hereby / with this
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hereby
Used in formal or written contexts to perform an action through the statement itself. It means something like 'through this letter' or 'with these words.'
Hiermit kündige ich meinen Mietvertrag fristgerecht zum nächstmöglichen Termin.
I hereby terminate my rental contract with proper notice at the earliest possible date.
'Hiermit' (hereby / with this) is a pronominal adverb made up of 'hier' (here) and the preposition 'mit' (with). It literally means 'with this' and replaces the combination 'mit + demonstrative pronoun' (with + demonstrative pronoun).
Das ist ein spezieller Stift. Hiermit kannst du auf Glas schreiben.
This is a special pen. With this, you can write on glass.
In the meaning 'with these words,' 'hiermit' (hereby) is very formal. You find it especially in official letters, contracts, certificates, or formal declarations.
Hiermit bestätigen wir den Eingang Ihrer Bewerbung.
We hereby confirm receipt of your application.
When 'hiermit' (hereby) appears at the beginning of a sentence, the conjugated verb follows directly in second position. The subject then moves behind the verb.
Hiermit widerspreche ich der Rechnung vom 15. Mai.
I hereby object to the invoice dated May 15.
The two meanings can be distinguished well: If 'hiermit' refers to a concrete tool or thing, the literal meaning 'with this' is meant. If it refers to the statement itself, the formal meaning is meant.
Ich habe einen neuen Schrubber. Hiermit wird der Boden schnell sauber.
I have a new scrubbing brush. With this, the floor gets clean quickly.
so that / with it
'damit' and 'hiermit' can both mean that you do something with this thing or by this means. The difference is in how they are used. 'damit' is very general and can refer to something mentioned earlier. It can also express a purpose. 'hiermit' often sounds more formal and refers more strongly to something that is being shown, given, or said right now. In letters or official texts, 'hiermit' is often used when the words themselves perform the action. Learners confuse the words because both can mean 'mit dieser Sache' (with this thing). In everyday language, however, 'damit' usually fits, while in formal statements 'hiermit' often fits.
Hiermit bestätige ich den Termin am Montag.
I hereby confirm the appointment on Monday.
Lisa hat ein neues Messer gekauft und schneidet damit das Gemüse.
Lisa bought a new knife and uses it to cut the vegetables.