What is lines of inquiry?

Parmenides, answered from the texts and cited to the page.

Parmenides answers

The goddess names them plainly at the outset: the ways of inquiry are those routes alone that reason can conceive.1 There are not many such routes — the poem is insistent on this — but only those defined by the formulae the goddess sets out, and the recognition that they are mutually exclusive and exhaustive is itself the only criterion for determining what is real.2

The first route is defined by the formula that a thing is, and that it is not for not being — this is the journey of persuasion, for persuasion attends on reality.3 To follow it is to deduce consequences from that formula, moving step by step until the character of what genuinely is stands fully revealed.4 The second route is defined by that a thing is not, and that it must needs not be — and this I tell you is a path wholly without report, for you can neither know what is not nor tell of it.5

What has no being cannot be recognized, cannot be named, cannot be the object of any inquiry at all.6 Then there is the third way, the way mortals actually walk: supposing a thing both to be and not to be the same and not the same, their journey turns back upon itself.7 This is not a genuine route of inquiry but the wandering that results when the first two are confused. The poem enumerates all three, then closes off two, so that only one story of the way remains: that it is.8

Sources

  1. those ways of enquiry which are alone conceivable
    The Fragments of Parmenides, pp. 55–57
  2. the recognition that they are mutually exclusive and exhaustive is represented (in opposition to the evidence of the senses) as itself constituting the only criterion (ἔλεγχον, ['test'] fr. 7, 5) for determining what is real
    The Fragments of Parmenides, pp. 20–21
  3. The one, that a thing is, and that it is not for not being, is the journey of persuasion, for persuasion attends on reality
    The Fragments of Parmenides, pp. 55–57
  4. To follow the way is to deduce consequences from the formula by which the way has just been defined
    The Fragments of Parmenides, pp. 292–293
  5. the other, that a thing is not, and that it must needs not be, this I tell you is a path wholly without report, for you can neither know what is not (for it is impossible) nor tell of it
    The Fragments of Parmenides, pp. 55–57
  6. if unity 'is not' in this sense, nothing can relate to it, so that there can be no knowledge, belief or perception of it, and it can have neither definition nor name
    The Fragments of Parmenides, pp. 292–293
  7. mortals stray, two-headed, since bewilderment in their own hearts directs their reason astray... by whom this thing has been supposed to be and not to be the same and not the same, and for all of whom their journey turns back upon itself
    The Fragments of Parmenides, pp. 288–289
  8. the mutually related expressions αἵπερ ὁδοὶ μοῦναι διζήσιός εἰσι νοῆσαι ['those ways of enquiry which are alone conceivable'], fr. 3, 2 and μόνος δ᾽ ἔτι μῦθος ὁδοῖο λείπεται ['only one story of the way is still left'], fr. 8, 1-2
    The Fragments of Parmenides, pp. 287–288