It means that Cook answered every objection/countered every objection with logical/persuasive arguments.
Student or Learner
Which of the following sentences is the closest in meaning to 'Cook met every objection with sound arguments'?1. With sound arguments Cook dealt with every objection.2. Cook met with (or: experienced) every objection that had sound arguments.or anything else?
I can see the ambiguity that you spotted. You need to use logic to work out whether it means:
What did Cook meet?
He met "objections which had sound arguments".
What did Cook use when he met objections?
He used "sound arguments against" every objection [that/which] he met.
The word usage is important. If the first meaning had been intended, it probably would have said "Cook met many objections, all of which had sound arguments".
Remember - if you don't use correct capitalisation, punctuation and spacing, anything you write will be incorrect.
Yes, we don't "meet" words or phrases or objections in American English. But I see it used enough ("I met this phrase") that I've gathered it's used elsewhere. But that meaning didn't cross my mind for this sentence.
I'm not a teacher, but I write for a living. Please don't ask me about 2nd conditionals, but I'm a safe bet for what reads well in (American) English.