is where the crux of your argument lies, so it's worth examining. I am about to argue that "all Standard Actions are not created equal" or "some Standard Actions are worth less (but never more) than the general case of the Standard Action."
I am now going to derive the specific case of the Double Move action from the general case of the Standard Action given my premises above. In the following example, "Standard Action(1)" represents the maximum potential "value" of a Standard Action.
Standard Action(1) = MEA + PA (thus far, we agree)
However, *I* arrive at the double move action by using my inequality substitution...
PA > MEA
MEA + PA > MEA + MEA - by addition (I add an MEA to both sides of the above inequality)
Standard Action(1) > MEA + MEA - by substitution
By definition, a Double Move action = MEA + MEA (I don't think I will get an argument here)
Standard Action (1) > Double Move Action - by substitution. Thus, a Double Move action is a "lesser use" of a standard action because you don't get your full value.
MEA + Partial A = MEA + MEA - by substitution
Partial A = MEA - by elimination
Standard A = Partial A + Partial A - by substitution
Because my argument is that your two selected standard actions are NOT equivalent, these three lines from your proof are not meaningful.
I believe your formulae should instead read:
MEA + PA > MEA + MEA - by substitution
PA > MEA - By elimination, which is really trivial since we already established this as an axiom in reading the definition of "partial action."
Standard A(1) = Partial A + MEA by definition, and PA > MEA, so we conclude that
Standard A(1) < Partial A + Partial A by substituion.
This is admittedly kind of a circular proof, since it tells me that a Double Move is not using the full potential of the Standard Action, but I included it as an exercise for completeness.
The problem is it is SO simple that people think that it cannot be so;
Partial Action = Move equivalent action
Again, this assumption results from your reading the rules (which I believe is an incorrect reading)...
I am explicitly told that I can change a PA to an MEA in the rules. NOWHERE in the rules am I told that I can change an MEA into a PA. This suggests not equality, but inequality. I can always do "less" than the maximum (i.e., since PA > MEA, I can change a PA to an MEA since it represents less than the maximum I could do) but I can never do "more" than the maxmium I can do (since trying MEA < PA indicates that if I went from an MEA to a PA I would be doing more than normal).
They deny this simple very concretely proven fact without fail; and sure enough they make those absolutely incredulous claims that you so convienently pointed out. I strongly reccomend reading the section on page 62 of the dmg simultinaeity in combat;
As I'm sure you are aware, there are two ways to dispute a logical argument - attack the logic OR attack the assumptions/premises on which the argument was made. Your logic is impeccable. Your assumptions are not... your entire argument rests on the assumption that a specific case of a single Standard Action (specifically, the Double Move action) is equal in "value" to the general case of all Standard Actions. You rest your entire argument on this assumption, as this is the "crux" of your "reverse-engineering" to declare that an MEA = a Partial Action.
I hope I need not remind you that comparing specific cases to general cases in logic is almost always done to DISPROVE a statement about the generality.
I deny this very "concretely proven fact" on the grounds that your assumptions are fallacious, and have simultaneouly demonstrated that I can "prove" the D&D rules as stated are self-consistent with what I consider the proper reading of the definition of the Move-Equivalent Action. If you accept my premise that going from a Partial Action "down" to a MEA is a "one-way trip," your argument is simultaneously disproved AND I can account for all the D&D rules as written (I can account for a Double Move Action being represented as a Standard Action and still explain why you cannot take two partial actions).
Stop thinking of actions in terms of category - think in terms of how much time they take.
full round actions and standard actions take 6 seconds
partial actions and move equivalent actions take 3 seconds
free actions and/or non actions take 1 second or less - or are part of some other action (i.e. when casting a spell, you prepare it's components).
now you have a universal way to look at time - in terms of how many seconds it takes to do something. Much much easier than all this "your turn" stuff. When you can only "start to do something new" on "you turn" - you learn where the word initiative came from - not "turn order".
This is based upon your interpretation of the rules. I can just as easily say...
Full Round Actions and Standard Actions take 6 seconds.
Partial Actions take 3.5 seconds
Move-Equivalent Actions take 2.5 seconds
Free Actions and/or Non-Actions take 1 second or less.
This is fully consistent with the rules as they stand... in the 6-second round, I cannot take two partial actions... that would take me 7 seconds. I *CAN* take a partial action and a MEA... that takes me 6 seconds. I *CAN* take a double-move action (takes me 5 seconds) but not a triple-move action (takes 7.5 seconds).
CONCLUSIONS:
I contend that your conclusion is false on the basis that your assumptions are flawed because you are (a) ignoring the implication in the definition of MEA that PA to MEA is a one-way trip and (b) you are attempting to make an inequality out of a general case and a specific case.
By using the assumption that "PA to MEA is a one-way trip, you can't go back" I can generate ALL of the actions listed in the SRD (specifically the Double Move Action) without generating "formulae" that allow me to turn a MEA back into a PA. In fact, using this rule, the rules are consistent as written. Using this rule, I do not get a situation where I can get 2 Partial Actions in a round.
Now, the "weak point" in my argument is my inference from the MEA definition that PA to MEA is a one-way trip. You can argue that I have read the rule incorrectly, in which case my entire argument is invalid and yours is correct. However, you cannot disprove my reading of the rule as false, just as I cannot disprove your reading of the rules as true.
I further conclude that since my reading of the MEA definition (the "one-way trip" version) produces conclusions that are completely in harmony with published rules, and your version (the "equivalency" version) produces conclusions that are out of harmony with the published rules, that my reading is the "more correct" reading.
Essentially, we are arguing about what is NOT stated in the rules... my argument boils down to, "I am told I can go PA to MEA but nowhere am I told that I can go MEA to PA, therefore, I conclude that it is a one-way trip." Your argument boils down to "nowhere am I told in the rules that I cannot go from MEA to PA, therefore I conclude that the two are interchangeable." Neither of us can be "proven" right or wrong - we can only show how our models are (or aren't) consistent with/have an effect on the rest of the rules. Now, if WotC issues errata confirming one position or the other, we will have a definitive answer of "who is right and who is dead" but until that time, the rules look like Schrodinger's proverbial cat... they give us two answers simultaneously.
When I am the GM, this inference of "one-way trips" is gospel truth. If you wish to use a different ruling in your games, that is your business (and perogative) as a GM.
I hope I have presented this in a manner so as not to be insulting, but rather as a deliberate, thorough, and well-reasoned attempt to show you the weak points of your argument and justify my point of view -- while simultaneously admitting the weaknesses inherent in my argument (I think it's only fair). Again, I understand both arguments and it boils down to your choice of premises (both sides of which are supported due to the rules' silence on this issue).
I hope you can see the validity of my argument through this proof, as I have admitted the validity of yours provided I accept your premises (I happen not to, but cannot *disprove* them, just as you cannot *disprove* mine). I merely point out that my premises produce conclusions consistent with the remainder of the rules. Whether that sways you into thinking that mine are somehow "better" is irrelevant.
Respectfully submitted,
--The Sigil