1.
A simple example: I could look at a sequence of numbers and make a new sequence of numbers in which each number is the difference between two consecutive numbers from the first sequence:
Then I could make sequence 3, with de differences between the numbers from sequence 2:
0, 2 (sequence 3: sequential subtraction, with as its object sequence 2)
Sequence 2 does with sequence 1 exactly the same thing (sequential subtraction) as what sequence 3 does with sequence 2. Both the operations carried out by 2 and 3 can be defined by the same name, sequential subtraction. We could say that 3 operates on a meta-level in relation to 2. We could say that 3 is meta-2. In logical language, we could define 2 as SeqSub(sequence 1), and 3 as SeqSub(SeqSub(sequence 1)). As you see, 2 is nested within 3.
On the other hand, consider grammar and normal language. Normal language describes the world, ideas, anything. Grammar describes, too, but it only describes language, not the world or anything. Grammar could be called meta-language. In this manner, we could call anything meta-x as long as x is a theory and meta-x is a theory about x, even if x and meta-x operate in different ways. X should be something vaguely similar to a theory, some abstract operation.
(You could even use meta- with things that aren't theories or abstract operations, but that is normally not done, except as a joke — suppose you had a brush to clean the floor, and a rag to clean the brush; then you might jokingly call your rag a meta-cleaner.)
Now what is the difference between grammar and sequence 3? We could say that grammar does not do exactly the same thing with language as what language does with its object, because grammar cannot, for example, refer to a physical thing directly. I think this what your quote means, the difference between identical operation on the one hand and similar-but-not-identical operation on the other. Sequential subtraction = sequential subtraction; grammar is a language, but language is not always grammar. In logical language, we could describe language as Describes(world), and grammar as GrammaticallyAnalyses(Describes(world)). They are nested, but in two slightly different ways.
Of course this distinction between "identical" and "similar" depends on definitions, which may be somewhat arbitrary. So I do not have full confidence in its strength and meaningfulness.
2.
I think I can feel what you mean, but I am not sure I'd phrase it like that.
I'd put metalogic in the second category mentioned above — similar to but not exactly the same as — but I am not sure. The reason why it is called "meta-" is that logic studies language and thinking, which makes logic an abstract operation and a theory; and metalogic studies logic, so that it is on a meta-level in relation to logic. Note that meta-x is always relative to its object: metalogic is not "meta-" in relation to, say, pottery.
3.
[Edited] In "metaphysics", the prefix "meta-" is used in its original sense in ancient Greek, which is here "after". Aristotle wrote the Physica, which were about the workings of nature: physis/phusis is Greek for nature. And he wrote the Metaphysica, which he called "The [books/bookrolls] about prime philosophy" — physics was the secondary philosophy. Later Greek scholars catalogued this work as "ta meta ta PHusika": "the things after the Physica", because they came after his Physica in their catalogue. Because his Metaphysica were about causality and other principles at work behind the physical world, it seems people later interpreted the "meta-" in Metaphysica as meaning "on a higher level than", and that is where our use of "meta-" came from.
4.
Stackoverflow is a website with questions and answers about programming. Meta-Stackoverflow is a website with questions and answers about Stackoverflow. So M-SO operates on SO the same way as SO operates on programming. In logical language, SO is SO(programming), and M-SO is SO(SO(programming)). You see how one is again nested within the other? That is why it is called Meta-Stackoverflow and not Newname.
Merriam-Webster's Eleventh Collegiate Dictionary (2003) lists several words in which the prefix un- negates a word that de- has previously negated:
- undeciphered
- undecomposed
- undefoliated
- undeformed
Such instances are quite rare, however, in comparison to the number of words that use re- to override a de- negation. Thus, for example, if a doctor decontaminates a wound with antiseptic, but bacteria later reappear, we say that the new germs "recontaminate" the wound, not that they "undecontaminate" it. Among the word pairs that follow this model are:
- decertify/recertify
- declassify/reclassify
- decolonize/recolonize
- decommission/recommission
- deconsecrate/reconsecrate
- deconstruct/reconstruct
- deemphasize/reemphasize
- deenergize/reenergize
- deescalate/reescalate
- dehydrate/rehydrate
- deinstitutionalize/reinstitutionalize
- delegitimize/relegitimize
- delist/relist
- demilitarize/remilitarize
- denationalize/renationalize
- depolarize/repolarize
- depopulate/repopulate
- deregulate/reregulate
- desegregate/resegregate
- deselect/reselect
- desensitize/resensitize
- destabilize/restabilize
- detach/reattach
Not all of these pairs are exact opposites (and certainly not in all senses of each term), but a number of them are very nearly so.
Still, noting the existence of this group of paired opposites is very far from saying that replacing a de- prefix with a re- prefix negates the word that de- was attached to. I suspect that Teresa, who contributed an earlier answer nominating en- as a possible negating prefix for words containing the de- prefix, had a similar (albeit smaller) set of opposed word pairs in mind:
- decamp/encamp
- decipher/encipher
- decode/encode
- decrypt/encrypt
- dethrone/enthrone
Unfortunately, the two instances that Teresa put forward to show en- in direct opposition to de- weren't good examples; but her argument about en- does have some validity, as the five word pairs above demonstrate.
Ultimately, the simple answer to the general question raised in the original post is that no prefix consistently and reliably undoes the negation that de- (or un- or dis-, for that matter) introduces.
Best Answer
Meta- comes from Ancient Greek, and in this sense means beyond, with, about or after. An example is Aristotle's Metaphysics which his editors placed after his Physics, and started readers thinking it might mean more than location. That suggests to me that the Latin pre- or Greek pro- could be possible antonyms used in a similar way.
But this would only help if people used it. Shall we try to push premetadata or prometadata? Or would actual data be more easily understood?