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Why does Dummett’s antirealism raise questions about the law of excluded middle?

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The Greek philosopher Plato — Realism Plato, marble portrait bust copied from an original of the 4th century bce; in the Capitoline Museums, Rome.

Michael Dummett’s antirealism challenges the law of excluded middle by rejecting the realist assumption that statements are true or false independently of our ability to recognize them as such.12

In a realist framework, the law of excluded middle holds that for any proposition p, either p or its negation, not-p, is true, regardless of whether we have evidence for either. Dummett, however, argues that the meaning of a statement should be explicated by its "assertibility conditions"—the grounds or proofs that allow us to recognize a statement as true.12

If truth is restricted to what can in principle be known or proved, the law of excluded middle loses its justification. For an arbitrary proposition, there is no guarantee that we possess a method to prove either p or not-p. Consequently, because classical mathematics relies on this principle, Dummett’s position supports revisionary, constructivist approaches such as intuitionism, which call into question large portions of classical mathematical practice.12