Herbrand analyzing functions

Herbrand-Confluence

We consider cut-elimination in the sequent calculus for classical first-order logic. It is well known that this system, in its most general form, is neither confluent nor strongly normalizing. In this work we take a coarser (and mathematically more realistic) look at cut-free proofs. We analyze which witnesses they choose for which quantifiers, or in other words: we only consider the Herbrand-d...

Herbrand Logic

Analyzing the selection of the herbrand base process for building a Smart Semantic Tree Theorem Prover

Traditionally, semantic trees have played an important role in proof theory for validating the unsatisfiability of sets of clauses. More recently, they have also been used to implement more practical tools for verifying the unsatisfiability of clause sets in first-order predicate logic. The method ultimately relies on the Herbrand Base, a set used in building the semantic tree. The Herbrand Bas...

The Herbrand Topos

We define a new topos, the Herbrand topos, inspired by the modified realizability topos and our earlier work on Herbrand realizability. We also introduce the category of Herbrand assemblies and characterise these as the ¬¬-separated objects in the Herbrand topos. In addition, we show that the category of sets is included as the category of ¬¬-sheaves and prove that the inclusion functor preserv...

Remarks on Herbrand Analyses

We show that for theories T + with function parameters, in general (1) T + ` A , A =⇒ / T + ` A (even if A does not contain function parameters and T + is an open theory), where A is the Herbrand normal form of A and A is a Herbrand realization of A . A similar result holds for first order theories T if the index functions (used in the definition of A from A) are allowed to occur in instances o...