FregeSystem

FregeSystem is a formal deductive system for classical first-order logic that follows the spirit of Gottlob Frege's Begriffsschrift. It provides a precise syntax for constructing well-formed formulas and a deductive apparatus for proving logical consequences.

Syntax: Terms and predicates; atomic formulas P(t1,...,tn); complex formulas built with negation, conjunction, disjunction, implication, and biconditional; quantifiers universal and existential; parentheses; variable binding; substitution rules; boundary conditions to avoid variable capture.

Proof theory: FregeSystem specifies axiom schemas and inference rules. Common choices include Hilbert-style axiom schemas plus Modus Ponens, along with rules for universal generalization and existential instantiation. Some variants use natural deduction styles with introduction and elimination rules for each connective and quantifier. Substitution must preserve variable binding.

Semantics and properties: The system is designed to mirror standard first-order semantics; with appropriate axiom schemas, FregeSystem is sound and complete with respect to classical truth-conditional semantics. Implementations may vary in proof strength and automation; many versions support meta-theoretic results such as consistency, soundness, and completeness proofs.

Use and history: FregeSystem traces its lineage to Fregean logic and is used in logic education and formalization tasks. It serves as a bridge between historical Begriffsschrift notation and modern formal languages, illustrating deduction principles and the interplay between syntax and semantics.