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:

1. Intuitionistic logic

Main article: Intuitionistic logic

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

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

Features:

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

Used in: Type theory, constructive mathematics.

2. Modal logics for constructive reasoning

Main article: Modal companion

Founder(s):

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

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

Further: Modern provability logics build on this.

3. Minimal logic

Main article: Minimal logic

Simpler than intuitionistic logic.

Founder: I. Johansson (1937)

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.

4. Intuitionistic type theory (Martin-Löf type theory)

Main article: Intuitionistic type theory

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 Coq, Agda.

5. Linear logic

Main article: Linear logic

Not strictly intuitionistic, but very constructive.

Founder: J. Girard (1987)

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

Features:

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

Used in: Computer science, concurrency, quantum logic.

6. Other Constructive Systems

• 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.

See also

• Constructivism (philosophy of mathematics)

References

• Brouwer, Luitzen Egbertus Jan (1908). Over de Grondslagen der Wiskunde (in Dutch). N.V. Uitgeversmaatschappij Sijthoff.

• Brouwer, Luitzen Egbertus Jan (1913). "Intuitionism and Formalism" (PDF). Bulletin of the American Mathematical Society. 19 (6): 191–194.

• Girard, Jean-Yves (1987). "Linear logic". Theoretical Computer Science. 50 (1). Elsevier: 1–101. doi:10.1016/0304-3975(87)90045-4.

• Gödel, Kurt (1986) [1933]. "Eine Interpretation des intuitionistischen Aussagenkalkiils". In Feferman, Solomon; Dawson, Jr., John W.; Kleene, Stephen C.; Moore, Gregory H.; Solovay, Robert M.; Van Heijenoort, Jean (eds.). Publications 1929–1936 (PDF). Collected Works. Vol. I. New York: Oxford University Press. ISBN 978-0-19-503964-1. / Paperback: ISBN 978-0-19-514720-9

• Heyting, Arend (1930). Die formalen Regeln der Intuitionistischen Logik (in German). Springer.

• Johansson, Ingebrigt (1937). "Der Minimalkalkül, ein reduzierter intuitionistischer Formalismus". Compositio Mathematica (in German). 4: 119–136.

• Kolmogorov, Andrey (1932). "On the Principle of Excluded Middle". Mathematical Logic Quarterly. 10: 65–74.