πίστις can be translated "faith" or "faithfulness" (Galatians 5:22-23) but in Plato, it's more commonly rendered "belief" although C.D.C. Reeve offers the rendering "folk-wisdom." Plato famously makes a distinction between opinion/εἰκασία (at the lower epistemic level) and belief/folk-wisdom (πίστις) at the higher epistemic level of knowing--the via cognoscendi. Conviction would also be a good way to render πίστις.
In this case, I'm discussing supposed logical movement from epistemic darkness to epistemic light (based on the Greek ἐπιστήμη). Plato seems to argue that some individuals perceive things at the level of opinion (εἰκασία) whereas others apprehend things mentally at the level of πίστις: yet πίστις is not knowledge since knowledge for Plato is likely (maybe) "justified true belief." Knowledge only materializes (according to the Republic) when one begins to reflect on abstract/intelligible objects. Therefore, one only begins progressing towards genuine knowledge when contemplating mathematical objects and subsequently reflecting on the Forms (εἶδος). Unless someone has passed through the various stages of knowing (the epistemic ladder) discussed by Plato, then one is only perceiving shadows or identifies objects via πίστις instead of apprehending those objects by dint of genuine knowledge (i.e., νόησις).
My understanding is that the cave and divided line in Plato's Republic should be read as close parallels. Granted, there are differences between the two analogies, but from what I recall, we should understand progression to be occurring in the divided line as one moves up the epistemic ladder.
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