In Japanese it’s all a matter of style. Whenever the language is used, a basic choice needs to be made between the formal way of speech, known as 敬体 (keitai, distal style), and the corresponding mode for informal speech, called 常体 (jōtai, direct style).
The basic feature of the distal style is that sentences end in either です (desu), the “be” copula that comes with nouns and adjectives, or ます (masu), the concurring ending for verbs. As for the latter, each and every Japanese verb has a plain form, ending in (r)u, from which a corresponding masu form is directly derivable. So much for the easy part.
The main problem is the relationship between the formal copula desu and its informal version da, which on closer inspection are not so, well, corresponding, at all. As said, these are used mainly when sentences end in nouns or adjectives. What makes them so tricky is that there are actually two types of adjectives in Japanese, and these require different treatment.
The first type are so-called na-adjectives, or 形容動詞 (keiyōdōshi). Copula-wise they function just like nouns, which means that formal and informal speech is indicated by the choice of desu or da, respectively. So when you want to communicate that something is 簡単 (kantan, simple), you’d either say 簡単です (kantan desu), in formal style, or 簡単だ (kantan da), in informal style.
But when it’s the second type of adjectives, so-called i-adjectives or 形容詞 (keiyōshi), the rules change. While desu still functions to mark the formal style, these adjectives won’t take da as a copula in informal style. Instead, for historical reasons we’d better not delve into, they’ll not take anything at all. And so when something is 難しい (muzukashii, difficult), formally you still say 難しいです (muzukashii desu), but informally you’d just say 難しい (muzukashii), period.
While native speakers of Japanese don’t seem to have any trouble with this asymmetry (what else would be the point in being native?), it can be quite nasty for non-natives who, in informal speech mode, will often have to suppress the urge to add a da where there shouldn’t be one. In linguistics, such misapplication of existing patterns — here, the transfer of copula rules for na-adjectives to i-adjectives — is called analogy.
Further contributing to the confusion is that native speakers on their part commonly seem to engage in some kind of analogy, too. However, they do it just the other way round, resulting in a high number of da-omissions where, according to common textbook lore, a da is required. And this works quite well in fact: Just compare the copula-excessive 難しいだ (muzukashii da), which every speaker of standard Japanese will tell you is ungrammatical, with the “de-copulated” 簡単, which, though syntactically incomplete, otherwise doesn’t stir any grammatical discomforts.
Omission of da is triggered even more by an underlying asymmetry in the plain style. Thus, even though both 難しい and 簡単だ are textbook examples of informal speech, the latter clearly feels more assertive than the former. That’s why such instances of “naked” da, as they are sometimes called, are normally avoided, and one way to do so is by withholding the da. In other words, and even though your textbook may say otherwise, the form that seems to correspond closest to the informal 難しい is not 簡単だ but just 簡単, period.
The deletion of da can have most dramatic effects when discourse particles such as よ (yo) or ね (ne) get involved. When attached to da, as in 簡単だよ (kantan da yo), the utterance normally becomes a bit less rough than with naked-da ending. Now when the da is deleted but the particle retained, as in 簡単よ (kantan yo), the result is that the utterance will suddenly sound assertively feminine. This makes the informal copula (and even more so, its omission) also an important gadget in Japanese gendered language.
It may be too early to make predictions, but it seems we are on our way to a more simplified distinction between the formal and the informal speech style, with only two sets of ground rules left: masu vs. (r)u for verbs, and desu vs. nothing for everything else. The informal copula da is certainly here to stay, but it may gradually take over different discourse functions, such as marking soliloquy-like asides, as in そうなんだ (sō nan da, I see) or brute force utterances like 今すぐやめるんだ (ima sugu yameru-n-da, You stop it right now).
Further hints for this development can be found in the speech of young children, who often fail to recognize da as a copula in the first place, for instance when saying 嫌だなの (iya da nano, I hate it), where da is reinterpreted as part of the adjective that precedes it. Or they may exploit the high assertiveness of da and use it as something like a discourse particle in its own right, when saying things like 言わないよーだ (iwanai yō da, I won’t tell you, na na na na boo boo).
Yes, da clearly has this nasty side to it, particularly when granted the important final slot in an utterance. In this respect, it isn’t to be taken as just a stylistic downgrade of its formal cousin, desu, but actually does a lot more than that. So be sure to handle with care, or there’ll be damage done with that darned da.
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