Intuitionistic Proof Rules
Context
Intuitionistic Logic is useful in the wiki as the proof-theoretic counterpart to classical truth-value reasoning. AbuseOfNotation2026 - The Case Against Boolean Logic motivates the topic philosophically, but the reusable technical payload is the constructive reading of proof judgments and connectives.
Formal Statement
An intuitionistic proof judgment has the form
where Gamma is a context of assumptions and P is the proposition constructed from that context.
Representative constructive clauses:
Negation is implication to contradiction:
Unrestricted excluded middle P lor not P and double-negation elimination not not P -> P are not accepted as general theorem schemas unless added as classical principles.
Derivation / Construction
The disjunction rules show the constructive difference most directly. A proof of P or Q is not merely evidence that at least one disjunct is classically true; it must provide either a proof of P or a proof of Q. To use a disjunction, both cases must lead constructively to the same result R.
Implication is similarly constructive: a proof of P -> Q is a transformation that takes any proof of P into a proof of Q. This is the proof-theoretic root of the Curry-Howard correspondence, where propositions act like types and proofs act like programs.
Implications
- Constructive proofs can carry algorithmic content, which is why intuitionistic logic appears naturally in proof assistants and type theory.
- Classical reasoning can be recovered by adding excluded middle or an equivalent principle, but doing so may erase constructive information.
- The technical issue is not vague “context dependence” alone; it is the stricter requirement that logical claims be supported by constructions available from a specified assumption context.