3.4 Proof systems for first-order logic

First-order logic proof systems build on propositional logic, adding quantifiers and predicates. Natural deduction and sequent calculus are two key approaches, using inference rules to construct proofs. These systems handle quantifiers through special introduction and elimination rules.

Axiom systems offer another proof method, using schemas as templates for generating axioms. Herbrand's theorem bridges first-order and propositional logic, while automated theorem proving techniques like resolution enable computer-assisted proof discovery in first-order logic.

Natural Deduction and Sequent Calculus

Inference Rules and Proof Systems

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• Natural deduction is a proof system that uses a set of inference rules to derive conclusions from premises
  • Consists of introduction and elimination rules for each logical connective
  • Allows for the construction of proofs in a way that closely resembles human reasoning (modus ponens, modus tollens)
• Sequent calculus is another proof system that operates on sequents, which are expressions of the form $\Gamma \vdash \Delta$
  • $\Gamma$ represents a set of formulas called the antecedent, and $\Delta$ represents a set of formulas called the succedent
  • Sequent calculus rules manipulate these sequents to prove the validity of logical arguments
• Inference rules in both systems specify how to derive new formulas from existing ones
  • Examples include $\wedge$-introduction, $\vee$-elimination, $\rightarrow$-introduction, and $\neg$-introduction

Quantifier Rules in Natural Deduction and Sequent Calculus

• Quantifier rules in natural deduction and sequent calculus handle the introduction and elimination of universal ($\forall$) and existential ($\exists$) quantifiers
• Universal quantifier introduction ($\forall$-introduction) allows for the generalization of a formula by introducing a universal quantifier
  • If a formula $\phi$ has been derived without any assumptions about the variable $x$, then $\forall x \phi$ can be derived
• Existential quantifier elimination ($\exists$-elimination) allows for the instantiation of an existentially quantified formula with a specific term
  • If $\exists x \phi$ has been derived, and a formula $\psi$ can be derived from $\phi[t/x]$ for some term $t$, then $\psi$ can be derived
• Similar quantifier rules exist in sequent calculus, adapted to work with sequents

Axiom Systems

Axiom Schemas and Substitution

• Axiom systems are another approach to proof theory, where a set of axioms is used as the starting point for deriving theorems
• Axiom schemas are templates for axioms, containing metavariables that can be instantiated with specific formulas
  • Example: the axiom schema $\forall x (\phi \rightarrow \psi) \rightarrow (\forall x \phi \rightarrow \forall x \psi)$ can be instantiated with specific formulas for $\phi$ and $\psi$
• Substitution is the process of replacing metavariables in an axiom schema with specific formulas to obtain concrete axioms
  • Allows for the generation of an infinite number of axioms from a finite set of axiom schemas

Herbrand's Theorem

• Herbrand's theorem is a fundamental result in proof theory that relates the validity of a first-order formula to the satisfiability of a potentially infinite set of propositional formulas
• States that a first-order formula is valid if and only if there exists a finite set of instances of the formula, obtained by substituting terms for variables, that is propositionally unsatisfiable
• Provides a bridge between first-order logic and propositional logic
  • Allows for the reduction of first-order validity to propositional satisfiability
• Has important implications for automated theorem proving, as it suggests a way to prove first-order formulas by considering their propositional instances

Automated Theorem Proving

Resolution and Proof Search

• Automated theorem proving refers to the use of computer programs to automatically find proofs of mathematical theorems
• Resolution is a powerful inference rule used in automated theorem proving for first-order logic
  • Operates on clauses, which are disjunctions of literals (atomic formulas or their negations)
  • Allows for the derivation of new clauses from existing ones by resolving complementary literals
  • Example: from the clauses $\neg P(x) \vee Q(x)$ and $P(a)$, resolution derives the clause $Q(a)$
• Resolution can be used as a refutation procedure, where the negation of the desired theorem is added to the set of clauses, and resolution is applied until a contradiction (empty clause) is derived
• Efficient proof search strategies, such as unit resolution and ordered resolution, are employed to guide the application of resolution inferences
  • These strategies help reduce the search space and improve the performance of automated theorem provers