Proof-theoretic in the context of Data structures

Proof-theoretic in the context of Data structures

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⭐ Core Definition: Proof-theoretic

Proof theory is a major branch of mathematical logic and theoretical computer science within which proofs are treated as formal mathematical objects, facilitating their analysis by mathematical techniques. Proofs are typically presented as inductively defined data structures such as lists, boxed lists, or trees, which are constructed according to the axioms and rules of inference of a given logical system. Consequently, proof theory is syntactic in nature, in contrast to model theory, which is semantic in nature.

Some of the major areas of proof theory include structural proof theory, ordinal analysis, provability logic, proof-theoretic semantics, reverse mathematics, proof mining, automated theorem proving, and proof complexity. Much research also focuses on applications in computer science, linguistics, and philosophy.

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Proof-theoretic in the context of Intuitionistic propositional calculus

Intuitionistic logic, sometimes more generally called constructive logic, refers to systems of symbolic logic that differ from the systems used for classical logic by more closely mirroring the notion of constructive proof. In particular, systems of intuitionistic logic do not assume the law of excluded middle and double negation elimination, which are fundamental inference rules in classical logic.

Formalized intuitionistic logic was originally developed by Arend Heyting to provide a formal basis for L. E. J. Brouwer's programme of intuitionism. From a proof-theoretic perspective, Heyting’s calculus is a restriction of classical logic in which the law of excluded middle and double negation elimination have been removed. Excluded middle and double negation elimination can still be proved for some propositions on a case by case basis, however, but do not hold universally as they do with classical logic. The standard explanation of intuitionistic logic is the BHK interpretation.

View the full Wikipedia page for Intuitionistic propositional calculus
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LINKED FROM (PARENTS)
← Intuitionistic propositional calculus ← Intuitionistic logic
LINKS TO (CHILDREN)
Mathematical logic → Theoretical computer science → Mathematical proof → Mathematical object → Recursive data type → Data structures → List (computer science) → Tree (data structure) → Axiom → Rule of inference → Syntax (logic) → Model theory → Formal semantics (logic) → Structural proof theory → Ordinal analysis → Provability logic → Proof-theoretic semantics → Reverse mathematics → Proof mining → Automated theorem proving → Proof complexity →