LTT CARGO FLT ISSUED VISUAL APCH CLRNC; BUT ENTERS FOG ON APCH.

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IT WAS THE END OF A LONG NIGHT OF FLYING AND THE LAST LEG OF THE WORK WK. BOS WAS FOGGED IN WITH HVY SEA FOG MOST OF THE MORNING. THIS WAS OUR 3RD ATTEMPT TO GET INTO BOS; HAVING ALREADY DIVERTED TWICE FOR FUEL AND TO WAIT FOR THE WX TO IMPROVE. FINALLY THE WX IMPROVED TO VFR. SO IT SHOULD HAVE BEEN AN EASY APCH. IN FACT; THE WX WAS CLR WHEN WE FLEW OVER THE ARPT ON DOWNWIND. ALL OF THE FOG WAS OVER THE BAY. APCH ASKED IF WE HAD THE ARPT. I WAS OPERATING THE RADIOS AND REPLIED THAT WE 'HAD HAD THE ARPT IN SIGHT AND SHOULD GET BACK IN SIGHT ON FINAL AND THAT WE ALSO HAD THE WDB (THAT WE WERE FOLLOWING) IN 'SIGHT.' APCH CLRED US FOR THE VISUAL TO RWY 4R AND TOLD US TO CONTACT TWR. TWR CLRED US TO LAND AND ASKED IF WE HAD THE RWY. I ADVISED THEM THAT 'WE DID HAVE IT; BUT WOULD NEED THE ILS.' THE CAPT WAS FLYING THE ILS WHICH WAS SET UP AS STANDARD PROC WHICH WAS GOOD; FOR THE SEA FOG MOVED BACK OVER THE ARPT. THE WDB ENDED UP MISSING THE APCH AND WE HAD TO SHOOT THE APCH TO MINIMUMS BEFORE GETTING THE RWY IN SIGHT. THE END RESULT WAS WE LANDED SAFELY; BUT SOME ERRORS WERE MADE THAT COULD HAVE AFFECTED THE SAFETY OF THE FLT. THE FIRST AND FOREMOST BEING THE MISUNDERSTOOD AND NONSTANDARD COM. THE SECOND BEING POOR CREW COM AND SLOPPY PROCS. ALL OF THESE ITEMS CAN BE ATTRIBUTED TO OUR BEING EXHAUSTED. WE WERE COMING UP ON OUR 14TH HR OF DUTY AFTER A LONG STRESSFUL WK. THIS RESULTED IN A 'GET HOMEITIS' MIND SET. NOT A GOOD THING WHEN FLYING.