One of the listed meanings of the 推量の助動詞 「む」 [on
Wikipedia](http://ja.wikipedia.org/wiki/%E5%8A%A9%E5%8B%95%E8%A9%9E_%28%E5%9B%BD%E6%96%87%E6%B3%95%29#.E6.96.87.E8.AA.9E)
is that of 反語, which I understand to mean "saying the opposite of what one
means for rhetorical emphasis" in this context (as opposed to just a literal
"antonym").

I have seen uses of most of the other meanings of 「む」 in my (limited) study of
文語, but have not encountered 「む」 used to mean 反語. What would an example of
this look like?
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classical-japanese subsidiary-verbs
What is an example of how the 推量の助動詞 「む」 can be used to mean 反語?
I edited the answer to make it clearer.

I think む itself does not seem to have this functions (反語). But it is often
used in rhetoric questions, which may make it sound like 反語. Such sentences
often contain か or や. If you check the dictionary, you will find か and や is
said to have this function too.

* * *

One explanation is that it's misleading to say む has the function of 反語
because itself does not have this function. It acquired the 反語 reading when it
appears as a part of 係り結び, that is

```

    NTか　笑はむ　(疑問)  The 疑問 function of か requires む to be in 連体形
    NTかは笑はむ　(反語)　The 反語 function of かは requires む to be in 終止形
    
```

The fact that む's 連体形 and 終止形 are the same leads to the misunderstanding.

However, this explanation has two problems:

  1. Both か and かは require 連体形 when used as 副助詞 and 終止形 when used as 終助詞.

  2. Both か and かは have 疑問 and 疑問 functions.

In addition, む is not required to enable the 反語's reading in this
constructions. No matter む appeare or not, you always need the context to
decide if a sentence is 反語.

**To conclude, it's hard to say that this type of む has the function of 反語.**

* * *

I think **the excessive use of む for 反語** might be an influence of 訓読. Chinese
definitely has more rhetoric questions than Japanese does. When they are read
in the Japanese way, except a few exceptions, **most of them are rendered as
んや**. If you write an article in 漢文調 style, then it will be likely that you
use んや for 反語.

Chinese rhetoric questions often contains certain adverbs, such as 安, 敢, 豈,
etc. Just like 係り結び, when you see these adverbs, the sentence must be ended
with んや. When you see them, the sentence is almost 100% 反語.

* * *

Examples for your reference:

> 觀百獸之見我 **敢** 不走 **乎**  
>  百獸の我を見て **敢て{あえて}** 走らざら **んや** を觀よ
>
> * * *
>
> 燕雀 **安** 知鴻鵠之志 **哉**  
>  燕雀 **安んぞ{いづくんぞ}** 鴻鵠の志を知ら **んや**
>
> * * *
>
> 臣以弑君、 **可** 謂仁 **呼**  
>  臣を以て君を弑す、仁と謂ふ **可{べ}け** **んや**
>
> * * *
>
> 子曰、其然。 **豈** 其然 **乎**  
>  **豈{あに}** 其然しから **んや**
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[This
example](http://kobun.weblio.jp/content/%E5%A6%82%E4%BD%95%E3%81%AF%E3%81%9B%E3%82%80)

いかが－は－せ－む 【如何はせむ】

Searched with "反語 む".

If speaker thought there is any way to solve the problem, the usage is
question.

If speaker thought there is no way to solve the problem, the usage is
rhetorical emphasis.
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The content below is an exact copy of [a deleted
answer](https://japanese.stackexchange.com/a/14956/3437) by [user
l'electeur](https://japanese.stackexchange.com/users/4032/l%C3%A9lecteur).

* * *

First, it is highly misleading to state that one of the meanings of 「む」 is
that of [反語]{はんご} because 「む」, all by itself, does not function as such.

It is when 「む」 is combined with particles 「やは」 or 「かは」 directly following the
subject of the sentence that the word combination can form a 反語 expression.
Furthermore, in 反語, the final 「む」 is of the [連体形]{れんたいけい} even though the
[終止形]{しゅうしけい} form is also 「む」. For the advanced learner, this phenomenon is
called 「[係]{かか}り[結]{むす}び」 or 「係り結びの[法則]{ほうそく}」.

Actual examples off the top of my head:

> 推量：「TNや笑はむ。」= "TN would/might laugh." The 「む」 is the 終止形.
>
> 反語：「TNやは笑はむ。」= "Would TN laugh? No, he would not." The 「む」 is the 連体形.
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