however / certainly
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however
Expresses a limitation or contrast to a previous statement. It is synonymous with 'jedoch' (however) or 'aber' (but).
Ich würde gerne mit ins Kino kommen, allerdings habe ich heute keine Zeit mehr.
I would like to come to the movies, however I don't have any more time today.
The two meanings of 'allerdings' (however / certainly) are easy to recognize from the sentence structure. As a limitation ('jedoch' = however), it connects sentences or parts of sentences. As strong agreement ('selbstverständlich' = of course), it stands alone as an answer.
Ich komme gerne mit, allerdings muss ich um 22 Uhr zu Hause sein.
I'm happy to come along, however I have to be home by 10 p.m.
As a limitation, 'allerdings' (however) can stand in position 1 (then the verb follows) or inside the sentence after the verb (then the word order does not change).
Wir wollten in den Park gehen. Allerdings fing es an zu regnen.
We wanted to go to the park. However, it started to rain.
In its function as a limitation, 'allerdings' (however) is often used in written language or in more formal conversations. It often sounds somewhat more refined than a simple 'aber' (but).
Ihre Bewerbung ist sehr überzeugend, allerdings fehlt Ihr Lebenslauf.
Your application is very convincing; however, your résumé is missing.
but
'aber' and 'allerdings' can both express a limitation or contrast. 'aber' is very neutral and very natural in everyday language. 'allerdings' often sounds a bit more precise or stronger and often appears at the beginning of a sentence or after an initial positive statement. Learners may confuse the words because they function similarly in many sentences. If you simply want to connect a contrast, 'aber' is usually appropriate. If you want to emphasize a limitation in particular, 'allerdings' sounds more natural.
Du kannst den Termin verschieben, allerdings musst du vorher im Büro anrufen.
You can postpone the appointment; however, you have to call the office first.
Im Park ist es schön, aber heute fahren viele Autos an der Straße vorbei.
The park is nice, but today a lot of cars are driving past on the road.