In Short (or tl;dr)
These all say the same thing (well, or can say the same thing; see below):
- It has been done so as to permit air to circulate freely.
- It has been done so to permit air to circulate freely.
- It has been done in order to permit air to circulate freely.
- It has been done to permit air to circulate freely.
- It has been done so (that) air can circulate freely.
(The fifth shows an alternate version with a finite clause instead.)
Of the to-infinitives, the fourth is best because it is shortest; the earlier ones are padded yet say the same thing as number four does.
In the second instance, done so to [verb], an ambiguity exists that cannot be completely resolved without more context being given. The first reading is given above, while the second reading equates these instead:
- It has been done so to permit air to circulate freely.
- It has been done like that to permit air to circulate freely.
- It has been done that way so that air can circulate freely.
In one reading, so to + infinitive is expressing simple purpose, just like in order to + infinitive does:
- We always lay a heavy log upon the hearth before we retire so to have hot embers awaiting us when we arise before dawn.
- We always lay a heavy log upon the hearth before we retire so as to have hot embers awaiting us when we arise before dawn.
- We always lay a heavy log upon the hearth before we retire in order to have hot embers awaiting us when we arise before dawn.
Or with that instead of to:
- We always lay a heavy log upon the hearth before we retire in such a way that we will have hot embers awaiting us when we arise before dawn.
- We always lay a heavy log upon the hearth before we retire. This is done (so) that we (may) have hot embers awaiting us when we arise before dawn.
- We always lay a heavy log upon the hearth before we retire so that we (may) have hot embers awaiting us when we arise before dawn.
- We always lay a heavy log upon the hearth before we retire in order that we (may) have hot embers awaiting us when we arise before dawn.
- We always lay a heavy log upon the hearth before we retire that we (might/shall) have hot embers awaiting us when we arise before dawn.
However, an alternate way to read so to + infinitive interprets so to mean thus or in this/that way; that is, it refers back to the matter that has come previously, making for these equivalences:
- We always lay a heavy log upon the hearth before we retire. We do this to have hot embers awaiting us when we arise before dawn.
- We always lay a heavy log upon the hearth before we retire; this way we have hot embers awaiting us when we arise before dawn.
- We always lay a heavy log upon the hearth before we retire. We do so so that we may have hot embers awaiting us when we arise before dawn.
In §VI of the OED2’s entry for so adv. and conj. (ɴᴏᴛᴇ ᴡɪᴛʜ ᴄᴀʀᴇ: the OED does not consider so to be a “preposition”!), the editors present the equivalence of the three conjunctions:
28. a. so··, or so··as, so as, followed by an infinitive denoting result or consequence.
The monumental An Historical Syntax of the English Language by Fredericus Theodorus Visser from 1979 covers the use of the infinitive as an adverbial adjunct quite well. In §965 on page 1019 of Volume I, Visser describes the type characterized by:
965 — Type ‘He had put on his globes so as to be ready’
Here the infinitive is preceded by so as to (viz. without any words between so as in the preceding section). Whether the infinitive group expresses result or purpose is not in all the examples easy to ascertain, the line of demarcation between these two notions being often extremely vague.
Occasionally the subject of the infinitive is expressed by inserting a (pro)noun beween so as and the to + infinitive. In later Modern English this (pro)noun is preceded by for in this case: [. . .]
There are a few examples of the use of so to instead of so as to to express result. This idiom is now obsolete: [. . .]
As I read them, both the OED and Visser feel that the omission of as in so (as) to is now considered either ‘irregular’ (OED) or ‘obsolete’ (Visser) when expressing a result. Whether they also consider the omission of as when expressing a purpose is not given.
I may not be reading something correctly here regarding obsolescence or irregularity. I say this because modern examples of (apparently) using so to to express purpose can be produced. Here are a few that I believe qualify:
- We don't call “moving in” marriage, since it is done so to intentionally avoid the obligations of the marriage vows.
- Plundering and undermining are done so to gain wealth and to discredit the faith and spiritual vitality of the people.
- Deceptions practiced are done so to achieve one or more of those goals.
- And the lists that we then compile, are done so to respond to the specific requests of the person asking for the information. . . .
- All the good works performed by the fraternity or sorority were done so to glorify Christ.
To me, these modern examples suggest that using so to with the sense of in order to remains a viable reading. It is for this reason that my in brevis abstract at the top of this posting equated the four versions that it did.
However, as Araucardia has kindly observed in comments, in at least some cases there may be wiggle room in deciding whether so is used anaphorically to refer back to the earlier part of the phrase.
Contemporary readers, especially ones unfamiliar with “older” English literature (say, through the 19th century, the last such citation given by the OED or Visser), may fail to recognize the dual readings possible with so to, and therefore only ever see it as meaning thus or in this way combined with in order to.
But while the majority of the modern instances of done so to are using so to mean in that way, others do not.
The original question does not give enough context to give one short and simple answer that is also correct, and complete.
Postscript: Preposition or Conjunction?
There has been some discussion in comments as to whether things like so as and in order are prepositions or conjunctions. The answer, substantiated below from four documented sources, is that these are indeed conjunctions, albeit subordinating ones not coördinating ones.
To start with, both the Oxford English Dictionary and the Oxford English Grammar of 1996 maintain that constructions like so as and in order are subordinating conjunctions, and that they can be followed either by non-finite verb clauses or by that and a normal subordinate clause.
From Oxford English Grammar of 1996 on page 158, with all italics in the orignal:
Some subordinators are restricted to certain types of non-finite or verbless clauses: for, in order to, in order for, so as, with, without.
[. . .]
- : I sometimes wonder whether Stephen actually went to prison in order to have something to talk about when he came on this show.
Next, this same position is also taken by Quirk’s Comprehensive Grammar of the English Language of 1985, which on page 1004 reads as follows, with all emphasis in the original. The first and last sentences are the salient ones here:
14.17 Subordinators for to-infinitive clauses
To-infinitive clauses may be introduced by several subordinators: as if, as though, for, in order, so as, whether . . . (or), with, and without. For in this function is restricted to infinitive clauses with their own subject, and indeed is often obligatory (cf 15.10):
- It would be absurd idea for them to move to another house at this state of their careers.
Since for may be combined with the subordinator in order to, it seems to be a device for introducing the subject rather than to be a true subordinator:
- In order for you to be eligible for a student grant, you parents must receive less than a stipulated annual income.
Whether (with or without correlative or) introduces a subordinate interrogative clause:
- I don’t know whether to put on the air-conditioning today.
In the absence of a subject in the subordinate clause, the subject is understood as identical with that of the matrix clause.
This shows that both Oxford and Cambridge include so as and in order among their list of subordinators (read, subordinating conjunctions), no matter whether these introduce a non-finite clause or a finite one.
As one final reference supporting this placement — and although I am rather shy about citing crowd-sourced material, particularly in the absence of duly footnoted references — I cannot help but observe that the Wiktionary page listing English prepositions includes none with so in them. They do of course include as there, too, while noting that as can also be an adverb or a conjunction — the selfsame categories into which so has traditionally fallen.
Apparently Messrs. Huddleston and Pullam have decided to reclassify subordinating conjunctions as prepositions when their argument is a non-finite verb clause but leave them as conjunctions when followed by a finite clause.
Makes no sense to me, but this would hardly be the first time I’ve caught Pullam barking up the wrong tree.