Analyze Properties of Logical System

Question

Introduction

I'm studying an ancient text which develops a legal system. The text is narrated by a narrator, who quotes earlier opinions of philosophers about the legal system and then tries to find the set of opinions that are free of contradictions. The legal system is derived with deductive logic; there are disagreements as to the axioms of the system, so depending on what the axioms are, a different legal system emerges. Therefore, this legal system is highly discrete, extremely non-subjective, and therefore modelable to an extent.

I'm interested in developing a formal logic system to model both the derivation of the legal system and the legal system itself. I have little background in formal logic but that does not concern me; my use for this logical system is pragmatic. I do not care whether it allows for something like Russel's paradox, or whether some laws in it cannot be derived in closed form. If the system works for all practical situations then it is sufficient.

My idea for the system is that it model logic with statements about set membership and generator functions which generate new statements from existing statements.

Ideas for Formalization

Let me try to explain what I mean. The logical model I have conceived of for formally modeling this legal process is built with the following ideas:

Let

a two-tuple of the form $(x, A)$ denote $x \in A$.

$S(t)$ be a set of statements at $t$

$F(t)$ be a set of functions at $t$

$F_{t}^{A \to B} \subset F$ be a subset of $F$ of functions that map an object of type $A$ to type $B$,

where $A$ and $B$ can be one of: a set of statements ($s$), or set of functions($f)$.

For example, the set of functions mapping a set of statements to a set of functions would be denoted by the symbol $F_t^{s \to f}$

A propagation rule for the set of statements: $$S(t+1) = \bigcup_{i,j} F(t)_j^{s \to s}(\mathcal{P}(S)_i) \bigcup_{i,j} F(t)_j^{f \to s} (\mathcal{P}(F)_i)$$

And a propagation rule for the set of functions: $$F(t+1) = \bigcup_{i,j} F(t)_j^{s \to f}(\mathcal{P}(S)_i) \bigcup_{i,j} F(t)_j^{f \to f}(\mathcal{P}(F)_i)$$

where $\mathcal{P}(S)$ is the power set function.

As an axiom:

The logical system is consistent. That means: for all $t \in \mathbb{N}$ and for an arbitrary object $x$ and a set $A$, there exists no two statements $s_1, s_2 \in S(t)$ such that $s_1 = (x \in A)$ and $s_2 = (x \in A^C)$

Logic Resolution

Now, assume that the narrator of the text is operating at the same level of scope as I am. Generally what will happen is that the narrator will arrive at what seems like a contradiction between two sets of statements and then reconcile the seeming contradiction by doing one of the following:

Showing the two sets of statements derive from different initial axioms
Showing that the system can be reformulated, i.e. certain statements and/or functions redefined, in such a way as to resolve just this inconsistency.

Question

I'm hoping someone can provide some guidance towards how I can analyze this system which I have created. - What field of math would study this system? - How do I approach the study of this system? - What are questions I should be asking myself about the formalized system I just conceived of in order to better understand the implications of the formalization? I need to develop an algorithm for resolving inconsistencies: - With my basic understanding of sets I can develop some rudimentary algorithms, but I want mine to be most generalizable. - Also, where do I learn how to develop these logical algorithms

Clarification

This is to certify that this is project is for my own hobby purposes. It is not related to my formal academic study whatsoever.

EDIT notice: I changed the question to make the question more clear, as I feared it might have been misunderstood by the earlier comments.

(1) Are you actually serious? (2) I think what you call "generator functions" are more commonly called "rules of inferences" And it is possible that when you say "order" you mean "arity" — DanielV, Jan 17 '17 at 13:33
I'm not trying to be rude, but you basically are asking "how can I convert casual human debate into formal logic" (assuming it is even possible) and then say "I have no background in these areas". It would be like if someone came onto the forum and said "I'd like help solving this millenia old physics problem, btw I have no background in physics". If this is for a school project (like grad school) then it sounds like you either bit off way more than a mortal person can chew, or perhaps you could lower your standards to something less than "model these debates computationally". — DanielV, Jan 17 '17 at 14:09
You may already know this, only mentioning because you used the word "debate" and said you have little background in the area: make sure what you actually are interested in is formal logic (the construction of mathematical argument to a level of algorithmic precision) and not rhetoric (the ability to persuade humans of a point). They are practically unrelated fields. — DanielV, Jan 18 '17 at 00:36

score 2 · Accepted Answer · edited Apr 13 '17 at 12:21

Honestly, as DanielV hinted at, if you really want to go through with this project you will need a significant investment of time and energy. I recommend you study (classical) first-order logic first (from any suitable resource such as those mentioned here).

And then after that you may wish to study both set theory and type theory, which are two largely incompatible attempts to formalize mathematics and logic. Set theory is the mainstream choice for modern mathematicians, while type theory is very common among theoretical computer scientists. There is also higher-order logic, which may be closer to what you are looking for, since you want to have higher-order rules that generate rules.

At the same time, if you want to capture natural language reasonably well you are bound to need some way of expressing and analyzing modal expressions, so you should learn about modal logic.

[After Question Edit]

Your edited question now focuses on whether there is a way to analyze formal systems in some sort of meta-system. Certainly most foundational systems of mathematics are capable of doing so, and that is precisely what logicians have been doing. For example the semantic-completeness theorem and negation-incompleteness theorems are proven in the meta-system about rather general formal systems, and usually apply to the meta-system itself!

Furthermore, if you want some kind of self-reflection, you might want to look at provability logic. But note that some key facts about provability logic are non-trivial to prove and will need a proper grasp of ordinary first-order logic.

Just for a taste of provability logic, you can prove: $ \def\prov{\square} $

$\neg \prov \bot \land ( \prov P \to \prov R ) \land ( \prov Q \to \prov \neg R ) \to \neg \prov P \lor \neg \prov Q$

which roughly says that if the system is consistent and from $P$ and $Q$ you can prove contradictory things, then either you cannot prove $P$ or you cannot prove $Q$.

Note that the default interpretation of your claim is different and false. $P$ and $Q$ may well be the same and still one may be able to prove contradictory things from them. Also, note that provability logic only works for finitely many assumptions, which should suffice for legal systems. If you want to handle infinitely many assumptions with some kind of self-reflection then you might wish to extend provability logic.