A320 FLT CREW INBOUND TO CMH EXPERIENCED TCASII RA WITH VFR TFC AFTER CTLR ASSUMED RPTR WAS PROVIDING POSITIVE SEPARATION.

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CMH WAS VFR WITH A SCATTERED TO BROKEN LAYER AROUND 6000 FT AND HAZE. I WAS PF WITH AUTOPLT AND AUTOTHROTTLES OFF. CLRNC WAS HDG 360 DEGS; DSND AND MAINTAIN 6000 FT. DURING DSCNT; A TA WAS ISSUED FOR VFR TFC AT OUR 11 O'CLOCK POS. WE ACQUIRED THE TFC ON TCASII AND PNF RESPONDED TO ATC; 'WE HAVE HIM ON TCASII.' SOMEWHERE AROUND 7000 FT IN THE DSCNT; TCASII UPGRADED TO A TA. EVEN THOUGH I WAS HIGH FOR THE APCH; REDUCED MY DSCNT RATE TO AVOID AN RA. WE WERE IN IMC DSNDING THROUGH THE HAZE AND CLOUD LAYER. WE NEVER VISUALLY ACQUIRED THE VFR TFC; BUT CONTINUED TO WATCH HIM ON TCASII. THE TFC WAS CONVERGING WITH US BOTH IN THE HORIZ AND VERT SENSE; SO I LEVELED OFF AT 6200 FT AND PREPARED MENTALLY FOR AN RA. JUST AFTER LEVELING OFF; TCASII UPGRADED TO A CLB RA. I SELECTED TOGA PWR AND PITCHED UP TO FOLLOW GUIDANCE. PNF ADVISED ATC OF OUR TCASII CLB AND GOT NO RESPONSE FROM ATC. I THEN PULLED THE PWR BACK TO MCT SINCE IT DIDN'T LOOK LIKE WE NEEDED ALL THE PWR. ALMOST INSTANTLY; TCASII UPGRADED AGAIN TO INCREASE CLB. I WENT BACK TO TOGA AND INCREASED CLB RATE. WE WERE CLR OF CONFLICT AT AROUND 6800 FT. ATC THEN GAVE US FURTHER CLRNC TO DSND AND TO INTERCEPT THE LOC RWY 10L. FLT LANDED WITHOUT FURTHER INCIDENT. ON THE GND; I EXPLAINED (VIA PHONE) WHAT HAPPENED TO THE CMH TRACON SUPVR. I WAS UPSET AT THE RA AND THE FACT THAT THE CTLR NEVER ACKNOWLEDGED OUR TCASII CLB. I THEN SPOKE WITH THE CTLR HIMSELF REGARDING THE TFC. HE SAID; 'BUT YOU SAID YOU HAD THE TFC ON TCASII?!' I EXPLAINED TO HIM THAT HAVING THE TFC ON TCASII IS NOT THE SAME AS CALLING TFC IN SIGHT. I THINK HE THOUGHT HE WAS NOT RESPONSIBLE FOR SEPARATION BY US STATING WE HAD THE TFC ON TCASII. THE TFC WAS A LIGHT VFR ACFT AND WAS NOT IN CONTACT WITH CMH APCH. HE EVIDENTLY OVERSHOT HIS TARGET ALT OF 5500 FT. IF HE WAS AT 5500 FT OR HIGHER; THERE WAS NO WAY HE WAS 500 FT BELOW THE CLOUD LAYER WE WERE IN. THE PNF NOTICED THE ACFT DSNDING (ON TCASII) AFTER WE STARTED OUR CLB. WE NEVER VISUALLY ACQUIRED THE TFC. I THINK THE VFR ACFT BUSTED VFR CLOUD SEPARATION MINIMA AND CAUSED OUR TCASII RA. ADDING TO THE PROB WAS THE FACT THAT; IN MY OPINION; THE CTLR WAS NO LONGER PROVIDING THE SEPARATION THAT HE SHOULD HAVE BEEN.