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What notions underlie 'off, away' and 'because of'?

ἀπό - Wiktionary

Etymology

From Proto-Indo-European *h₂epó (“off, away”). Preposition

ᾰ̓πό (apó) (governs the genitive)

  1. from, away from
  2. because of, as a result of

The Online Liddell-Scott-Jones Greek-English Lexicon

III. OF ORIGIN, CAUSE, etc.:

6. of the cause, means, or occasion from, by, or because of which a thing is done

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    Could you clarify what you are looking for? It's not incredibly difficult to formulate a backwards-looking link between the two meanings (e.g. "A cause is that from which an effect proceeds."), but all this needs to be taken with a grain of salt: there isn't some kind of a priori necessity that language develops in a certain way. I'm not sure what deeper reason you're looking for in this and similar questions. – brianpck Jan 8 at 5:20
  • @brianpck I know there's no a priori necessity that language develops in a certain way. I'm seeking surmises like yours, as in (e.g. "A cause is that from which an effect proceeds." – AYX.CLDR Jan 8 at 5:22
  • Of course, Latin ex can also have both sets of meanings, and the logic is presumably similar to what brianpck has given for apo: an effect arises out of a cause, etc.. – cnread Jan 8 at 19:51
  • @brainpck the theme of these questions is clear: the same rigor as for the phonetic derivation needs to be applied to the semantic side of things. Except that semantic change is less likely total. Therefore, all the more weight lies on specific examples, cognates, typological arguments or any other systematic approach to establish plausibility, because, indeed, it's often too easy to fabulate about semantics. – vectory Jan 10 at 8:00
  • compare po#Albanian "yes; used ... to show a continuous action.", "From Proto-Albanian *apā, from Proto-Indo-European *e-, *ē- (“then, at that time”). Compare German ob (“if, whether”), Dutch of (“or, whether, but”), English if.", see po#Lithuanian < *h2po- further down, for a sense "for" is implied in #ido < Ru po, ymmv. – vectory Jan 10 at 8:13
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As pointed out by brianpck and Shootforthemoon above, there is a general explanation for your apparently specific question on Ancient Greek.

To put it in cognitive linguistic terms (e.g., in Lakoff & Johnson's (1980, 1999) famous theory of conceptual metaphors), the explanation of the polysemy involved in your question is explained by the following conceptual metaphor: CAUSES ARE SOURCES (i.e., the abstract cognitive domain of CAUSES is {understood/expressed} in terms of the more physical domain of SOURCES).

Some examples and comments from Lakoff & Johnson (1999: 213) follow: "She got rich from her investments. He got a sore arm from pitching too many innings. Harry died from pneumonia. I'm tired from working all day (...). In the most basic kind of causation, a physical force is applied to move something or change its appearance. In such cases, typically whatever exerts the force must move from an initial source location to a position in which force is exerted. In such a situation, there is a correlation between the application of the causal force and motion from an initial location. This correlation is the basis of a metaphor for causation in which causes are conceptualized as sources and the word from expresses source".

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  • Following the concept of conceptual metaphors I call it a huge pile of [insert metaphor] that ignores possible explanations through internal reconstruction of the particle. Lakoff & Johnson 's is mainly concerned with synchrony, which is a good start, whereas we are interested in diachrony, from synchrony's sake. – vectory Jan 10 at 8:21
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I think the answer is very simple and resides in the same meaning of such preposition as indicating origin and provenience. It literally means "from" and introduces the complement of "moving from", but also all the figurative usages of this complement.

Concretely, someone moves from a place to another.

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Figuratively, an action moves from [the will of] someone towards someone or something else.

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