🤔Proof Theory Unit 4 – Natural Deduction and Sequent Calculus

Key Concepts and Definitions

• Natural deduction is a proof calculus that closely models the informal reasoning used in mathematical proofs
• Sequent calculus is a formal system for deriving logical conclusions from premises using inference rules
• Inference rules specify how new formulas can be derived from existing ones in a logically valid way
• Logical connectives include conjunction (∧), disjunction (∨), implication (→), and negation (¬)
• Proof trees visually represent the structure of a proof, with the conclusion at the root and the premises at the leaves
• Soundness ensures that if a formula is provable in the system, it is logically valid
• Completeness guarantees that if a formula is logically valid, it can be proven within the system

Historical Context and Development

• Natural deduction was developed by Gerhard Gentzen in the 1930s as a more intuitive alternative to Hilbert-style systems
• Gentzen aimed to create a proof system that closely resembled the way mathematicians actually reason and construct proofs
• Sequent calculus, also introduced by Gentzen, provides a more symmetric treatment of logical connectives
• Gentzen's work laid the foundation for the field of proof theory and influenced the development of automated theorem proving
• Dag Prawitz further refined natural deduction in the 1960s, introducing the concept of normal proofs
  • Normal proofs eliminate unnecessary detours and redundancies in the proof structure
• Sequent calculus has been extended to various logics beyond classical logic, such as intuitionistic logic and linear logic

Natural Deduction: Rules and Structure

• Natural deduction consists of introduction and elimination rules for each logical connective
• Introduction rules specify how to introduce a connective, while elimination rules describe how to use or eliminate it
• The →-introduction rule (→I) allows the derivation of an implication by assuming the antecedent and deriving the consequent
• The ∧-elimination rules (∧E₁ and ∧E₂) enable the derivation of individual conjuncts from a conjunction
• The ∨-introduction rules (∨I₁ and ∨I₂) permit the introduction of a disjunction by proving either disjunct
• The ¬-introduction rule (¬I) allows the derivation of a negation by assuming the negated formula and deriving a contradiction
• Proofs in natural deduction are typically presented as proof trees or nested subproofs, with assumptions discharged by the inference rules

Sequent Calculus: Principles and Notation

• Sequent calculus operates on sequents, which are expressions of the form $Γ ⊢ Δ$, where $Γ$ and $Δ$ are sets of formulas
• The sequent $Γ ⊢ Δ$ can be read as "from the assumptions $Γ$, we can derive at least one of the conclusions in $Δ$"
• Sequent calculus rules are divided into left and right rules, depending on which side of the turnstile (⊢) the principal formula appears
• The left rules describe how to use an assumption, while the right rules specify how to derive a conclusion
• The cut rule allows the combination of two sequents by using a formula on the right side of one sequent as an assumption in the other
• Sequent proofs are often presented in a tabular format, with the conclusion at the bottom and the premises above, separated by horizontal lines
• The structural rules of weakening, contraction, and exchange manage the context of assumptions and conclusions in a sequent

Comparison of Natural Deduction and Sequent Calculus

• Natural deduction focuses on the introduction and elimination of logical connectives, while sequent calculus maintains a symmetry between the left and right sides of the turnstile
• Proofs in natural deduction tend to be more readable and intuitive, as they resemble the structure of informal mathematical proofs
• Sequent calculus proofs often have a more uniform and modular structure, making them easier to manipulate and transform
• The cut rule in sequent calculus allows for the composition of proofs, enabling proof search and automated theorem proving techniques
• Sequent calculus is more suitable for proving meta-theoretical properties, such as cut elimination and consistency
• Natural deduction is more commonly used in educational settings and for modeling human reasoning, while sequent calculus is favored in proof theory research and automated reasoning

Proof Strategies and Techniques

• Backward reasoning starts with the desired conclusion and works backwards to the premises, applying inference rules to reduce the goal to simpler subgoals
• Forward reasoning begins with the given premises and applies inference rules to derive new formulas until the desired conclusion is reached
• Proof by contradiction assumes the negation of the desired conclusion and derives a contradiction, allowing the conclusion to be inferred by ¬-introduction
• Proof by cases considers different possibilities and shows that the desired conclusion holds in each case, often using ∨-elimination
• Induction is used to prove properties of recursively defined structures, such as natural numbers or lists
  • The base case proves the property for the simplest instance, while the inductive step shows that if the property holds for a given case, it also holds for the next case
• Lemmas are intermediate results that are proven separately and used as building blocks in larger proofs, helping to modularize and simplify the overall proof structure

Common Challenges and Misconceptions

• Choosing the appropriate inference rule to apply at each step of the proof can be challenging, requiring a deep understanding of the rules and their interactions
• Keeping track of assumptions and their scope is crucial, especially when working with nested subproofs or multiple branches in a proof tree
• Discharging assumptions correctly is essential to ensure the validity of the proof, as undischarged assumptions can lead to incorrect or incomplete proofs
• Recognizing when to apply proof strategies like contradiction or induction can be difficult, as it requires identifying patterns and structures in the problem
• Translating between natural language statements and formal logical notation can be confusing, particularly when dealing with complex propositions or quantifiers
• Understanding the limitations and strengths of each proof system is important to choose the most appropriate approach for a given problem or domain

Applications and Real-World Relevance

• Proof theory provides a rigorous foundation for reasoning about the correctness and properties of mathematical statements and computer programs
• Automated theorem provers and proof assistants, such as Coq and Isabelle, rely on the principles of natural deduction and sequent calculus to verify proofs and ensure the correctness of software and hardware systems
• Formal verification techniques, based on proof theory, are used in safety-critical domains like aerospace, transportation, and medical devices to guarantee the absence of errors and bugs
• Programming languages and type systems often incorporate concepts from proof theory to ensure type safety and catch potential errors at compile time
• Proof theory has applications in artificial intelligence and machine learning, enabling the development of explainable and trustworthy AI systems that can provide justifications for their decisions
• Philosophical and foundational questions in mathematics, such as the consistency and independence of axioms, are studied using the tools and techniques of proof theory