There are Gödel's incompleteness theorems which applies strong limitations to systems that can describe basic arithmetic.
My experience is that "systems that are full of caveats" tend to arise when people try to prove the unprovable and mangle the language to do it. However, my experience is that these mangled systems full of caveats also tend to be really close to something simple and good, so it's worth seeing what is nearby. To choose your example religions, it might be helpful to un-bundle the Abrahamic religions, and look at the individual groups' beliefs. After all, Westborough Baptists are technically of Abrahamic roots, though there is a strong movement within Christians to isolate them because they don't find Westborough's beliefs to be representative.
So, the existence of caveats and exceptions does not automatically question the validity of a religion. However, there is a strong correlation between such caveats and exceptions and greater issues which may be as unsurmountable as Whitehead and Russel's principia mathematica. That correlation may be a cause for such mistrust
By my readings of Gödel, there are certain behaviors that must appear if a FOL system can describe arithmetic. As a result, any religion (or theory in general), if successfully rendered down to First Order Logc(FOL), must have one of the following characteristics. One can pick one or more, but never zero:
- Incorrect (must be wrong, somewhere)
- Incomplete (must not have an answer, somewhere)
- Unprovable (this is the usual solution for religions, and there's nothing wrong with it)
- Intractable (If the rules are not recursively enumerable, Gödel doesn't apply)
- Illogical (breaks the rules of logic)
- Describes a world which contradicts the basic rules of arithmetic (always an interesting choice)
If the system can only be rendered down to Second Order Logic(SOL), Gödel showed that it is unprovable, which is also on the list. It is also valid for a religion to choose a logic outside of FOL and SOL, but it becomes difficult to use other logics when conversing with non believers because FOL and SOL are the most accepted formal logics out there.
It is very easy to accidentally admit arithmetic, especially if a religion has a statement about its own truthfulness (doing so has a strong tendency to admit a set-theory derived Peano arithmetic). Once this happens, it is hard to tear it out. However, it is very common for people to find "inconsistencies" in their interpretation and seek to resolve them. This creates ballooning caveats and exceptions. I find this often happens when a member of a religion tries to explain something from their text using FOL that was not truly representable in FOL in the first place. In my personal experience, almost ever one of these caveats or exceptions has arisen, not from the original corpus of the religion, but from an interpretation of the religion seeking to apply FOL.
I would like to draw an example from Jewish tradition, not to find fault in them, but merely because they provide some of the clearer examples I have seen of this behavior. Consider Exodus 31:12-17:
And the LORD spoke unto Moses, saying: 'Verily ye shall keep My
sabbaths, for it is a sign between Me and you throughout your
generations, that ye may know that I am the LORD who sanctify you. Ye
shall keep the sabbath therefore, for it is holy unto you; every one
that profaneth it shall surely be put to death; for whosoever doeth
any work (melakha—מְלָאכָה) therein, that soul shall be cut off from
among his people. Six days shall work be done; but on the seventh day
is a sabbath of solemn rest, holy to the LORD; whosoever doeth any
work in the sabbath day, he shall surely be put to death. Wherefore
the children of Israel shall keep the sabbath, to observe the sabbath
throughout their generations, for a perpetual covenant. It is a sign
between Me and the children of Israel for ever; for in six days the
LORD made heaven and earth, and on the seventh day He ceased from work
In the middle is the Hewbrew word, "melakha." The text here is not inconsistent. In fact, to the best of my knowledge, there is no mathematical reason to distrust anything here. However, the rabbis have had to define melakha, which is glossed to English as "work," but very clearly has a more exacting meaning than its gloss would suggest.
Rabbis have spent thousands of years clarifying what melakha means. As the ages go on, they have gotten more and more complicated as they seek to be consistent. If you go to Mi Yodea, you can find exacting discussions as to whether "walking in front of the sensor of an automatic light" qualifies as melakha, because the use of electricity has been declared to be melakha.
I will draw attention to a particularly interesting contrasting opinion. Dan Willard's work circa 2000 regarding self-verifying systems that start from the Universe and are divided down from there (instead of building up from zero and one) are particularly interesting. They circumvent Gödel's theorems entirely by refusing to admit diagonalization. They would form a particularly interesting set of religions which admit arithmetic (very desirable for a math major!). However, I distrust anyone short of a PhD in mathematics to successfully write any religious document which adheres to the fine line Willard walked with his mathematical proofs (it's really easy to accidentally admit diagonalization when you aren't looking).