5 research outputs found
Double-Negation Elimination in Some Propositional Logics
This article answers two questions (posed in the literature), each concerning
the guaranteed existence of proofs free of double negation. A proof is free of
double negation if none of its deduced steps contains a term of the form
n(n(t)) for some term t, where n denotes negation. The first question asks for
conditions on the hypotheses that, if satisfied, guarantee the existence of a
double-negation-free proof when the conclusion is free of double negation. The
second question asks about the existence of an axiom system for classical
propositional calculus whose use, for theorems with a conclusion free of double
negation, guarantees the existence of a double-negation-free proof. After
giving conditions that answer the first question, we answer the second question
by focusing on the Lukasiewicz three-axiom system. We then extend our studies
to infinite-valued sentential calculus and to intuitionistic logic and
generalize the notion of being double-negation free. The double-negation proofs
of interest rely exclusively on the inference rule condensed detachment, a rule
that combines modus ponens with an appropriately general rule of substitution.
The automated reasoning program OTTER played an indispensable role in this
study.Comment: 32 pages, no figure
Investigations into Proof Structures
We introduce and elaborate a novel formalism for the manipulation and
analysis of proofs as objects in a global manner. In this first approach the
formalism is restricted to first-order problems characterized by condensed
detachment. It is applied in an exemplary manner to a coherent and
comprehensive formal reconstruction and analysis of historical proofs of a
widely-studied problem due to {\L}ukasiewicz. The underlying approach opens the
door towards new systematic ways of generating lemmas in the course of proof
search to the effects of reducing the search effort and finding shorter proofs.
Among the numerous reported experiments along this line, a proof of
{\L}ukasiewicz's problem was automatically discovered that is much shorter than
any proof found before by man or machine.Comment: This article is a continuation of arXiv:2104.1364
Proof-theoretic Semantics for Intuitionistic Multiplicative Linear Logic
This work is the first exploration of proof-theoretic semantics for a substructural logic. It focuses on the base-extension semantics (B-eS) for intuitionistic multiplicative linear logic (IMLL). The starting point is a review of Sandqvist’s B-eS for intuitionistic propositional logic (IPL), for which we propose an alternative treatment of conjunction that takes the form of the generalized elimination rule for the connective. The resulting semantics is shown to be sound and complete. This motivates our main contribution, a B-eS for IMLL
, in which the definitions of the logical constants all take the form of their elimination rule and for which soundness and completeness are established
Automated Deduction – CADE 28
This open access book constitutes the proceeding of the 28th International Conference on Automated Deduction, CADE 28, held virtually in July 2021. The 29 full papers and 7 system descriptions presented together with 2 invited papers were carefully reviewed and selected from 76 submissions. CADE is the major forum for the presentation of research in all aspects of automated deduction, including foundations, applications, implementations, and practical experience. The papers are organized in the following topics: Logical foundations; theory and principles; implementation and application; ATP and AI; and system descriptions