Everyone uses all the functions, it's the ones we favor that determine our types. Additionally, it isn't a balance between F and T that results in making axioms. All types will do this, but it relies heavily on main functions.
Axioms are going to be developed by both primary functions. An SF will probably not "make" as many Axioms as much as they'll submit to already established ones. NFs will create many axioms as they don't have to be based in sensing and all for more whimsical endeavors. An NF may not necessarily make logical axioms as it might satisfy their "self" (F) and they don't need to objectify their axioms.
An ST is going to make the least axioms and accept the already established ones. STs are the ultra practical and crafty people. They like having hands on activities and tinkering, so they want to know how something traditionally works in order to tinker with it, that doesn't necessarily mean inventiveness though. They want practical use and they want things to work. Alternatively, SFs want to satisfy both the practical side and the personal side. They may be crafty in more sentimental ways or put their desire for practical practice and personal achievement into something like medicine. In other words, an SF may scrapbook, but an ST will probably want to fix something. Their axioms are more community established.
In the case of NTs, we have N- which is the theorizing side. I use to say creative, but from a very literal standpoint that's confusing. The process of actually creating something is more of a sensing feature, as they like to interact with things and are known to be crafty.
NTs intake with Intuition, which is the theorizing. So the world around us is interpreted in theories and abstractions. Bu then we have T as our decision making function. This means that we make decisions objectively. Therefore, the creating of systems is very much an NT thing to do.
NTs create axioms by saying, well here is something abstract and interesting, but how can I work this into something rational or practical or useful. If we can make things work in a system we'll use it, if not we may throw it out, though it is more typical of P types to hang onto things, just in case. J types are more likely to find one thing to build on.
I don't think you're example is F/T. I think it's just usual NT stuff. The P expression may confuse us the most, as it makes our decision making vague and difficult.
An NF is going to create axioms based on personal feeling towards it. With a P that makes decision making more difficult, but Fs are more likely not to bother systematizing everything and may rely more heavily on "feeling"- which in this case is actually intuition. This makes NFs the most reliant on intuition.
I think this is often confusing for INTPs, as the combination of NP can be haphazard and create "F" look-a-likes.
NP is the randomness generator. It doesn't get anymore "let's change it up" than NP.
So, what I'm saying (this is getting long), is that F isn't creating axioms. Our input method is (S/N), so N is making abstractions all the time, the combination of P make for the new random things.
A balanced F/T would suggest making axioms on a case by case basic. If the axioms are more often based on the overall picture, the ultimate goal, the objective idea, then that's the T decision making. If the axiom is based on personal understanding of individual situations and adjusting to humane solutions for individuals, that is the F decision making.
Depending on how dramatic the N/S F is, an F may be endlessly forgiving or they might decide getting together with an ex is a lost cause after the third breakup. The same is true for T types. If the N is very dramatic (probably highly P), one exception to the rule may change our understanding of the situations. An ST is more likely to say, "The majority is still this way and needs to change".
If someone was close on the S/N T scale they might be insane. They may have the practical, follow the rules sense or the idea to change the rules and with P it would make them like a ticking time bomb. SPs do occasionally have a "need to escape" and run off to Vegas or do something risky to fulfill their need for spontaneous action.
INTJs don't make as many assertion "public". INTJs tend to analyze a system and how well it works, they may tinker with it, but what what they express is the end result of their work. They express Thinking, their decision making side and this is their secondary function. INTPs express out Intuition and thus we aren't usually decided on what we're actually thinking. It takes us a long time to come to a conclusion. We express intuition, as our secondary function.
Opposite to this is our extroverted counterparts. An ENTP expresses thinking, as their primary function. This makes ENTPs the ultimate debaters as their intuition is working behind the scenes and translating the exterior world for them and their decision making function is expressed. ENTPs will express thinking regardless if they're wrong, as they simply like to debate and they learn from being argued with. Because of how their minds work they're very quick and can probably clean up a mistake in their logic before some people can spot it. Being NPs their axioms can rapidly change, but I think their extroversion gives them more confidence in their conclusions, as they express them.
ENTJs express their Intuition, which makes them socially compatible with INTPs. ENTJs make good leaders because they express their intake function and this makes them open to suggestion and ideas. With their secondary thinking function they take ideas and work them into a plan. ENTJs have axioms like a fill-in the blanks. Due to their extroversion I think they can adjust to different people and situations, but I think the basis for their axioms don't change, even though they appear to accept other people's ideas.
It is in the structure that I understand the formation of axioms.
My axioms, when I'm at my "best", are based on all the likely influences. My T looks for structures or potential for structures (the decision making process is putting things into structures). My N looks for oddities and variations, to consider a larger influence (I desire new input). NP allows for sudden changes.