We have already met one way to make a suggestion.

タバコをやめたほうがいいです。
It’s better that you quit smoking.

べき expresses a similar meaning, but is much more forceful—to the point where it is almost an order. While you may hear it in speech it is a considerably hard expression and is more often used in writing.

タバコをやめるべきです。
You should quit smoking.
約束やくそくまもべきだ。
You should keep promises.
あしいたいなら医者いしゃてもらうべきだとおもいます。
If your leg hurts I think you should have the doctor take a look at it.

When used with the verb する it is usually abbreviated.

するべき → すべき

留学りゅうがくべきかどうかについて、おや真剣しんけん相談そうだんします。
Seriously discuss with parents whether or not you should study abroad.

Unlike はず no particle is required to modify a noun.

絶対ぜったいまもべき約束やくそく
A promise you should definitely keep.

べき can also be used to express a likely result, as well.

おどろべき発見はっけん
Surprising discovery.

べき can be changed to the negative by conjugating the auxiliary verb.

くるまがなくても赤信号あかしんごうの時に、わたべきではない
Even if there aren’t any cars you shouldn’t cross the road when the lights are red.
このまち治安ちあんわるいからよるいえ出掛でかけるべきではない
This town isn’t safe and so at night you shouldn’t leave the house.

A more literary way of forming the negative is with べきからず.

イギリスでは高速道路こうそくどうろ時速じそく70マイル以上いじょう運転うんてんべからず
In England you should not drive at over 70mph on the motorways.

Incidentally, べし is the archaic form of べき. You may occasionally see it writing, but it is rarely ever used in speech.

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