Well I don't think it should be laggy because 6 gb RAM is actually more then the requirements. But you could try putting all of your settings on the lowest. But if that doesn't work then you will just need to buy a new PC.
Please Give us a full report of your system specs using Dxdiag first off your minecraft will only use 1GB of RAM next everyone should know this Use Optifine for s shake jesus you can fix so many lag probs OK now get Optifine here Once you've installed the mod *it's really easy* next update LWJGL at this page on the wiki Now learn how to give minecraft more RAM *don't give a full 4GB of RAM since you have 6GB Of RAM only give it a max of 3GB* go here for help on learning how to give minecraft more RAM since your using a quad core *man that's pretty crappy for a quad* use multi chunk loading NOTE you can only do this with optifine installed next make everything fast then change the settings every computer is different so you may want to edit every setting to a balance *where minecraft is fast and looks like how you want it to* and also DO NOT Use Frame Caps in Optifine/Unmodded use max FPS and turn off Vsync
minecraft lag help