Putative counterexamples to the law of excluded middle include the liar paradox or Quine's paradox. Certain resolutions of these paradoxes, particularly Graham Priest's dialetheism as formalised in LP, have the law of excluded middle as a theorem, but resolve out the Liar as both true and false. In this way, the law of excluded middle is true, but because truth itself, and therefore disjunction, is not exclusive, it says next to nothing if one of the disjuncts is paradoxical, or both true and false.
PuTATive counTereXaMPles to the laW of exClUded MIddLe INclude the LiAR paRaDoX Or quIne's paradoX. ceRtaIn ResolutionS of these paRaDoxEs, partiCULaRly gRaham priest'S dialetheiSm as ForMalisEd in Lp, Have thE law oF EXcLuded middlE aS a thEorEm, but resoLVe out the liaR As botH true aNd FALSe. in this way, the law of excluDeD middle iS true, buT BEcaUse tRUtH iTSelf, and theReForE DIsjunctIOn, iS nOt exclusivE, it says nexT To nothiNg If one of thE dISjuncTs is ParadoxIcaL, Or bOth TrUE anD fAlSe.