TITLE: Au revoir, law of excluded middle?
QUESTION [13 upvotes]: In what way and with what utility is the law of excluded middle usually disposed of in intuitionistic type theory and its descendants? I am thinking here of topos theory and its ilk, namely synthetic differential geometry and the use of topoi in algebraic geometry (this is a more palatable restructuring, perhaps), where free use of these "¬⊨P∨¬P" theories is necessarily everywhere--freely utilized at every turn, one might say. But why and how are such theories first formulated, and what do they look like in the purely logical sense?
You will have to forgive me; I began as a student in philosophy (not even that of mathematics), and the law of excluded middle is something that was imbibed with my mother's milk, as it were. This is more of a philosophical issue than a mathematical one, but being the renaissance guys/gals that you all are, I thought that perhaps this could generate some fruitful discussion.

REPLY [2 votes]: A version of Russell's paradox provides a failure of "law of excluded middle". I don't know if this directly answers your question, but it illustrates the constraints on getting this "law" to work.
Define a "logical function" to be one that returns 'yes' or 'no' when given an input. "law of excluded middle" asserts that if the 'yes' inputs are specified then assigning 'no' to everything else defines a logical function. Consider  

$R[f]$:= If( $f$ is a logical function) then not($f[f]$), else 'no')

But $R$ cannot be a logical function, for if it were we would have $R[R]= not(R[R]$. The error must be that "($f$ is a logical function)" is not a logical function. If it is not 'true' then it is 'wrong' in the sense that the reasoning is invalid. Usually this is seen as invalidating "unlimited comprehension" (naive set theory). 
Most mathematics requires excluded-middle arguments, so we interpret this problem as showing inputs have to be restricted for these arguments to work. In a sense the job of set theory is to provide safe contexts for excluded-middle arguments. This also suggests why set theory is complicated: we cannot describe "safe contexts" using logical functions.