Word that describes a problem that is missing a precondition
Solution 1:
You seem to have two different kinds of examples there. In the first example the solution set is ill-defined:
A set is well-defined if any given object either is an element of the set, or is not an element of the set
but it's never clear whether a given number a is an element of the solution set to the equation; whether it is or not depends (as you state) on the value of b.
In your second example, the solution set (that is, the set of solutions to the problem posed) is well-defined, but empty because the conditions are impracticable:
(Definition 2) not practicable : incapable of being performed or accomplished by the means employed or at command
Solution 2:
Two words commonly used in engineering (especially where mathematical analysis is the basis of that engineering) are
- Underspecified
- Indeterminate
There are various ways to describe what these mean in technical detail, for example, as covered by this wikipedia article. Generally they mean that there is no unique solution; a unique solution cannot be determined. (There may even be no solution.) But their literal meanings are clear enough:
Indeterminate:
1) not determinate; not precisely fixed in extent; indefinite; uncertain.
2) not clear; vague.
3) not established.
4) not settled or decided.
5) Mathematics
(of a quantity) undefined, as 0/0.
(of an equation) able to be satisfied by more than one value for each unknown.
Underspecified means insufficiently specified; not enough information is provided.
I would have to say that your letter to Cambodia example is a stretch, but it if you assume it's missing a precondition (as your title says), then the words that fit are underspecified and indeterminate. (I would use another word in that case, like impossible.)
Solution 3:
In philosophical terms, the suppression or elision of a necessary precondition is known as a "enthymematic argument", and the hidden or missing precondition is the "enthymeme".
From Lander University's Philosophy Dept:
An enthymeme is an argument in which one proposition is suppressed—i.e., it's missing for one reason or another.
In some cases, the missing proposition is not stated because it is obvious.
In other cases, if the missing proposition were present, the argument might lose rhetorical force.
Occasionally, the proposition is suppressed in an effort to conceal the unsoundness or the invalidity of the argument.
Related, and better known, is the fallacy of the "loaded question", where one of the implicit premises of a question is controversial or unjustified. From Wikipedia's article on the loaded question fallacy, for example:
A loaded question or complex question fallacy is a question which contains a controversial or unjustified assumption (e.g., a presumption of guilt).
This better matches your delivery example: the unfounded, implicit, assumption is that it is possible to get from here to Cambodia in 10 minutes or less. If this assumption were made explicit, it could be easily challenged, rendering the broader question immaterial.
Solution 4:
What is your intention for use (excuse me if you already stated it, I didn't see it)?
The reason I ask is that I don't know how useful my suggestion will be to you, as you said you want to avoid tech jargon and my suggestion is philosophical jargon (while it isn't tech jargon, it is still jargon [although it has a long history before its current usage]). I don't think it's exactly what you're looking for (I'm not even sure if the Marxist dialectics really apply to purely logical system with no real-world counterpart composed of rival processes), and I realize it's 9 months after the question was asked, but I though I should leave my two cents here since I think my suggestion is accurate, and, while not one word, very short in length (the explanation is quite a bit longer though). So I thought it might still be useful (mainly for others who possibly come across this question sometime in the future).
I believe the phrase that best delineates the relationship you are describing between the incompatible criteria is that they are a "dialectical contradiction". Two opposing and irreconcilable forces are competing against each other (a contradiction in the jargon of Marxist dialectics) and there is no resolution that doesn't either: Negate one of the sides of the contradiction (in this case that would most likely mean ignoring one of the criteria) --- or --- the preconditions are changed, so the problem resolved is a fundamentally different question.
Further explanation (using your first example):
Deliver this physical letter from our location in the US to this address in Cambodia.
It must arrive within 10 minutes.
One side of the contradiction is the criterion that the physical letter must travel a distance that takes longer than 10 minutes to travel. The other side of the contradiction is the criterion that it must arrive within 10 minutes.
There is no resolution to the contradiction unless one side is negated, which is not an option. Each side of the contradiction is equally important. One has no power over the other. The dialectics dictate that this means that nothing changes, nothing ever happens. There is no answer to this problem and there never will be.
Addressing the "preconditions" point (this is paramount to the dialectics):
One of the preconditions is that we only have the current technology available. One seemingly logical way out of this contradiction is to teleport (or deliver some other way that is fast enough) the letter to Cambodia. However, this is not a resolution to this contradiction, it ignores one of the preconditions (the implied precondition that we only have present technology available, since we are in the present), it resolves a problem that wasn't the one posed. The only way that resolution solves the problem is if the preconditions are changed.
Remember the only possible solutions are ones that negate one side of the contradiction. For example, if the letter is sent via fax machine or email via scanner, this negates the side of the contradiction that constitutes the physical letter being delivered a long distance (it ignores the criterion), but we are not allowed to do that with your particular problem.
As you can see, the term "dialectical contradiction" very accurately describes your scenario. Any supposed solution implies either the negation of one side, which is not allowed in this instance,