し合う vs 互いに. What nuance do they add or what is the nuance when one is removed from a sentence?


My understanding of 互いに 'mutually' or 'each other', while the auxiliary verb
合う usually means 'to/with each other'. I have seen sentences with them used
together but also with only one and I'm wondering what the difference is.

For example:

> あの二匹の犬は互いに追いかけ合っています
>
> あの二匹の犬は互いに追いかけています。
>
> あの二匹の犬は追いかけ合っています。

Or:

> 私たちは月に一回互いにメール交換をし合っています。
>
> 私たちは月に一回互いにメール交換をしています。
>
> 私たちは月に一回メール交換し合っています。

Or:

> 彼らは互いに話し合っています。
>
> 彼らは互いに話しています。
>
> 彼らは話し合っています。




From goo辞書:
[互いに](https://dictionary.goo.ne.jp/word/%E4%BA%92%E3%81%84%E3%81%AB/)

> ［副］双方が同じようなことをしあうさま。また、同じような状態にあるさま。「互いに顔を見合わす」「互いに助け合う」

And I think 合う: _"to/with each other."_ is valid here.

Without contexts, it is bit hard to explain though, especially first example
about dogs. Sorry for not being concise.

My interpretation is,

  * あの二匹の犬は互いに追いかけ合っています : 

Two dogs are chasing each other in a way of circling around or back and forth,
etc.

  * あの二匹の犬は互いに追いかけています。:

This one could be two dogs doing the same thing i.e in the same state. So, it
might be two dogs chasing the same thing ex) Toy-Bones for dogs or something
dogs tend to chase. So, it can be dogs are actually not chasing each other.
But it still depends on the context. We know at least two dogs are "chasing".

  * あの二匹の犬は追いかけ合っています。: 

I think this is the same as No.1. But in my perception of the reality, I tend
to imagine this sentence as more static than the first one i.e. Dogs taking a
rest and trying to catch the back of another dog if one has been caught and
start the chasing-game again.

* * *

  * 私たちは月に一回互いにメール交換をし合っています。
  * 私たちは月に一回互いにメール交換をしています。
  * 私たちは月に一回メール交換し合っています。

I think the these three are equivalent in most cases. Using メール交換 makes what
they are doing is more clearer. The last one is bit ambiguous. It can be
exchanging e-mails to other than agents of the subject.

* * *

  * 彼らは互いに話し合っています。
  * 彼らは互いに話しています。
  * 彼らは話し合っています。

I think this one is close to No.1. First one describes one group having a
conversation. Second one is bit ambiguous without the context. They are just
talking to someone else other than themselves. i.e. in the same state. The
last one can be same as first one, but it might be as same as dogs' example.
It might not be dynamically talking to each other such as real-time
conversation, but it may mean "negotiating". So, it can be more static and
used as the concept.

