Depends on the child. Have a chat with him and listen carefully to his feelings, motivation and motives, etc. Is he showing off, do you think? Or, is he sufficiently mature and disciplined to organise and handle his daily practice of all 4 instruments without any supervision? If he's self-motivated and self-disciplined, then, I don't see why not since at his age, there usually isn't much school homework.
However, if he is really serious about becoming a musician someday (which some kids are, despite their early age like Daniel Hope, for example) then you will need to think twice/thrice. Junior conservatoire teachers and also music specialist school teachers invariably advised strongly against taking MORE than two instruments.
According to the head of a London JC, taking multiple instruments dilute practice time and therefore the quality (of play) and he added that at grade 8 level, you are expected to excel at your first instrument or so before they allow you to take up a third one. Parents were told in clear terms that taking up 3 istruments is not encouraged. The child must excel in the first one. Later, he elaborated excellence as getting for example, AT LEAST 140 marks, if not more for G8, for example!
Menuhin School is similar in attitude - most of their kids are G5 distinction level at age 8 (although they do look at potential when identifying future musicians) and they DO expect 3 hrs of practice daily on the first instrument so it seems you can kiss goodbye to being a soloist/chamber musician if you have to keep up the practice for all 4. We are looking at 0.5 hr each daily for the other 3. Where is the child's time to stand and stare?
You are his mother so you are the best person to know whether he loves music or breathes it as well.
Some kids do live and breathe music in a manner of speaking. Our daughter has to be prevented at times from over-practice as she plays after we've gone to bed and dreams of being a musician. On the other hand, if he isn't that Daniel Hope type, and he enjoys it and has the necessary innate self-discipline, then, provided you can afford it, why not? Let him have some fun.

He has some years yet before school workload will become an issue.