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... Column B is either a "T" or an "F" for true or false. In Column C, I am trying to build a formula that will take each T and convert it to a "1", each F and convert it to a "0" ...
There is an inherent flaw in your logic and depending on what you plan to do with the 1'a and 0's it could have detrimental effects.
The problem as I see it will be totaling the 1'a and 0's since you haven't provided for the eventuality of a blank cell which will likely read a 0 . In any event, here are a couple of possibilities,
=IF(B1="T", 1, IF(B1="F", 0, ""))
... or,
=IF(B1="T",TRUE, IF(B1="F",FALSE, ""))
... or,
=B1="T"
Remember that a boolean value of TRUE evaluates to a 1 and a FALSE evaluates to a 0.