Welcome to the Reasonable Faith Forum.
Properly Basic Beliefs (PBB) are claims of knowledge most notably made by simple and single lines of inference. Multiple lines of inference would be "advanced" or "complex" or "systematic" beliefs. PBBs are often derived from First Principles or Axioms, such that Inner World Skepticism (doubting everything you know) can be defeated. The most famous example of a PBB is "I think therefore I am" (Descartes, Meditations on First Philosophy) which is the same thing as saying: subjective thoughts are real in the sense that we own our cognitive agency. By agency I mean "the driver of the car" such that your Mind is the command and control headquarters of everything you experience, think, do, know, or say.
Just about all of philosophy can be broken down into three categories: Nominalism (defining the meaning of terms), Epistemology (the process of obtaining knowledge), and Ethics (prescribing a way to live life).
So one aspect of PBB is the role your conscience plays as the governor of your Moral Agency. For example, feeling guilt over an action could be defined as a PBB because you know for certain that you feel guilt. Throwing a tennis ball and watching it bounce of a wall reinforces the PBB that "tennis balls can bounce." There are countless examples, but the best way to think about it is to use very short sentences or phrases to describe reality.
Often, philosophers like to use syllogisms to deduce PBB, in the form of Premises + Conclusions. For example:
Premise 1 -- I see sunlight illuminating the sky.
Premise 2 -- If there is daylight, then the sun must exist.
Conclusion -- The sun exists, because I see daylight.
Often, the Skeptic will point out that it is possible you could be the victim of an elaborate hoax (somebody could have set up a bunch of really bright lights all around your house), or you're hallucinating/dreaming, or you're senses are distorted, and on and on and on ad infinitum. I think there is a time and place for doubts, but sometimes Skeptics can go way overboard, by suggesting that we could be living in The Matrix. Skepticism born out of speculation should be skewered in my opinion, because it doesn't help one get to PBB. The job of the positive philosopher is to defeat skepticism, not use it in vain to doubt everything into absurdity. In my opinion, if you want to be a good philosopher, all you need to do is ask three questions over and over:
1. What do you mean by that?
2. How do you know this?
3. So what?
In my experience, Skepticism has a hard time overcoming #3. So what if we live in The Matrix? What are the implications? It's kind of silly when you take a step back from all of it, and come back to the real world that has more pressing demands, but philosophy without some sort of a goal is foolishness.
In terms of the Inner Witness of the Holy Spirit (IWHS) this would be something like moments of clarity, insight, or some kind of epiphany over a truth claim. It's difficult to get into without sounding pseudo-scientific, but the reality is, the IWHS is pseudo-scientific and paranormal. For instance if you said something mean to a friend and after the fact didn't really care, but later on you feel this overwhelming sense to repent and seek out reconciliation, this would be IWHS for sure. Try talking to people who have had an "altar call" experience and ask them about what they were thinking and feeling moments before they decided to go up. Chances are most of them will say things that a very irrational, such as "I felt this overwhelming sense that I was a sinner, and I needed the grace of Christ."
My #1 suggested reading for you is Proper Confidence by Leslie Newbigin. It's a very quick read (100+ pages). It's ironic in that it was written in 1995, at the dawn of the New Atheism movement, but it's remarkable how relevant it is when it comes to Christianity and Modernity. Newbigin gets to the heart and core issues of how and why Western civilization has been completely consumed by the Enlightenment to the point that we don't know how to think any differently - especially when it comes to faith.