Syntactic rule for the "⇔" character?

Question

The character "⇔" is used in the definitions in 国語{こくご}辞典{じてん} to specify antonyms. I have questions:

(1) Is "⇔" a formal character in Japanese syntax? Or, is it sort of an ad-hoc way to specify antonyms? At least 2 online 国語辞典 use "⇔" in the same manner, so it looks like a formal part of Japanese syntax.
(2) But, my hardcopy 国語辞典 uses the "↔" character (the filled-in version). So, if both "↔" and "⇔" are formal syntactic characters, they seem to have the same properties.

Both "↔" and "⇔" are really only used to specify antonyms merely as a convention (perhaps because of a historical reason), right?

The characters would never be used in any other context to specify antonyms, or anything else, right?

If there are any good websites that have the rules of Japanese syntax, I'd love to have them. I've never really studied syntax.

Answer 1

These arrows aren't part of the Japanese language, and therefore aren't part of Japanese syntax. They're written symbols, and they aren't used to correspond to any particular spoken utterance.

If you did consider ⇔ or ↔ syntactically, you could call it a unary prefix operator, taking a single operand which follows the operator itself:

⇔はやい

Although to fit with its semantics, you might pretend it's a binary infix operator with the first operand obligatorily deleted, with the missing information recoverable from context:

おそい⇔はやい

But these analyses would only be useful in particular contexts, as in the dictionaries where you found them. As you noticed, these conventions do differ from dictionary to dictionary. Most dictionaries explain how they use special symbols in a section inside the front cover or available online. For example, see the 凡例 for 大辞林. (However, these symbols aren't usually described in syntactic terms.)

Answer 2

1. ↔ ⇔, etc. are not a part of Japanese.
2. Convention.

Turn to the 「凡例」 part of any dictionary and you will find the the notation and convention it uses. Obviously, there are differences among them.

大辞泉・凡例

5. 語義解説の末尾には対義語・対語を↔で示した。
あが・る【上がる／揚がる／挙がる】…↔おりる。
だい‐じょう【大乗】…↔小乗。

6. 参照する項目などについては、語義解説の末尾に → で示した。
あい‐そ【愛想】…→愛嬌(あいきよう)［用法］

7. 語義解説のすべてを別の項目にゆだねるときは、⇨を用いてその見出しを示した。
きゃく‐し【客思】⇨かくし（客思）
こくさい‐ろうどうきかん〔‐ラウドウキクワン〕【国際労働機関】⇨アイ‐エル‐オー（ＩＬＯ）

大辞林・凡例

3. 子見出しとなる慣用句・ことわざなどの句項目は、[句] → の後ろに行を改めて、漢字仮名交じりで示した。
あげ く　[0]【 挙（げ）句・揚（げ）句 】
　　　　［句］ → 挙げ句の果て
あし　[2]【 足・脚 】
　　　　［句］ → 足を洗う

---

(10) 対義語は ↔ を用いて示した。対義語が二つ以上の語義区分に共通する場合は ▽ でまとめて示した。

(11) 参照項目は→で示した。

(12) 解説をすべて他の見出しで行なった場合は、その見出しを⇒ のあとに示した。

新英和大辞典

cf.　参照すべき語句を示す.
⇒　その先の語句に詳しい説明があることを示す.
(←→)　訳語の後で対照語(句)を示す.

故事ことわざ・慣用句辞典 does not use ⇔ or ↔.

Answer 3

Symbols like ⇔ and ↔ are, well, just symbols. They are formal enough that Unicode assigns distinctive code points, but it's certainly not in the same league as ひらがな or かたかな, and I wouldn't consider them to be a part of the syntax of Japanese.

Dictionaries almost always develop their own shorthand notations to pack more information into the same amount of pages, and use of ⇔/↔ as antonyms are just one of them. The use of such notations have to be consistent within a given dictionary, but I don't think there's any universal convention across major dictionaries. So one dictionary using ⇔ while another using ↔ doesn't surprise me. Every dictionary has a legend in the beginning describing what all those shorthands mean, so if you want to learn more about them, just check the first few pages.

I don't think these shorthand notations are unique to Japanese dictionaries, either. I believe English dictionaries refer to nouns simply as "n.", for example.