Constructive logic

Constructive logic is a family of logics where proofs must be constructive (i.e., proving something means one must build or exhibit it, not just argue it “must exist” abstractly). No “non-constructive” proofs are allowed (like the classic proof by contradiction without a witness).

The main constructive logics are the following:

Founder: L. E. J. Brouwer (1908, philosophy)¹² formalized by A. Heyting (1930)³ and A. N. Kolmogorov (1932)⁴

Key Idea: Truth = having a proof. One cannot assert “${\displaystyle P}$ or not ${\displaystyle P}$” unless one can prove ${\displaystyle P}$ or prove ${\displaystyle \neg P}$.

Features:

• No law of excluded middle (${\displaystyle P\lor \neg P}$ is not generally valid).
• No double negation elimination (${\displaystyle \neg \neg \ P\to P}$ is not valid generally).
• Implication is constructive: a proof of ${\displaystyle P\to Q}$ is a method turning any proof of P into a proof of Q.

Used in: type theory, constructive mathematics.

Founder(s):

• K F. Gödel (1933) showed that intuitionistic logic can be embedded into modal logic S4.⁵
• (other systems)

Interpretation (Gödel): ${\displaystyle \Box P}$ means “${\displaystyle P}$ is provable” (or “necessarily ${\displaystyle P}$” in the proof sense).

Further: Modern provability logics build on this.

Simpler than intuitionistic logic.

Founder: I. Johansson (1937)⁶

Key Idea: Like intuitionistic logic but without assuming the principle of explosion (ex falso quodlibet, “from falsehood, anything follows”).

Features:

• Doesn’t automatically infer any proposition from a contradiction.

Used for: Studying logics without commitment to contradictions blowing up the system.

Founder: P. E. R. Martin-Löf (1970s)

Key Idea: Types = propositions, terms = proofs (this is the Curry–Howard correspondence).

Features:

• Every proof is a program (and vice versa).
• Very strict — everything must be directly constructible.

Used in: Proof assistants like Rocq, Agda.

Not strictly intuitionistic, but very constructive.

Founder: J. Girard (1987)⁷

Key Idea: Resource sensitivity — one can only use an assumption once unless one specifically says it can be reused.

Features:

• Tracks “how many times” one can use a proof.
• Splits conjunction/disjunction into multiple types (e.g., additive vs. multiplicative).

Used in: Computer science, concurrency, quantum logic.

• Constructive set theory (e.g., CZF — Constructive Zermelo–Fraenkel set theory): Builds sets constructively.

• Realizability Theory: Ties constructive logic to computability — proofs correspond to algorithms.

• Topos Logic: Internal logics of topoi (generalized spaces) are intuitionistic.