Weather permitting, we have arranged to play tennis on Saturday. May I ask why the last word is allowed?

First of all, you should know non-predicate verbs and master them well.

If this hypothesis holds, then you can look down:

The main sentence of this sentence is that we have ... Saturday, in which have arranged is the predicate verb, so it is impossible to have another predicate verb without a conjunction.

Therefore, the form of permit must be the non-predicate verb structure -ed to do.

Weather permitting, the relationship between weather and permission is active, so v-ing structure is used.

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