That Aristotle also viewed the two major phases of Parmenides' poem as dual accounts of the same entity in different aspects is perhaps most apparent in his characterization of Parmenides, in the course of the discussion at Metaphysics 1.5.986b27–34, as having supposed that “what is is one in account but plural with respect to perception.” Theophrastus likewise seems to have adopted such a line. Alexander of Aphrodisias quotes him as having written the following of Parmenides in the first book of his On the Natural Philosophers:

Coming after this man [sc. Xenophanes], Parmenides of Elea, son of Pyres, went along both paths. For he both declares that the universe is eternal and also attempts to explain the generation of the things that are, though without taking the same view of them both, but supposing that in accordance with truth the universe is one and ungenerated and spherical in shape, while in accordance with the view of the multitude, and with a view to explaining the generation of things as they appear to us, making the principles two, fire and earth, the one as matter and the other as cause and agent (Alex.Aphr. in Metaph. 31.7–16; cf. Simp. in Ph. 25.15–16, D.L. 9.21–2).

Parmenides was the first philosopher rigorously to distinguish what must be, what must not be, and what is but need not be.

In the crucial fragment 2, the goddess says she will describe for Parmenides “which ways of inquiry alone there are for understanding” (fr. 2.2). The common construal of this phrase as tantamount to the only conceivable ways of inquiry has been one of the principal spurs for readings according to which only two, not three, paths feature in the poem, for it is natural to wonder how the goddess can present fragment 2's two paths as the only conceivable paths of inquiry and nonetheless in fragment 6 present still another path, that along which mortals are said to wander. Two-path interpretations respond to this apparent difficulty by identifying the path of mortal inquiry with fragment 2's second path (though implausibly so, as noted above, sect. 2.2). Parmenides' goddess in fact has good reason to distinguish the two ways of inquiry presented in fragment 2 from the way subsequently presented in fragment 6. The two ways of fragment 2, unlike the third way, are marked as ways “for understanding,” that is, for achieving the kind of understanding that contrasts with the “wandering understanding” the goddess later says is characteristic of mortals. The use of the Greek datival infinitive in the phrase, “there are for understanding” (eisi noêsai, fr. 2.2b; cf. Empedocles fr. 3.12 for the identical construction) distinguishes the two ways introduced in this fragment from the one subsequently introduced in fragment 6, as ways for understanding. That the goal is specifically understanding that does not wander becomes clear when she subsequently presents the third way as one followed by “mortals who know nothing” (fr. 6.4), which leads to “wandering understanding” (plagkton nöon, fr. 6.6). Comparison with fr. 8.34–6a's retrospective indication that “understanding” (noêma, to noein), by which is apparently meant trustworthy thought (cf. fr. 8.50), has itself been a major goal of the inquiry suggests that a way for understanding is one along which this goal of attaining trustworthy understanding might be achieved.

The two ways of inquiry that lead to thought that does not wander are: “that [it] is and that [it] is not not to be” (fr. 2.3)—i.e., “that [it] is and that [it] cannot not be”—and “that [it] is not and that [it] must not be” (fr. 2.5). Each verse appears to demarcate a distinct modality or way of being. One might find it natural to call these modalities, respectively, the modality of necessary being and the modality of necessary non-being or impossibility. Parmenides conceives of these modalities as ways of being or ways an entity might be rather than as logical properties. If one respects the organizing metaphor of the ways of inquiry, one can, even at this stage of the goddess' revelation, appreciate what it means for “that [it] is and that [it] cannot not be” to define a way of inquiry. This specification indicates that what Parmenides is looking for is what is and cannot not be—or, more simply, what must be. Pursuing this way of inquiry requires maintaining a constant focus on the modality of the object of his search as he tries to attain a fuller conception of what an entity that is and cannot not be, or that must be, must be like. To remain on this path Parmenides must resolutely reject any conception of the object of his search that proves incompatible with its mode of being, as the goddess reminds him at numerous points.

The goddess tells Parmenides to stick with "what is". The goddess tells Parmenides to avoid "what is not". The one leads to knowledge of the eternal world; the other one leads nowhere, but does show what not to do which is a valid thing to know. The third way is the world of "wandering understanding", the way of ignorant mortals. This is the world that appears to be the case for those who choose to accept what they see in front of their very eyes, the world of "appearances."

" mortals mistakenly suppose that an object of genuine understanding may be subject to the variableness implicit in their conception of it as being and not being the same, and being and not being not the same. This is not to say that the things upon which ordinary humans have exclusively focused their attention, because of their reliance upon sensation, do not exist. It is merely to say that they do not enjoy the mode of necessary being required of an object of unwandering understanding. The imagery in fr. 6.4–7 that paints mortals as wandering blind and helpless portrays them as having failed entirely to realize that there is something that must be that is available for them to apprehend if only they could awaken from their stupor. Even so, the goddess does not say that mortals have no apprehension. Understanding that wanders is still understanding."

"The goddess reveals to Parmenides, however, the possibility of achieving understanding that does not wander or that is stable and unchanging, precisely because its object is and cannot not be (what it is). The third way of inquiry can never lead to this, and thus it is not presented by the goddess as a path of inquiry for understanding. It directs the inquirer's attention to things that are (what they are) only contingently or temporarily: they are and then again are not, or they are a certain way and then again are not that way. The problem with this path is not, as too many interpreters have understood it to be, that nothing exists to be discovered along this way. There are innumerably many things that are (and exist) in the manner specified at fr. 6.8–9a (and fr. 8.40–1). However, since their being is merely contingent, Parmenides thinks there can be no stable apprehension of them, no thoughts about them that remain steadfast and do not wander, and thus no true or reliable conviction. According to Parmenides, genuine conviction cannot be found by focusing one's attention on things that are subject to change. This is why he has the goddess repeatedly characterize the cosmology in the second phase of her revelation as deceptive or untrustworthy. The modal interpretation thus makes it relatively straightforward to understand the presence of the poem's cosmology. It is an account of the principles, origins, and operation of the world's mutable population. It is Parmenides' own account, the best he was able to provide, and one firmly in the tradition of Presocratic cosmology. At the same time, however, Parmenides supposed there was more to the world than all those things that have grown, now are, and will hereafter end (as he describes them in fragment 19). There is also what is (what it is) and cannot not be (what it is)."

My simile is that the world, divided into everything or nothing, is like a water hydrant spewing at full blast or completely turned off.
This is like Parmenides and his investigations of what is and what is not.
Religion is like the "wandering understanding." It is like sticking a straw into the spewing hydrant and the truth that it draws off in the dribs and drabs, it considers "real".