Here's why I do and don't care if you don't:
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Actually, I just wanted an excuse to post some new glamour shots of some of my basses. But since we're here, I've noticed for the past few years that the venerable PJ combination has been slammed for being neither fish nor fowl, and I am here to say that it is, indeed, both fish and fowl.
Let's address the most common complaints:
---It doesn't sound as good when you solo the P pickup vs not having the J on a straight-up P Bass. 100% myth, poppycock, balderdash, and bovine waste products. If a $100 SX PJ with the P soloed can sound the same as a $100 SX P as long as they have the same strings, then there is no reason every bass in the world can't have PJ's that sound the same as a regular P when the P's soloed. I used to have sonic proof of this, but it got lost in a computer crash a few years ago.
---You don't get the best of both worlds with a PJ. OK, so let's assume the sound is so different that you'd rather stick with the J. Fine by me. But the sound is not completely different by any means. I've played plenty of Jazz basses and basses with J pickups in my day and I'm well familiar with what a typical Jazz sounds like. It's close enough for the few times that I go for a J sound that I'm not sweating what little difference there is in a good balanced set of PJ's. Throw pickups in there willy-nilly, sure, it probably won't sound all that great. Just might, though.
And to reverse the situation, a good J with the neck pickup soloed may not sound identical to a good P, but it will do a convincing job as well. I have a preference for a good split P pickup in that position, but I've heard plenty of convincing P sounds out of J's. The first Precisions are single coils, too, and I never heard anyone complain that they don't sound like P's.
---60 cycle hum or whatever frequency it is in your country. Can happen, but only on rare occasions is it an issue you can't do anything about. Many bass and guitar players use single coils with no hum problems all over the world. Granted, I prefer hum-cancelling pickups for both the P and the J, but to say that it can't work because the P's a humbucker and the J is single coil is myth.
So that's why I made PJ's my jam...preferring the split P neck pickup as my main sound, but needing a more scoopy J sound on occasion. And I get it, too, and any complaints that it doesn't sound like a real J are just nitpicking.