Criticisms
Many modern logic systems reject the law of excluded middle, replacing it with the concept of negation as failure. That is, there is a third possibility: the truth of a proposition is unknown. The principle of negation-as-failure is used as a foundation for autoepistemic logic, and is widely used in logic programming. In these systems, the programmer is free to assert the law of excluded middle as a true fact; it is not built-in a priori into these systems.
Mathematicians such as L. E. J. Brouwer and Arend Heyting contested the usefulness of the law of excluded middle in the context of the modern mathematics
Stéphane Lupasco (1900-1988) has also substantiated the logic of the included middle, showing that it constitutes "a true logic, mathematically formalized, multivalent (with three values: A, non-A, and T) and non-contradictory". Quantum mechanics is said to be an exemplar of this logic, through the superposition of "yes" and "no" quantum states; the included middle is also mentioned as one of the three axioms of transdisciplinarity, without which reality cannot be understood.
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