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Everyone has agreed upon that Japanese has three sets of characters - hiragana, katakana, and kanji.
Yet the Latin alphabet is considered to be one character set, even though the capitals and lowercase letters differ just as much as the two kana.
Why is this the case?
(1) First of all, Japanese uses 5 separate scripts, not 3. They are:
- Hiragana
- Katakana
- Kanji
- Romaji
- Arabic Numerals
All of these scripts are used frequently in Japanese, so it is not correct to say it only uses 3 scripts.
(2) Secondly, you ask why romaji are not considered to be two separate scripts since they have upper case and lower case. You imply that hiragana/katakana are analogous to upper case/lower case, but this is also incorrect. The upper case and lower case variants of alphabet letters are known as allographs (see here for more details). This means that they are simply variants of the same letter. In contrast, although hiragana and katakana may represent the same phonemic sounds, they have distinctly different characteristics. For example, Hiragana are used to represent verbal inflections whereas Katakana do not. Katakana are used to represent loan words in Japanese, whereas Hiragana are not. In short, there are many important linguistic differences between the two scripts and since they have these different functions, they are not considered to be just variants of the same symbols (as is the case with upper case/lower case).
The question would be difficult to reach an answer because there is no other similar case, to my knowledge, commonly using two distinct set of letters that have equivalent phonetic values outside those bicameral writing systems and Japanese. In both cases, relationship between sets are conventionally and discretely defined, not such that a general ground has been established.
In defense of this traditional idea about hiragana and katakana, I can at least say that they are different in graphical origin. Though they both derive from kanji, hiragana is made from whole-character cursive forms, while katakana taken from partial forms of kanji. All other systems where multiple styles might be mixed, including Latin alphabet, Georgian alphabet and Chinese characters, are historically unilinear i.e. one style developed into another and so on. In this sense, German before WWII might be qualified to be a mixed digraphia in a certain degree, but they share the same graphical origin. A more accurate analogy to hiragana and katakana is letters in Coptic alphabet, which are from Greek and Demotic, two separate descents of Hieroglyph, except that they merged both sets into one rather than keeping two tracks.
Note that although hiragana and katakana have been standardized to be almost parallel in the 19th century like the way they are now, their difference was historically much more substantial. The set of currently called hiragana was far bigger than katakana due to homophonous variants (変体仮名); they had different collection of orthographic ligatures (合略仮名) because of graphical difference; they were used in distinct social settings and not usually mixed together.
