Εἶτα, ἦν δ᾿ ἐγώ, οὐ καλὸν ἄρτι ὡμολόγεις τὴν σωφροσύνην εἶναι;
Πάνυ γ᾿, ἔφη.
Οὐκοῦν καὶ ἀγαθοὶ ἄνδρες οἱ σώφρονες;
Ναί.
Ἆρ᾿ οὖν ἂν εἴη ἀγαθόν, ὃ μὴ ἀγαθοὺς ἀπεργάζεται;
Οὐ δῆτα.
Οὐ μόνον οὖν ἄρα καλόν, ἀλλὰ καὶ ἀγαθόν ἐστιν.
Ἔμοιγε δοκεῖ.
For the logical conclusion that he draws from this to make sense, I would have expected something like “Is there anything that is not good that produces good men?” which I guess would be Ἆρ᾿ οὖν ἂν εἴη τι οὐκ ἀγαθόν, ὃ ἀγαθοὺς ἀπεργάζεται;
As far as I can see, the logic flows smoothly. Something has to be ἄγαθος to produce τὸ ἄγαθον, right? If σωφροσύνη is the quality of a good person, then it must be ἀγάθη as well as καλή. Is there good which does not produce good people?
It’s just bad logic. Ex.:
Nothing that has not given birth can be a mother rabbit.
Everything that has given birth is a bunny.
Example of correct logic logic:
A: No ἀγαθοί are produced by something not ἀγαθόν
B: Everything that produces ἀγαθοί is an ἀγαθόν
Unlike the first, A does imply B. I just have trouble seeing Plato making a mistake like this. He does ignore precise logic often enough, but usually more artfully. And maybe even more out of character, it means the dialogue doesn’t quite flow smoothly at this point, if you’re paying close attention. I think something has gone wrong here.
Okay, what am I missing here? How are your good examples different from the enthymeme Ἆρ᾿ οὖν ἂν εἴη ἀγαθόν, ὃ μὴ ἀγαθοὺς ἀπεργάζεται; I read that as “Therefore is there any good which does not produce good people?”
Well, let me put my math hat on for a few moments.
Socrates: Is there any good which does not produce good people? Charmides: No way.
Written as a conditional statement, this becomes “if something does not produce good people then it is not a good”. The contrapositive of a true statement is always true, so “if something is a good, then it produces good people.”
But we don’t want that. What we want is the inverse of the original statement: “only goods produce good people.” More formally: “if something produces good people then it is a good.”
Unfortunately, logic implies nothing about the inverses of conditionals. Only the contrapositive of a statement can be relied on as true.
The logical fallacy that we’re looking at even has a name: “Assuming the inverse.” Some people (but not Plato) just can’t see this stuff. For example look at these two statements:
“If it’s raining, then Sam will meet Jack at the movies”
“If it’s not raining, then Sam will not meet Jack at the movies.”
A good chunk of the population thinks those two statements are logically equivalent. But they are not at all equivalent.
Now I get it! Thanks. I didn’t want accidentally to overthink something and ruin my reputation. I think the statement works well enough for the actual argument Socrates is represented as making, but you do have a point.