It's raspberries that produce on 2-year-old vines. Blackberries produce on the new vines. Semi-trailing blackberries can be trained on a trellis with upright canes that are clipped to produce sideways runners that produce on 1-year-old vines, that are trained like grapevines.
The erect blackberries you can mow flat, if you want, in the fall, and they will start again each spring and produce on the new vines. Sometimes for disease control people mow them in the fall and burn all of last year's vines.
For what it's worth, my husband has his berry bushes and I have mine. He refuses to cut his, and so be it. He gets plenty of berries, but his are now 8 feet by 8 feet by 8 feet, and the berries only produce on the upper 18 inches. He can't reach the center, he can't reach the backside, he almost can't reach the top, which is now up against a shed. He tries to reach for those and falls into scratchy vines.
Healthy berries will keep putting out shoots over and above the old vines that will produce berries each June and sometimes a second crop in the fall. It's just that a lot of berries are out of reach, the bottom becomes barren and a zone for rabbits and spiders. It's a tangled mess that is cutting off light to new growth, the drippers are no longer in the right place, and it's so sad to lose so many berries because you can't reach them.
If they didn't produce well, then it's their circumstance that needs to be looked at, not whether they are 1 or 2-year-old vines.
Unless there's a kind that's new and I'm forgetting about them....just enjoy the heck out of them!
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Life goes on within you and without you - George Harrison