What is an adjective for "weaker than weak"? I defined a notion (say, some kind of equivalence) in three forms, the first implies the second, which in turn implies the third.
I would like to use "strong", (nothing), and "weak" to describe them.  But some one of great importance already used "weak" for something strongly related to the second form.  To be compatible with previous studies, I have to refer to the first as "strong", to the second as "weak".
The third is weaker than the weak.  I need an adjective to describe it.  "Weaker" is not good enough.  The notion also induce an adjective (e.g. equivalent) and a verb (e.g. equal), so I also need an adverb. "Weakerly" sounds strange.
Is there a standard adjective to describe the third notion? I now use "feeble".
I also see the possibility of defining a "stronger than strong" form.  Suggestions are also welcome for this.
 A: I would say "pathetic" or "puny".
A: In PDE theory, the term "very weak solution" is in wide use. Google results.
A: Very weak, according to xkcd:

But, more seriously, you can also put it the other way. 
For example, in case of the Riemann Hypothesis, the Riemann Hypothesis is implied by the Generalized Riemann Hypothesis, which is in turn implied by the Grand Riemann Hypothesis. 
But that maybe overkill.
A: I don't think there is a standard adjective to describe this. If there is, we would need to know the context of the terms stronger and weaker to answer. It sounds like you are defining this weaker-er notion in your paper (since you have to introduce a new term), so it is really on you to give it a name. Now to compile a list of suggestions:


*

*subweak

*weakerer

*superweak

*ultraweak

*extraweak

*weak' (weak prime)

*weak*

*weak$^2$ (weak squared, or weak two, or too weak)

*feeble

*anemic

*fragile

*puny

A: I can't resist adding this one to the list: what do you call a principle weaker than Weak Konig's Lemma?
Funny you should ask . . . (Weak Weak Konig's Lemma)
Ayup, we're a creative bunch. :P

Oh my goodness: page 18, after proposition 9.1. It's merely suggested, but: "Weak Weak Weak Konig's Lemma."

And heaven forbid we be at a loss to describe something not as weak as weak! (Page 8, definition 4.5.) "Strong weak truth table reducibility"
A: An interesting challenge!
A couple of additional ones I didn't see in previous answers:
strong > weak > light (adverb 'lightly')
strong > weak > delicate (adverb 'delicately')
strong > weak > wimpy (adverb 'wimpily') - sounds less formal but has precedents in technical areas.
strong > weak > tentative (adverb 'tentatively') - likely less useful, has overtones of non-strength attributes (eg: hesitation).
A: How about double weak and triple weak? 
Advantages:


*

*Clearer in rank than normal intensifiers (sub, very, super, extra, ultra). A triple weak is clearly weaker than a double weak, while it is not clear whether a super weak is weaker than a very weak or not. If we have a stronger version of a weak proposition, but weaker than the normal one, then we can just renumber the ranks and all are good.

*More natural in language than weak² or weak*, and less redundant than weak weak

*Reduce cognitive load for the naming, and leave more energy to actually studying it

