Select the option that completes the sentence CORRECTLY. We do not accept any…

Concept: A preposition links a noun to the rest of the sentence, and many nouns and verbs take one fixed preposition by convention (collocation). The noun and verb “liability” / “be liable” are conventionally followed by “for” to name the thing one is answerable or responsible for — “responsibility for”, “liable for”, “liability for”.

Application: Here the sentence says the speaker does not accept responsibility for the damage or losses. Inserting the conventional partner gives “We do not accept any liability for damage or losses”, which is grammatical and idiomatic.

Contrast with the other choices:

• “at” marks a point in place or time; “liability at damage” is not an English collocation.

• “with” signals accompaniment or means; “liability with damage” does not express responsibility and is not idiomatic.

• “against” means opposition or protection from; “liability against damage” would suggest acting in opposition to the damage, which is not the intended meaning.

Result: the preposition “for” correctly completes the sentence.