we have not seen him yet, and we have removed him John: ακόμα δεν τον είδαμε, Γιάννη τον εβγάλαμε.
The actually meaning of the phrase is rather more transparent: "we have not seen him yet, [but] we have [already] named him John". It's said to refer to rushing to conclusions, assessing an outcome before all the facts are in.
But what I have translated as "named" really is literally "removed". Why is naming associated with removing?
What's actually happened of course is that the verb βγάζω is used more generally than "remove". This can be explained through an underlying difference between Greek and English: βγάζω does not mean just to take something out of a set; it also means to put that something into a bigger set. Not just "remove from", but "expose to"; not just "take out from", but "put out to".
English doesn't make that conflation, which is why the GTAGE translation can use remove incorrectly. You can βγάζω εφημερίδα, which is to put out a newspaper (publish it), whereas βγάζω την εφημερίδα with the definite article would be taken as taking out the newspaper (remove it physically). English does allow you to take someone out (e.g. to dinner); but that's always about accompanying someone into public view. The corresponding idiom in Greek is να σε βγάλω στην κοινωνία "to put you out into society"—to make them aware of the ways of the world.
Putting something out means exposing it to the world, which means making it known to the world. And the main way of making something known is by giving it a name for the world to use. The fuller expression for "naming someone" is τού έβγαλα το όνομα Γιάννης, "I put out for him the name John". The point of naming is not that I have something to call John by; it's publishing the name, so that the rest of the world has something to call him by too. The expression has been compacted into the small-clause ditransitive you see, τον έβγαλα Γιάννη "I put him out John"—i.e. "I published him (to be) John", just like "I painted him (to be) white".
The same notion of "putting out" as public knowledge can be seen in:
He/she took them out arseholes: Τους έβγαλε μαλάκες
This is not quite "he named them arseholes" (though it can mean that too); but it is "he put them out (to be) arseholes", "he publicised them (to be) arseholes". The sense here is "proved them (in public) to be"; we dodge the "naming" sense here, because "arseholes" is not a name, it's a description, and descriptions put out in public have to be proved to be accepted.
The same principle motivates the expression μη με βγάλεις ψεύτη, "don't put me out (to be) a liar", "don't prove me a liar": namely, "don't make it a commonly accepted judgement that I am a liar". The expression corresponds to English don't let me down. The premiss is that I have publicly sung your praises as a person able to see something through; you should "not make the public think I was lying", by not living up to those expectations.
So we have names being put out, and descriptions being put out; but putting out the name of someone means something different again:
Eye-removal beats name-removal: Κάλλιο να σου βγει το μάτι παρά το όνομα
Literally, "Better that your eye be put out than your name." The expression is playing with the double meaning of βγάζω, because βγάζω το μάτι σου really does mean removing someone's eye. What does putting out the name of someone mean? It can't be giving someone a name: the definite article implies this is not a newly coined name (it's not: "than that someone makes up a name for you").
This context is more obscure, but "name" here stands in for "reputation": people with a good or bad reputation are said to have a good or bad name (καλό, κακό όνομα). Which makes sense, since the name is the public label for the person, and is what public judgement of the person is attached to. "To have a name put out" means to get a reputation, to have one's name put out in circulation for public judgement. And in closed societies, having one's name put out for public judgement is not normally a good thing.
… Did you work all that out? Or, to quote GTAGE,
Have you removed edge; Έβγαλες άκρη;
Surprisingly perhaps, we're back to "taking out", rather than "putting out". The image is of a complicated shape, standing in for the problem, which the person is trying to work out. Working out the shape means extricating its edges from the jumble: not necessarily removing them physically from the jumble, but distinguishing them—removing them conceptually. The corresponding phrase in English, though with a slightly different image, is get to the bottom of this.
Although getting to the bottom of things is a responsibility not everyone is eager to undertake, if they have to live with the consequences:
Will I extract the snake from the hole; Εγώ θα βγάλω το φίδι από την τρύπα;
For a change, the translation is straightforward. It makes it to GTAGE because it's an image unfamiliar from English, but the image is clear: getting the snake out of the hole is to the public benefit of the farmworkers, otherwise the snake could strike them at any time. But the speaker doesn't want to be the one to have to do it; and the inclusion of the personal pronoun is emphatic. ("Will I have to get the snake out of the hole?")
If you don't live up to your expectations, though,
I don't have face to exit in society: Δεν έχω πρόσωπο να βγω στην κοινωνία
This unintentionally makes sense in English, because English borrowed from Chinese the concept of "face" as public reputation. Greek does not use πρόσωπο in that way as much, but this template of putting one's face out in public as confidence does turn up elsewhere; a variant of the expression is με τι μούτρα θα βγεις την κοινωνία; "with what sort of a face will you go out into society", using the vulgar word for "face".
The verb is not βγάζω "take out, put out", but the related βγαίνω "go out", though with the same kind of ambiguity GTAGE exploits for βγάζω: going out of a place (exiting), vs. going out in public. "Face" here is used as a vehicle of shame: if a person is embarrassed, their face will be blushing, and will give the person's shame away. It will not be a face that one can get away with showing in public, "that one can go out with in society". I'm assuming the concept of blushing is behind the Chinese notion of Face as well.