Well, thanks for such a considered response drug_mentor, I will do my best to respond where I can.
Regardless of whether you accept or reject S5, it was not formulated in order to 'hide illegal operations' or to 'prove ridiculous falsities'.
I don't doubt (or at least, don't doubt the possibility) that the purpose of it's formulation was an intellectually honest one, but intuitively I am not convinced that it cannot be used for the purpose of "proving" ridiculous falsities by an illegal operation which is not actually logical, but which is made to seem so by the current formulation. This is however a position which I accept I may struggle to defend.
Why would you suppose modal logic is less rigorous than algebra if you don't know anything about it? It seems like a fairly groundless assumption; prominent mathematicians like Tarski and Godel have done significant work in modal logic.
Admittedly, I suppose this because of an inherent distrust of the utility of certain areas of philosophical debate, although I concede that this is not likely to be a logically defensible position in any sense and that reference to utility is also problematic. So, I have read a little about modal logic, but it would seem to me still that algebraic mathematics is inherently more rigorous (or at least, more likely to be more rigorous) than modal logic, because algebraic mathematics (it seems to me) is essentially an application of classical logic, as it does not deal in possibilities (except in probabilities, of course). I think that the introduction of possibility into any argument necessarily makes it a less rigorous discussion - or, possibly, only that it is probably less rigorous - because it increases the complexity of that argument.
Well, for the most part a 'proof' in logic is more certain than one in science, since logic uses deductive reasoning and science uses inductive reasoning.*
I think that I accept the truth of this as a general statement, so I concede that my own statement may be at least partly incorrect or incomplete, but...
The steps in the proof are a series of logical operations, it isn't clear why mathematical operations should have a privileged epistemological status over logical ones - especially when you consider that numerous prominent mathematicians and philosophers of mathematics have thought mathematics is reducible to logic. Not so many people think that now, but it is possible to axiomatise arithmetic with second-order logic. (Look up Frege's theorem if you are interested; note that I am not claiming all of mathematics can be derived from second-order logic.)
Ultimately maths and logic are reducible to various sets of axioms, and formulas which follow from said axioms - can you offer some reason for thinking that the mathematical axioms ought to enjoy a privileged epistemological status compared to the logical ones?***
My reason for thinking this is the same as the point I made (or, tried to make) above regarding algebraic mathematics and/or classical logic versus modal logic. To be clear though again, I am not saying that mathematics is always going to be a more reliable measure of truth than modal logic, only that it seems to me more likely to be true - at least, in those areas to which it can be applied. I accept that classical logic and mathematics are far more equivalent in terms of their epistemological privilege, so to speak, even classical logic might be arguably slightly higher on the scale.
I did look up Frege's theorem and honestly I don't understand it but I think that is interesting, and may lend weight to the possibility that higher order logic may be closer to mathematics and classical order logic (in terms of epistemological privilege, likelihood of leading to a true statement, and/or perhaps the trustworthiness of the particular method of application of human reason that is in question... slightly losing track of what I am trying to argue here 8().
Do you have anything other than a vague intuition that modal logic is less rigorous than mathematics to back up your assertion?
To be blunt - at the moment, no.
You seem to be suggesting that logic should only be applied to mathematical arguments.
I don't believe that logic should only be applied to mathematical arguments - in fact quite the opposite. I would like to believe that logic can be applied to practically everything in the vast sphere of human experience, even if in some areas it is not currently clear how best to do this.
I am feeling like ultimately I have used a lot of words to say very little in this post, and concede that you have rightfully called me out on trying to argue something about which I know little to nothing. Admittedly, my original stance is based on a bias towards science and maths over practically every other method of seeking truth - I have generally found this to be a fairly reliable position to hold but obviously acknowledged bias is not in itself a defense of anything. I have since tried to get at least a basic understanding of the differences between different order types of logic and other terms you have used in order that I might construct a somewhat more coherent response, worthy of your own fairly thorough critique of my original argument, and I think that it is likely that I would consider S5 to be too strong to be applied to this particular argument, as you mentioned. Although, to be honest I am not 100% clear what the word "strong" means in this context either.
When I have more time I will look into modal logic further and have another look at your original post when I can (hopefully) understand the notation. It is interesting that modal logic has been used by mathematicians and may be applicable to arithmetic, this does change my view of it somewhat.