Glossary

9.3 Proof systems for intuitionistic logic

4 min read•august 7, 2024

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

These proof systems have important properties like and . They also connect to type theory through the , 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

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  • 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 Γ⊢Δ\Gamma \vdash \Delta, where Γ\Gamma and Δ\Delta are sequences of formulas
    • Γ\Gamma represents the assumptions (antecedent) and Δ\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 is a proof procedure for intuitionistic logic based on the idea of refutation
    • To prove a formula φ\varphi, the tableau method attempts to construct a countermodel for ¬φ\neg \varphi
    • If the construction of the countermodel fails, then φ\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
    • 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 Γ⊢φ\Gamma \vdash \varphi and φ,Δ⊢ψ\varphi, \Delta \vdash \psi, then Γ,Δ⊢ψ\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

  • is a notion of logical consequence in intuitionistic logic
    • A formula φ\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 φ∨ψ\varphi \lor \psi is intuitionistically provable, then either φ\varphi or ψ\psi is intuitionistically provable
    • The existence property states that if ∃xφ(x)\exists x \varphi(x) is intuitionistically provable, then there is a term tt such that φ(t)\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
    • (∧)(\land) corresponds to the product type (×)(\times), where a proof of φ∧ψ\varphi \land \psi is a pair of proofs of φ\varphi and ψ\psi
    • Implication (→)(\to) corresponds to the function type (→)(\to), where a proof of φ→ψ\varphi \to \psi is a function that maps proofs of φ\varphi to proofs of ψ\psi
    • Disjunction (∨)(\lor) corresponds to the sum type (+)(+), where a proof of φ∨ψ\varphi \lor \psi is either a proof of φ\varphi or a proof of ψ\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

Key Terms to Review (21)

