2.3 Proof systems: axiomatic, natural deduction, and tableau methods

Proof systems are the backbone of formal logic, providing methods to verify the validity of arguments. Axiomatic, natural deduction, and tableau methods each offer unique approaches to constructing proofs, with their own strengths and weaknesses.

These systems build on the syntax and semantics of propositional logic, allowing us to derive logical conclusions. By understanding these proof methods, we gain powerful tools for reasoning and analyzing complex logical statements.

Axiomatic Systems and Inference Rules

Fundamental Components of Axiomatic Systems

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• Axiomatic systems provide a formal framework for deriving logical conclusions from a set of axioms and inference rules
• Axioms serve as the foundational statements or premises that are assumed to be true within the system
• Inference rules define the valid ways in which new statements can be derived from existing axioms or previously derived statements
• Logical symbols and connectives ($\land$, $\lor$, $\rightarrow$, $\neg$) are used to express and manipulate logical statements within the axiomatic system

Derivations and Proofs

• Derivation is the process of deriving new statements or conclusions from the axioms and inference rules of an axiomatic system
• Each step in a derivation must be justified by applying an inference rule to the previous statements or axioms
• The sequence of steps in a derivation forms a proof, demonstrating the logical validity of the derived conclusion
• Proofs in axiomatic systems are typically presented in a linear format, with each step numbered and justified by the applied inference rule (modus ponens, modus tollens)

Advantages and Limitations

• Axiomatic systems provide a rigorous and formal approach to logical reasoning, ensuring the validity of derived conclusions
• The use of axioms and inference rules allows for the systematic construction of proofs and the verification of logical arguments
• However, axiomatic systems can be complex and may require a significant amount of effort to construct and understand
• The choice of axioms and inference rules can impact the expressive power and limitations of the axiomatic system (propositional logic, first-order logic)

Natural Deduction

Introduction and Elimination Rules

• Natural deduction is a proof system that closely resembles the way humans reason and construct arguments
• Introduction rules specify how to introduce logical connectives and construct new statements (conjunction introduction, disjunction introduction, implication introduction)
• Elimination rules define how to eliminate or simplify logical connectives and derive new statements from existing ones (conjunction elimination, disjunction elimination, implication elimination)
• The introduction and elimination rules for each logical connective form the basis of natural deduction proofs

Proof Construction and Proof Trees

• In natural deduction, proofs are constructed by applying introduction and elimination rules to derive the desired conclusion from the given premises
• Proofs can be represented using a proof tree, where each node represents a statement, and the edges represent the application of inference rules
• The root of the proof tree is the conclusion, and the leaves are the premises or assumptions used in the proof
• Proof trees provide a visual representation of the logical structure and dependencies within a natural deduction proof (subproofs, assumptions, discharging assumptions)

Advantages and Applications

• Natural deduction proofs closely resemble human reasoning patterns, making them more intuitive and easier to understand compared to other proof systems
• The introduction and elimination rules provide a systematic way to construct proofs and analyze logical arguments
• Natural deduction is widely used in various areas of logic, mathematics, and computer science (program verification, type theory, theorem proving)
• The proof tree representation allows for the visualization and analysis of the logical structure and dependencies within a proof

Tableau Method

Tableau Construction and Closure

• The tableau method is a proof procedure that determines the satisfiability or validity of a logical formula
• To construct a tableau, the negation of the formula to be proved is assumed, and the formula is broken down into its constituent parts using tableau expansion rules
• The expansion rules are applied to each branch of the tableau until a contradiction is reached (closed branch) or no further expansions are possible (open branch)
• A closed tableau, where all branches are closed, indicates that the original formula is valid or unsatisfiable (proof by contradiction)

Proof Procedure and Strategies

• The tableau method involves systematically applying the expansion rules to the formulas on each branch of the tableau
• Common expansion rules include conjunction expansion ($\alpha$-rule), disjunction expansion ($\beta$-rule), and negation expansion ($\gamma$-rule)
• Strategies for efficient tableau construction include selecting the most promising formulas for expansion and applying the expansion rules in a specific order (depth-first, breadth-first)
• The proof procedure continues until all branches are closed (indicating validity) or an open branch is found (indicating satisfiability)

Advantages and Limitations

• The tableau method provides a systematic and mechanical procedure for determining the satisfiability or validity of logical formulas
• Tableaux can be used to prove the consistency or inconsistency of a set of formulas, making them useful for logical analysis and reasoning
• The proof tree structure of tableaux allows for the visualization and understanding of the logical dependencies and case distinctions within a proof
• However, the tableau method can be computationally expensive for large and complex formulas, as the number of branches can grow exponentially (proof complexity)