(Dual) Hoops Have Unique Halving

  title={(Dual) Hoops Have Unique Halving},
  author={Rob Arthan and Paulo Oliva},
  booktitle={Automated Reasoning and Mathematics},
Continuous logic extends the multi-valued Łukasiewicz logic by adding a halving operator on propositions. This extension is designed to give a more satisfactory model theory for continuous structures. The semantics of these logics can be given using specialisations of algebraic structures known as hoops and coops. As part of an investigation into the metatheory of propositional continuous logic, we were indebted to Prover9 for finding proofs of important algebraic laws. 
1 Citations

On First-Order Model-Based Reasoning

This paper surveys a selection of semantically-guided or model-based methods that aim at meeting aspects of this challenge of reasoning semantically in first-order logic, concluding with the recent Model-Constructing satisfiability calculus.



Distributivity in Łℵ0 and Other Sentential Logics

Certain distributivity results for Łukasiewicz's infinite-valued logic Łℵ0 are proved axiomatically (for the first time) with the help of the automated reasoning program OTTER and a wide variety of positive substructural logics are established by the use of logical matrices discovered with the automated model-finding programs MACE and MAGIC.

Hoops, Coops and the Algebraic Semantics of Continuous Logic

This paper defines the notion of continuous hoop, or coop for short, and shows that coops provide a natural algebraic semantics for continuous logic, and characterise the simple and subdirectly irreducible coops and in-igate the decision problem for various theories of coops.

Axiomatization of the infinite-valued predicate calculus

  • L. Hay
  • Mathematics, Computer Science
    Journal of Symbolic Logic
  • 1963
For the proposed axiomatization, a property akin to but weaker than completeness is proved; it has been shown that the set of valid formulas of the infinite-valued predicate calculus is not recursively enumerable.

More Proofs of an Axiom of Łukasiewicz

  • J. Slaney
  • Computer Science
    Journal of Automated Reasoning
  • 2004
It is shown that one of the standard axioms of the denumerable-valued pure implication logic of Łukasiewicz becomes derivable from the remainder in the presence of negation, and is similarly derivable using conjunction and disjunction instead ofNegation.

A proof of completeness for continuous first-order logic

This article shows that in continuous first-order logic a set of formulae is (completely) satisfiable if (and only if) it is consistent, and shows that if Σ⊧φ, then proofs from Σ, being finite, can provide arbitrarily better approximations of the truth of φ.

An Application of Automated Equational Reasoning to Many-valued Logic

A new set of axioms of an algebaric nature, similar to those given by J.Hsiang for the Boolean Algebra, are presented, for the many-valued logic of Lukasiewicz.

Algebraic analysis of many valued logics

This paper is an attempt at developing a theory of algebraic systems that would correspond in a natural fashion to the No-valued propositional calculus(2). For want of a better name, we shall call

On the structure of hoops

Abstract. A hoop is a naturally ordered pocrim (i.e., a partially ordered commutative residuated integral monoid). We list some basic properties of hoops, describe in detail the structure of


. 1 Involutive Pocrims and their Subreducts The purely intensional fragments of affine linear logic (i.e., linear logic with the weakening axiom p → (q → p)) are known to be algebraizable in the

Metamathematics of Fuzzy Logic

  • P. Hájek
  • Philosophy, Computer Science
    Trends in Logic
  • 1998
This paper presents a meta-analysis of many-Valued Propositional Logic, focusing on the part of Lukasiewicz's Logic that deals with Complexity, Undecidability and Generalized Quantifiers and Modalities.