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Garrigou-Lagrange, in his work entitled Reality, a synthesis of Thomistic thought (which can be found online), when discussing act and potency according to Aristotle, states that this distinction is necessary to refute Parmenides' argument against the reality of change.

Then, he spells the argument out:

If a thing arrives at existence it comes either from being or from nothing. Now it cannot come from being (statue from existing statue). Still less can it come from nothing. Therefore all becoming is impossible. This argument is based on the principle of contradiction or identity, which Parmenides thus formulates: Being is, non-being is not; you will never get beyond this thought.

Written in this form the argument is completely unintelligible to me. What does it mean to 'come from being' or to 'come from non-being'? What is the meaning of 'to come from' in this case? And does 'being' here denote existence? I.e., when he says 'cannot come from being', does he mean 'cannot come from something that exists'?

Can this argument be put in a more rigorous (terminologically) and clear form?

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Okay in pieces:

If a thing arrives at existence it comes either from being or from nothing.

I don't think this particular bit is too hard to follow. Anything that exists exists either because it can from something else or because it just popped into existence. (if it eternally existed then it never arrived).

If A, then B or N

Now it cannot come from being (statue from existing statue).

Parmenides rejects a principle of change for existing things. He thinks they are always what they are and thus rejects B.

Still less can it come from nothing.

Most of us think it's kind of weird for something to come from nothing. Thus, he rejects N. (I think the only exception is the Christian doctrine from Augustine from creatio ex nihilo).

Therefore all becoming is impossible.

This follows by modus tollens when we've denied both B and N.

1. If A, then B or N          A
2. Not B                      A
3. Not N                      A
4. Not B and Not N            ^I  2,3
5. Not (B or N)               DeM 4
6. Not A                      MT 1,5  

This argument is based on the principle of contradiction or identity, which Parmenides thus formulates: Being is, non-being is not; you will never get beyond this thought.

For Parmenides, being just keeps on being whatever it is, and non-being by definition does not exist.

I'm not quite sure why the author thinks "you will never get beyond this thought." Aristotle has a nice argument and position that is beyond this thought, which is that things can change in multiple ways, they can both enter and leave existence, and they can change modally, like the cheeseburger that becomes a part of my body or the dirt that becomes a part of a tree.

But it still remains nearly axiomatic that "something cannot come from nothing". What's largely denied at this point is Parmenides rejection of B -- because we distinguish between coming into being and matter changing forms. (the former is governed by the law of conservation of matter/energy; the latter is not a problem for chemistry and physics).

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Though I cannot give you a truly good answer, I may be able to add a bit of direction.

First, as you may know, Parmenides survives only through a very few ancient fragments, ancient references, and Plato's narrative. Yet strangely, much of modern physics remains "Parmenidean" in its treatment of spacetime and the mathematical "framing" of motion or change.

Parmenides asks where "change" or "the new" comes from. Presumably nothing can come out of "nothing," ex nihilo. How can something be contained in or deduced from....nothing? Similarly, if there exists some "thing," how can some "other thing" come out of it? You are, I believe, correct to assume that Parmenides means by "being" roughly existence or "is-ness." Some thing "is." Such concepts are so basic, so omnipresent, that the difficulty lies in pulling them into our awareness. Where something "new" comes from seems completely unproblematic to most of us.

But if you accept basic rules of noncontradiction or the excluded middle, etc., then we start off at the most fundamental level with an enormous paradox. Things "exist" or "don't exist." But how do they pass from one to the other? It is well worth reading the opening of Hegel's "Science of Logic" to ponder this problem. And "ponder" is the right word. You don't just "work it out" logically.

The problem then is explaining or defining flux, transformation, or "motion" of any sort. This is hardly as arcane or as settled a matter as some think. It is the whole career of modern science since Galileo. It impinges on calculus and physics and measurement problems at many, many levels. And it is difficult because it is way, way too basic.

The reference to Aristotle's "potentia" brings in one of many intermediating concepts designed to handle this "is/is not" transition problem. We introduce "becoming" or "possibility" or "could" or "dialectic" or "Trinity" or some other acceptable covering term for the officially "excluded middle." We accept that identity is simply unstable.

When you say you have difficulty grasping this, welcome to the club. If you are asking within the context of philosophy studies, I would argue, at the risk of "moralizing," that philosophy is not all logic and computing, "either/or" but also requires sinking into possibly unanswerable questions in language and the living history of human consciousness. Parmenides remains in our texts because he raised a question so fundamental it still perplexes modern physics, while also generating endless answers.

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I think that the argument may be put like this:

  1. The statements

b exists, and

b does not exist,

cannot be both true.

  1. The above is the fact regardless of context, i.e. that the statements

b exists (in some context), and

b does not exist (in another context),

cannot be both true.

  1. To state that b "arrives at existence" entails that

b does not exist (at some time), and

b exists (at another time)

are both true.

Which contradicts (2.)

  1. Therefore, a thing cannot "arrive at existence"

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