Indecomposability (intuitionistic logic)

In intuitionistic logic, indecomposability refers to a property of formulas or, more generally, constructions where certain attempts to break them down or analyze them structurally do not succeed in providing a full understanding of their justification or proof. It highlights a key difference between intuitionistic and classical logic: in intuitionistic logic, a formula is only considered true if we have a constructive proof of it.

The concept often arises in the context of disjunctions and existential quantifications.

• Disjunction (A ∨ B): In classical logic, A ∨ B is true if either A is true or B is true. Intuitionistically, we require a specific proof of either A or B. The truth of A ∨ B is not established merely by knowing that one of them must be true; we must know which one. A disjunction is considered indecomposable, in a sense, if we cannot determine constructively which disjunct holds. We need to be able to exhibit a proof of A, or a proof of B, to assert A ∨ B.

• Existential Quantification (∃x P(x)): Similarly, in classical logic, ∃x P(x) is true if there exists at least one x for which P(x) holds. Intuitionistically, we require a constructive method for finding such an x, and a proof that P(x) holds for that specific x. Knowing that some x exists satisfying P(x) is not enough; we must be able to exhibit such an x and prove P(x) for that x. If we cannot explicitly find such an x, the existential quantification remains, in this sense, indecomposable. We cannot reduce it to the verification of P(c) for some constant c.

Therefore, indecomposability underscores the constructive nature of intuitionistic logic. It emphasizes that to assert a complex statement, we need to be able to provide a concrete construction or proof that demonstrates its truth, rather than relying on abstract or indirect arguments. The focus is on providing an actual method for verifying the truth of a statement, instead of relying on the mere possibility of its truth.