Arend Heyting: Arend Heyting was a significant figure in the development of intuitionistic logic, particularly known for formalizing the syntax and semantics of this logical system. His work laid the groundwork for understanding how intuitionistic logic differs from classical logic, especially regarding proof systems and the underlying philosophical foundations. Heyting’s contributions highlight the importance of constructivism in mathematics, where the existence of a mathematical object is tied to the ability to construct it explicitly.
Axiom of Choice: The Axiom of Choice is a fundamental principle in set theory stating that given a collection of non-empty sets, there exists at least one way to choose an element from each set. This axiom plays a crucial role in various areas of mathematics, including analysis and topology, influencing the structure and behavior of mathematical objects.
Classical completeness: Classical completeness refers to the property of a logical system where every semantically valid formula can be proven using the system's axioms and inference rules. This means that if something is true in all models of the logic, there exists a formal proof within that logic. Understanding classical completeness is crucial for exploring how proof systems operate in various logical contexts, particularly when contrasting with intuitionistic logic, which may not exhibit the same form of completeness.
Conjunction: In logic, a conjunction is a compound statement formed by connecting two or more propositions using the logical connective 'and', symbolized as $$\land$$. This operation is fundamental as it defines how multiple statements can be combined to yield a true or false value based on their individual truth values.
Constructive existence: Constructive existence is the principle that for a mathematical or logical statement to assert the existence of an object, one must be able to provide a method or construction that demonstrates the object's existence. This concept emphasizes the need for tangible proof, aligning with intuitionistic logic, which rejects non-constructive proofs such as those relying on the law of excluded middle.
Constructive Proof: A constructive proof is a type of mathematical proof that not only demonstrates the existence of a mathematical object but also provides a method for actually constructing such an object. This approach emphasizes the idea that existence claims must be supported by explicit examples or algorithms, distinguishing it from non-constructive proofs which may rely on indirect arguments. Constructive proofs are foundational in various areas of logic and mathematics, particularly in contexts that prioritize intuitionistic logic and constructive mathematics.
Curry-Howard correspondence: The Curry-Howard correspondence is a deep connection between logic and computation, showing how propositions correspond to types and proofs correspond to programs. This correspondence reveals that intuitionistic logic can be viewed through the lens of type theory, and it highlights how constructive proofs can be translated into executable algorithms.
Cut Elimination: Cut elimination is a fundamental process in proof theory that aims to simplify proofs by removing 'cut' rules, which allow for the introduction of intermediate assertions not originally present in the premises. This concept is crucial for understanding the efficiency and structure of proofs, as it transforms proofs into a more direct form, aligning them closer to intuitionistic principles and providing a clearer representation of derivations.
Implication: Implication is a fundamental logical connective that describes a relationship between two propositions, typically expressed as 'if P, then Q', where P is the antecedent and Q is the consequent. This relationship signifies that if the antecedent is true, then the consequent must also be true, establishing a crucial link in logical reasoning. Understanding implication is key in various logical systems, as it influences the structure of arguments, proofs, and the semantics of different logics.
Intuitionistic tableau method: The intuitionistic tableau method is a proof system used to determine the validity of formulas in intuitionistic logic through a tree-like structure of derivations. This method emphasizes the constructive nature of proofs, allowing only those proofs that can be effectively constructed or realized. It contrasts with classical logic by rejecting certain principles like the law of excluded middle, focusing on the necessity of explicit witness or evidence in the proof process.
Intuitionistic validity: Intuitionistic validity refers to a principle in intuitionistic logic where a statement is considered valid only if there is a constructive proof of it. In this framework, the focus is on the existence of evidence for statements rather than merely their truth value, distinguishing it from classical logic. This has significant implications for proof systems as it leads to different rules and interpretations of logical connectives.
Kripke semantics: Kripke semantics is a framework used for interpreting modal and intuitionistic logics through the use of possible worlds and accessibility relations. It connects the truth of propositions to various contexts, represented by these possible worlds, which can vary in their relationships, allowing for a nuanced understanding of necessity and possibility in logic.
L.E.J. Brouwer: L.E.J. Brouwer was a Dutch mathematician and philosopher known for founding intuitionism, a philosophy of mathematics emphasizing constructivism and the rejection of the law of excluded middle. His work significantly influenced the development of proof theory, particularly in the context of intuitionistic logic, which contrasts with classical logic by requiring that mathematical objects be constructively defined.
Modus ponens in intuitionistic logic: Modus ponens is a fundamental rule of inference in intuitionistic logic that allows one to derive a conclusion from a conditional statement and its antecedent. Specifically, if you have a statement of the form 'If P, then Q' (P → Q) and you know that P is true, you can conclude that Q is true as well. This rule plays a crucial role in the proof systems for intuitionistic logic, emphasizing the constructive nature of proofs.
Natural Deduction: Natural deduction is a proof system used in logic that allows one to derive conclusions from premises using a set of inference rules in a structured and intuitive way. It emphasizes the natural reasoning process, enabling proofs to be constructed through the application of these rules without the need for additional axioms or complex structures, making it particularly useful in various fields of mathematics and philosophy.
No double negation elimination: No double negation elimination is a principle in intuitionistic logic that states that a statement cannot be inferred from its double negation. This means that from $$ eg eg A$$, one cannot conclude $$A$$. This principle distinguishes intuitionistic logic from classical logic, where double negation elimination is generally accepted as valid. It reflects a more constructive approach to truth in which a proof of a statement must provide explicit evidence rather than relying on the absence of contradictions.
Normalization: Normalization refers to the process of transforming a proof into a standard or simplified form, often to achieve a unique or canonical representation. This concept is central to proof theory, as it helps establish consistency and ensures that proofs are free from unnecessary complexities or redundancies, making them easier to analyze and compare across different logical systems.
Provability: Provability refers to the property of a statement being demonstrable within a formal system based on its axioms and inference rules. It connects deeply with various foundational aspects of mathematical logic, emphasizing the importance of formal proofs in establishing the truth of mathematical propositions.
Sequent Calculus: Sequent calculus is a formal proof system designed for deriving sequents, which express the validity of implications between formulas in logic. It serves as a key framework for studying proof theory, enabling structured reasoning about logical statements and their relationships, particularly in first-order logic and intuitionistic logic.
Topological semantics: Topological semantics is an approach used in logic to interpret modal and intuitionistic logics through the lens of topology, which studies the properties of space. By associating logical formulas with open sets in a topological space, this framework helps to clarify the relationships between different logics and their semantics. It plays a vital role in understanding proof systems and their interrelations, particularly in how intuitionistic logic differs from classical logic.
Truth Values in Intuitionistic Logic: Truth values in intuitionistic logic refer to the principles of how propositions are evaluated based on their constructibility or provability, rather than simply being true or false. In this system, a statement is considered true only if there is a constructive proof of its truth, which contrasts with classical logic where every proposition is assigned a definite truth value regardless of proof. This nuanced approach allows intuitionistic logic to capture the idea that not all truths are known or can be proven without an explicit construction.
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