For example and for instance are synonymous phrases, and both mean "for example".
When you see the for in the two phrases, you can't help but think that the meaning should be "for." But the "for example" language is weird.
For conventional expressions, "examples" are meant as "means", not "ends". In fact, many dictionaries interpret for example and for instance as an example, which is also the evidence of the "means" theory.
In addition, for example and for instance are not followed by articles, while the typical semantics of example and instance are countable nouns.
According to the current dictionary, both phrases are idiomatic expressions and cannot be parsed, but looking at their for, I am still a little haunted, and I still want to look up the information.
A search of information revealed that there is a phrase exemplī grātiā in Latin. The preposition in Latin is not as developed as in English, and there is no preposition for. The last ā of grātiā means "multi-case", which is equivalent to the English for, that is, grātiā is equivalent to for the sake of.
exemplī grātiā means for the sake of an example, which is for example in modern English. It may be abbreviated because the word "for example" is often used and for the sake of an example is very long, so it is shortened to for example.
Note that the abbreviation for for example in English is not Fe., but back to the Latin version of eg.(Initials of exemplī grātiā).
for example is an abbreviation, which does not respect the syntax at all (otherwise example is an article).
for instance is an analogy to the usage of for example.
for example instances are all "sentence adverbials", i.e., modifying sentences (not verbs or adjectives in the sentence).
Let's talk about Sake again.
Sake is derived from the Old English sacu, which can be traced back to the Proto-Germanic word *sako, which means affair, thing, charge, accusation. Further on is the Proto-Indo-European root *sag, which means"to investigate, seek out。Now the English word seek is also derived from *sag.
The original semantics of sake are mostly replaced by case and cause, and it is rarely used in the singular in modern English. There are phrases that have persisted tenaciously, such as:
1.for the sake of somebody something
2. for god’s/christ’s/goodness’/he**en’s/pete's sake (for God's sake).
3.for the sake of it
4.for its own sake
5.for the sake of argument