Intuitionistic logic uses special proof systems to reason without assuming every statement is true or false. These systems, like natural deduction and sequent calculus, have unique rules that reflect the constructive nature of intuitionistic reasoning.

These proof systems have important properties like normalization and cut elimination. They also connect to type theory through the Curry-Howard correspondence, linking logical proofs to computer programs. This connection has big implications for both math and computer science.

Intuitionistic Proof Systems

Natural Deduction and Sequent Calculus

• Natural deduction is a formal proof system that closely resembles the informal proofs used in mathematical reasoning
  • Consists of a set of inference rules that allow deriving conclusions from premises
  • Proofs are constructed as trees, with the conclusion at the root and the premises at the leaves
  • Intuitionistic natural deduction excludes the law of excluded middle and double negation elimination (classical logic rules)
• Sequent calculus is another formal proof system for intuitionistic logic
  • Operates on sequents, which are expressions of the form $\Gamma \vdash \Delta$, where $\Gamma$ and $\Delta$ are sequences of formulas
  • $\Gamma$ represents the assumptions (antecedent) and $\Delta$ represents the conclusions (succedent)
  • Inference rules in sequent calculus manipulate sequents to derive new sequents
  • Intuitionistic sequent calculus restricts the succedent to at most one formula

Intuitionistic Tableau Method

• The intuitionistic tableau method is a proof procedure for intuitionistic logic based on the idea of refutation
  • To prove a formula $\varphi$, the tableau method attempts to construct a countermodel for $\neg \varphi$
  • If the construction of the countermodel fails, then $\varphi$ is intuitionistically valid
• Tableaux are trees labeled with formulas, where each branch represents a possible model
  • The tableau rules decompose complex formulas into simpler ones, generating new branches
  • Closure rules identify contradictory branches, which are then closed (marked with a cross)
• The intuitionistic tableau method differs from the classical tableau method in the rules for negation and implication
  • The intuitionistic rules ensure that the constructed models are intuitionistically valid

Properties of Intuitionistic Proofs

Normalization and Cut Elimination

• Normalization is a process of transforming proofs into a standard form (normal form)
  • In natural deduction, a proof is in normal form if it contains no "detours" (redundant steps)
  • Normalization eliminates the detours by applying reduction rules to the proof tree
  • The normalization theorem states that every proof in intuitionistic natural deduction can be reduced to a normal form
• Cut elimination is a fundamental property of sequent calculus
  • The cut rule allows the use of an intermediate lemma in a proof: if $\Gamma \vdash \varphi$ and $\varphi, \Delta \vdash \psi$, then $\Gamma, \Delta \vdash \psi$
  • The cut elimination theorem states that any proof in intuitionistic sequent calculus can be transformed into a proof without the cut rule
  • Cut-free proofs have a simpler structure and are easier to reason about

Intuitionistic Validity

• Intuitionistic validity is a notion of logical consequence in intuitionistic logic
  • A formula $\varphi$ is intuitionistically valid if it is provable in an intuitionistic proof system (e.g., natural deduction or sequent calculus)
  • Intuitionistic validity is weaker than classical validity: every intuitionistically valid formula is classically valid, but not vice versa
• The disjunction and existence properties are characteristic of intuitionistic logic
  • The disjunction property states that if $\varphi \lor \psi$ is intuitionistically provable, then either $\varphi$ or $\psi$ is intuitionistically provable
  • The existence property states that if $\exists x \varphi(x)$ is intuitionistically provable, then there is a term $t$ such that $\varphi(t)$ is intuitionistically provable

Connections to Type Theory

Curry-Howard Correspondence

• The Curry-Howard correspondence, also known as the proofs-as-programs interpretation, establishes a deep connection between intuitionistic logic and type theory
  • Propositions in intuitionistic logic correspond to types in a typed lambda calculus
  • Proofs of propositions correspond to lambda terms (programs) of the corresponding types
  • The inference rules of intuitionistic natural deduction correspond to type constructors and term constructors in the lambda calculus
• The correspondence allows the interpretation of logical operations as programming constructs
  • Conjunction $(\land)$ corresponds to the product type $(\times)$, where a proof of $\varphi \land \psi$ is a pair of proofs of $\varphi$ and $\psi$
  • Implication $(\to)$ corresponds to the function type $(\to)$, where a proof of $\varphi \to \psi$ is a function that maps proofs of $\varphi$ to proofs of $\psi$
  • Disjunction $(\lor)$ corresponds to the sum type $(+)$, where a proof of $\varphi \lor \psi$ is either a proof of $\varphi$ or a proof of $\psi$, tagged with a label indicating the choice
• The Curry-Howard correspondence has far-reaching consequences in both logic and computer science
  • It provides a foundation for constructive mathematics and the development of proof assistants (e.g., Coq, Agda)
  • It enables the extraction of certified programs from proofs, ensuring their correctness with respect to the specified properties