I'm having a thought that I would like more expert opinion on, as this crosses boundaries between philosophical logic and political science.
My premise is that a political ideology can be represented as (or, I prefer, actually is) a logical system. We know that certain logical systems are undecidable such as first-order predicate logic. Logical systems can have other meta-logical properties as well, such as soundness, consistency, completeness, and their respective negations. So this challenges my understanding of meta-logic as well, which I'm not completely fluent in.
Examples of political ideologies which I would represent as logical systems are things like Marxism, as well as particular systemizations of feminism, scientific racism, etc (Note: The intent isn't to equivocate any of these, just that these are the best examples I can think of that are clear enough to understand formally. I wouldn't consider things like liberalism or conservatism as ideologies, because they aren't based on a system of ideas, but seem to represent a coalition of interests. Valid?). Scientific theories can also be formalized, but differ from ideologies they don't aim to change things politically or culturally (I know that in some quarters this is controversial).
If someone wants a reference, I'll dig for it, but I've recently read that basically the more expressive a logical system is, the more difficult are it's meta-logical properties...I think decidability itself is mentioned here. Another premise to my thinking is that when you analyze a body of work (an ideology in this case) into a logical system, the more expressive that logical system is the more arguments you might find that are valid. For instance, "All men are mortal. Socrates is a man. Therefore, Socrates is mortal." is invalid if analyzed at the level of propositional logic. I assume that this pattern continues the more expressive the logic is, as long as the more expressive logic is a strict superset of it. For instance, second-order logic will find more validities than first-order logic, etc. I beg for some charity in trying to understand my gist here.
So my question is whether this argument makes sense either logically or philosophically. Basically, I see most of the political ideologies that we know of are either logically undecidable, or are on the path to becoming undecidable, as the ideology is refined through iteration, dealing with contradictions (either external or internal) and political opposition by becoming more subtle and nuanced in the ideas and propositions they express. For instance, Marxism itself represents an advanced iteration of intellectualized socialism. I can go to white supremacist websites and find someone with lengthy logical attempts to argue for the truth of their views.
My issue is with the idea I said before that an argument's validity depends on the logical system, and you can often analyze the same argument using more or less expressive logical systems. I'll use the term "correct" to mean that the argument was found valid after the "best" analysis of the natural language argument. So what does it mean for an argument to be "correct" when it requires an undecidable logical system to analyze the statement? Well, it means that we may never be able to prove that the argument is invalid, even if the proponents of the ideology assert the truth and rationality of it in the best conscience?
I also wonder about arguments that require something like second-order logic, or something of similar complexity, that we may say are sound, but are incomplete. So the ideology might be right, but we can never prove it? Or what if the ideology is logically inconsistent, would we ever be able to establish that?
It seems to me, that if there is any basis to this, then it justifies quite a bit of skepticism of a political process based on rational argument and logical debate. Am I right?