ACR X FALSE GPWS WARNING.

Narrative

THE COPLT WAS FLYING THE ACFT. OUR RTE WAS SEATTLE TO SAN JOSE. THE WX IN THE BAY AREA WAS SCATTERED TO BROKEN CLOUDS; WITH WINDS FROM THE S. WE WOULD LAND ON RWY 12 AT SAN JOSE. WE WERE CLRED FOR THE BRINY 1 ARR. SHORTLY AFTER COMMENCING THE ARR; WE WERE CLRED DIRECT TO WOODSIDE VOR. WE WERE ALSO GIVEN A DSCNT. FROM WOODSIDE VOR WE WERE CLRED TO FLY THE 130 DEG RADIAL AND WERE CLRED TO 4500 FT MSL. I DON'T RECALL THE DISTANCE FROM WOODSIDE WHEN WE WERE GIVEN A HEADING THAT POINTED US BASICALLY AT SAN JOSE ARPT. WE WERE ASKED TO RPT THE ARPT IN SIGHT FOR A VISUAL APCH. THERE WERE CLOUDS BELOW US AND BTWN US AND THE ARPT. WHILE IN CLEAN LEVEL FLT; THE GPWS ACTIVATED WITH 'TERRAIN; TERRAIN.' THE COPLT COMMENCED AN IMMEDIATE CLB. SHORTLY THEREAFTER 'WHOOP; WHOOP; PULL UP' SOUNDED. WE LEVELED OFF AT 5500 FT AND RPTED THE INCIDENT. THE CTLR INFORMED US 'THE MINIMUM VECTORING ALT IN THAT AREA IS 4500 FT.' ABOUT THIS TIME; WE RPTED THE ARPT IN SIGHT AND WERE CLRED FOR A VISUAL APCH. NO FURTHER CONTACT WITH BAY APCH TOOK PLACE. WE COULDN'T SEE THE GND BELOW US WHEN GPWS ACTIVATED; AND I DON'T KNOW THE EXACT ELEVATION OF THE TERRAIN BELOW.