Abs-E (or, speak only in the positive)

The wires one immediately sounds very plausible. I think its because "stay away from the wires" gives me a clear instruction I can immediately start following. "Don't touch the wires" instead gets logged away as something I will hopefully remember if I start thinking about the wires again.

There seems to be a body of evidence supporting the idea that positive statements require less cognitive load to process than negative statements. These are a couple of examples I could pull up:

https://doi.org/10.1037/mac0000057

https://link.springer.com/article/10.1007/s10979-007-9103-y

I remember in a Tali Sharot talk, sorry I can't remember the specific one, where she noted it was easier to get people to press a button at the right time when an incentive is offered than it is to dissuade people from pressing a button with a penalty. Which broadly seems corroborated by Skinnerian "Schedules of Reinforcement" research.

positive statements like "Stay away from the wires" are more effective than negative statements, like "Don't touch the wires," because your brain basically ignores the negative part of it. "*mumble mumble* touch the wires? Don't mind if I do!"

That's what I was going for with

When reading or hearing a negation used in language, you must first process the positive form it contains to understand the entire statement. For example, to understand "the sky is not green", you must first understand "the sky is green", then negate it. Usually, this happens quickly and subconsciously, but it can harmfully slow down or weaken understanding by making you first consider a false idea.

I find that editing my writing to use positive statements does make it better. I feel doubtful I could easily take it to the extent of making all positive statements. This might be an interesting use of LLM rewrites: negative->positive rephrasing feels like something within GPT-4's capabilities, and it would let you quickly translate a large corpus to read & evaluate. (I dislike the current name Abs-E and by analogy to E-Prime, suggest 'E+' - short for 'English-positive'.)

This would also combine well with 'Up-Goer-Five' style writing. In fact, I think Up-Goer-Five writing is already mostly E+ writing because of the need to say what something is rather than is not.

"Grass will pass through you without providing energy" : "without providing energy" seems little different to "not providing energy", it's still at heart a negative claim

That one seems easy to do if you go more quantitative. What is 'energy'? I mean, by e=mc^2, some grass embodies a lot of energy. You mean calories. "Grass provides 0 calories" is a positive assertion, which is more correct and still reasonably natural English. "Oh, I meant for humans". Fine, your first two versions failed this ('indigestible' for whom, exactly?) but easily revised: "Grass provides 0 calories to humans." 0 is not a negation, but a specific number, and so is valid, and correctly expresses the intent while not being overly universal and implying false things about herbivores.

"Only food that can be easily digested will provide calories"

That statement would seem to also be obviously wrong. Plenty of things are 'easily digested' in any reasonable meaning of that phrase, while providing ~0 calories. Water, for example. Or artificial sweeteners. Minerals like calcium. (Chiral molecules, if you want to go really exotic.)

On further consideration, and by analogy to "is immortal" being functionally equivalent to "will live forever" (so if it's interchangeable wording, does that mean that "is immortal" is actually equally a positive statement?)

This example might be considered a benefit of the style. People can mean rather different things by 'immortal' if they are simply defining it by negation as 'not dying'. One common definition is 'not aging' (ie. the probability of annual mortality being the same each year indefinitely); the other common one is some sort of 'indestructible and will exist to the end of the universe'. The former is fairly ordinary and mundane and describes, say, naked mole rats; the latter is purely imaginary and found only in fictional works like comic books or sacred scriptures. If the former, you might say something like 'has constant mortality rate', and if the latter, 'existing forever'.

So banning 'im-mortal' (which etymologically, turns out to be what you'd assume: Latin in - mortalis, "not-mortal") could be useful. (You do see IRL people sometimes object to longevity discussions on dumb grounds like "you can't become immortal, what about accidents?!"...)

In constructivist logic, proof by contradiction must construct an example of the mathematical object which contradicts the negated theorem.

This isn't true. In constructivist logic, if you are trying to disprove a statement of the form "for all x, P(x)", you do not actually have to find an x such that P(x) is false -- it is enough to assume that P(x) holds for various values of x and then derive a contradiction. By contrast, if you are trying to prove a statement of the form "there exists x such that P(x) holds", then you do actually need to construct an example of x such that P(x) holds (in constructivist logic at least).

You almost always have some information to concentrate your priors. Between mutually-helpful speakers, implicit with an answer to a question is that the answer gives all the information you have on the question that could benefit the questioner. E.g.

What will the closing price of Apple be at the end of the year?

"Almost certainly somewhere between $150 and $250."

I predict that it mostly gets worked around, by using only a few extra words.

"The sky is something other than blue" and "I will be somewhere else tomorrow" are both semantically-equivalent to the forbidden forms.  Even "I deny that the sky is blue" is a positive-form negation of the object-level statement.

I suspect all such workarounds depend on one of a relatively small set of negation-enabling words, such as "other", "else", and "deny", as you demonstrate. Prohibiting more words should eventually block all workarounds, while making writing more annoying.

It's interesting that Esperanto is an artificial language, and its paucity of antonyms is by deliberate design. Orwell's Newspeak has the same feature ("ungood" = bad), and was in part a satire on Esperanto and similar artificial languages.

I suspect that natural languages generally have primitive antonyms for the most common words. Anna Wierzbicka analysed the language of thought into a small number of "semantic primes", originally 14 and currently 65. She arrived at these through the study of natural languages, searching for a set of concepts with which any sentence of the target languages could be expressed. Despite the limited inventory, she found reason to include several antonym pairs: same/other, live/die, good/bad, do/don't, few/many, etc. Even want/not-want was in the original set of 14, although I see that not-want is not in the current set.