GPWS SYSTEM ACTIVATED JUST AFTER LIFT OFF. ACFT IN CLIMB ATTITUDE AND NORMAL SPEEDS.

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1/88 TKOF RWY 36 DCA TO MSP 3 KT TAILWIND; 25 DEG TKOF; ENG ANTI ICE. TKOF PWR FUEL FLOW BEYOND THE WHITE MARKS LIFT JUST BEYOND RED & WHITE ALTERNATING CENTERLINE LIGHTS. PITCH TO 12 DEGS; V1 127 NORMAL. GEAR UP POSITIVE RATE AND UPON GEAR RETRACTION THE STANDBY GEAR UNSAFE WARNING AND PULL UP LIGHT AND HORN WARNING FROM GPWS. I SCANNED THE INSTRUMENTS AND CONFIRMED PITCH 12 DEGS MARKER BUG +12 (NORMAL) AND POSITIVE RATE ON IVSI AND PWR WAS NORMAL. COPLT CALLED 'YOU'RE CLBING.' I WAS AFRAID TO TURN LEFT IN CASE I WAS ACTUALLY SINKING AND WOULD GO INTO THE BRIDGE. THE NOISE LEVEL WAS VERY DISCONCERTING AND I HAD DIFFICULTY ASSESSING WHAT THE PROB WAS. I CONTINUED STRAIGHT AHEAD BUT REALIZED THE MONUMENT WAS OUT THERE SO I STARTED A LEFT TURN AND AT 400' CALLED FOR FLAPS 15 DEGS. AS SOON AS THE FLAPS RETRACTED FROM 25 TO 15 DEGS; ALL WARNINGS CEASED. ESTABLISHED ACFT IN 328 DEG RADIAL DCA AND POPPED OUT OF OVCST AT 700' AND BREATHED A SIGH OF RELIEF. SUSPECT AIR/GND SQUAT SWITCH MALFUNCTION BECAUSE SENSOR IN THE AIR THOUGHT LNDG FLAPS WERE DOWN. SHOULD HAVE RECEIVED TKOF WARNING ON GND IF IT SENSED LAND FLAPS. STEADY GEAR WARNING IN AIR SENSED LAND FLAPS. NO DES GEAR UP TKOF SO GPWS SENSED AND NO GEAR. INITIALLY FELT BAD THAT I DIDN'T REACT QUICK ENOUGH (ADD PWR AND PITCH UP); BUT LATER REALIZED I FLEW THE ACFT UNTIL WE COULD IDENTIFY THE PROB. THE PROB WENT AWAY BEFORE WE FIGURED IT OUT. NOTIFIED MAINT; DISPATCH AND FLT STANDARD PLT NARRATIVE TO CHIEF PLT ALSO PULL FLT RECORDER. CALLBACK CONVERSATION WITH RPTR REVEALED THE FOLLOWING: ACFT HAD NOT BEEN DEICED. WX CONDITIONS WERE OVCST WITH RESTRICTION VIS DUE TO FOG BUT NO RAIN OR SNOW. ALL RWYS AND TXWY CLR. MAINT ADVISED THE GPWS COMPUTER WAS CHANGED AND THAT SEEMED TO SOLVE THE PROB. RPTR STILL FEELS HE SHOULD HAVE GOTTEN A TKOF WARNING HORN AND IS NOT CONVINCED THE PROB WAS SOLVED. PRESS SYS WAS NORMAL.