[Paper Review] Algebraic proof theory for LE-logics

Giuseppe Greco, Peter Jipsen+3 more|arXiv (Cornell University)|Aug 14, 2018

Advanced Algebra and Logic57 references4 citations

TL;DR

This paper extends algebraic proof theory to normal lattice expansion (LE) logics by introducing functional D-frames as a generalization of residuated frames, enabling uniform proofs of semantic cut elimination and the finite model property (FMP) for display calculi D.LE and their extensions with analytic inductive axioms. The key contribution is a unified framework linking proof-theoretic cut elimination to algebraic properties like MacNeille completion and canonical extensions.

ABSTRACT

In this paper we extend the research programme in algebraic proof theory from axiomatic extensions of the full Lambek calculus to logics algebraically captured by certain varieties of normal lattice expansions (normal LE-logics). Specifically, we generalise the residuated frames in [34] to arbitrary signatures of normal lattice expansions (LE). Such a generalization provides a valuable tool for proving important properties of LE-logics in full uniformity. We prove semantic cut elimination for the display calculi D.LE associated with the basic normal LE-logics and their axiomatic extensions with analytic inductive axioms. We also prove the finite model property (FMP) for each such calculus D.LE, as well as for its extensions with analytic structural rules satisfying certain additional properties.

Motivation & Objective

• To generalize algebraic proof theory from substructural logics to arbitrary normal LE-logics, extending results on cut elimination and finite model property.
• To establish a uniform connection between proof-theoretic properties (cut elimination, FMP) and algebraic properties (MacNeille completions, canonical extensions).
• To generalize residuated frames to functional D-frames for arbitrary signatures of normal lattice expansions.
• To demonstrate that analytic inductive axioms can be safely added to D.LE calculi without losing cut elimination.
• To provide a unified framework for metalogical results across two-valued and many-valued LE-logics using duality theory and canonical extensions.

Proposed method

• Introduce functional D-frames as a generalization of residuated frames, where relations for connectives are functional, enabling algebraic semantics for arbitrary LE signatures.
• Construct complex algebras from polarity-based relational structures (D-frames), generalizing the complex algebra construction from modal logic.
• Use canonical extensions and constructive canonicity of analytic inductive inequalities to establish semantic cut elimination.
• Prove that validity of analytic structural rules is preserved from D-frames to their complex algebras, ensuring cut elimination in D.LE calculi.
• Apply the finite embeddability property and D-frame constructions to prove the finite model property (FMP) for D.LE and extensions with analytic structural rules.
• Leverage duality theory and the generalized Sahlqvist framework to unify proof-theoretic, algebraic, and semantic results across LE-logics.

Experimental results

Research questions

• RQ1 Can the algebraic proof theory framework for substructural logics be extended to arbitrary normal LE-logics beyond the full Lambek calculus?
• RQ2 How can residuated frames be generalized to support arbitrary signatures of normal lattice expansions?
• RQ3 What is the semantic role of functional D-frames in capturing cut elimination for display calculi?
• RQ4 Under what conditions do analytic inductive axioms preserve cut elimination in D.LE calculi?
• RQ5 Can the finite model property be uniformly established for D.LE calculi and their extensions with analytic structural rules?

Key findings

• Semantic cut elimination is established for the display calculus D.LE and its extensions with analytic inductive axioms, using the preservation of validity from D-frames to their complex algebras.
• The finite model property (FMP) is proven for D.LE calculi and their extensions with analytic structural rules satisfying additional conditions, via finite embeddability and D-frame constructions.
• Functional D-frames are introduced as a generalization of residuated frames, enabling a uniform treatment of arbitrary signatures of normal lattice expansions.
• The construction of complex algebras from polarity-based D-frames generalizes MacNeille completions and provides the algebraic semantics underlying cut elimination.
• The canonical extension of D-frames ensures constructive canonicity of analytic inductive inequalities, which is key to proving cut elimination and FMP.
• The framework unifies proof-theoretic, algebraic, and semantic results across two-valued and many-valued LE-logics, enabling parametric proofs of metalogical properties.